cmra.v 23.6 KB
Newer Older
1
Require Export algebra.cofe.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

Class Unit (A : Type) := unit : A  A.
Instance: Params (@unit) 2.

Class Op (A : Type) := op : A  A  A.
Instance: Params (@op) 2.
Infix "⋅" := op (at level 50, left associativity) : C_scope.
Notation "(⋅)" := op (only parsing) : C_scope.

Definition included `{Equiv A, Op A} (x y : A) :=  z, y  x  z.
Infix "≼" := included (at level 70) : C_scope.
Notation "(≼)" := included (only parsing) : C_scope.
Hint Extern 0 (?x  ?y) => reflexivity.
Instance: Params (@included) 3.

Class Minus (A : Type) := minus : A  A  A.
Instance: Params (@minus) 2.
Infix "⩪" := minus (at level 40) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
20
21
22

Class ValidN (A : Type) := validN : nat  A  Prop.
Instance: Params (@validN) 3.
23
24
Notation "✓{ n } x" := (validN n x)
  (at level 20, n at level 1, format "✓{ n }  x").
Robbert Krebbers's avatar
Robbert Krebbers committed
25

26
27
Class Valid (A : Type) := valid : A  Prop.
Instance: Params (@valid) 2.
28
Notation "✓ x" := (valid x) (at level 20) : C_scope.
29
30
Instance validN_valid `{ValidN A} : Valid A := λ x,  n, {n} x.

31
Definition includedN `{Dist A, Op A} (n : nat) (x y : A) :=  z, y {n} x  z.
Robbert Krebbers's avatar
Robbert Krebbers committed
32
33
34
Notation "x ≼{ n } y" := (includedN n x y)
  (at level 70, format "x  ≼{ n }  y") : C_scope.
Instance: Params (@includedN) 4.
35
Hint Extern 0 (?x {_} ?y) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
36

37
Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus A} := {
Robbert Krebbers's avatar
Robbert Krebbers committed
38
  (* setoids *)
39
40
  mixin_cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x);
  mixin_cmra_unit_ne n : Proper (dist n ==> dist n) unit;
41
  mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n);
42
  mixin_cmra_minus_ne n : Proper (dist n ==> dist n ==> dist n) minus;
Robbert Krebbers's avatar
Robbert Krebbers committed
43
  (* valid *)
44
  mixin_cmra_validN_S n x : {S n} x  {n} x;
Robbert Krebbers's avatar
Robbert Krebbers committed
45
  (* monoid *)
46
47
48
49
  mixin_cmra_associative : Associative () ();
  mixin_cmra_commutative : Commutative () ();
  mixin_cmra_unit_l x : unit x  x  x;
  mixin_cmra_unit_idempotent x : unit (unit x)  unit x;
50
51
  mixin_cmra_unit_preservingN n x y : x {n} y  unit x {n} unit y;
  mixin_cmra_validN_op_l n x y : {n} (x  y)  {n} x;
52
  mixin_cmra_op_minus n x y : x {n} y  x  y  x {n} y
Robbert Krebbers's avatar
Robbert Krebbers committed
53
}.
54
Definition CMRAExtendMixin A `{Equiv A, Dist A, Op A, ValidN A} :=  n x y1 y2,
55
56
  {n} x  x {n} y1  y2 
  { z | x  z.1  z.2  z.1 {n} y1  z.2 {n} y2 }.
Robbert Krebbers's avatar
Robbert Krebbers committed
57

Robbert Krebbers's avatar
Robbert Krebbers committed
58
59
60
61
62
63
64
65
66
67
(** Bundeled version *)
Structure cmraT := CMRAT {
  cmra_car :> Type;
  cmra_equiv : Equiv cmra_car;
  cmra_dist : Dist cmra_car;
  cmra_compl : Compl cmra_car;
  cmra_unit : Unit cmra_car;
  cmra_op : Op cmra_car;
  cmra_validN : ValidN cmra_car;
  cmra_minus : Minus cmra_car;
68
69
70
  cmra_cofe_mixin : CofeMixin cmra_car;
  cmra_mixin : CMRAMixin cmra_car;
  cmra_extend_mixin : CMRAExtendMixin cmra_car
Robbert Krebbers's avatar
Robbert Krebbers committed
71
}.
72
Arguments CMRAT {_ _ _ _ _ _ _ _} _ _ _.
73
74
75
76
77
78
79
80
81
82
83
Arguments cmra_car : simpl never.
Arguments cmra_equiv : simpl never.
Arguments cmra_dist : simpl never.
Arguments cmra_compl : simpl never.
Arguments cmra_unit : simpl never.
Arguments cmra_op : simpl never.
Arguments cmra_validN : simpl never.
Arguments cmra_minus : simpl never.
Arguments cmra_cofe_mixin : simpl never.
Arguments cmra_mixin : simpl never.
Arguments cmra_extend_mixin : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
84
Add Printing Constructor cmraT.
85
Existing Instances cmra_unit cmra_op cmra_validN cmra_minus.
86
Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT (cmra_cofe_mixin A).
Robbert Krebbers's avatar
Robbert Krebbers committed
87
88
Canonical Structure cmra_cofeC.

89
90
91
92
93
94
95
96
(** Lifting properties from the mixin *)
Section cmra_mixin.
  Context {A : cmraT}.
  Implicit Types x y : A.
  Global Instance cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x).
  Proof. apply (mixin_cmra_op_ne _ (cmra_mixin A)). Qed.
  Global Instance cmra_unit_ne n : Proper (dist n ==> dist n) (@unit A _).
  Proof. apply (mixin_cmra_unit_ne _ (cmra_mixin A)). Qed.
97
98
  Global Instance cmra_validN_ne n : Proper (dist n ==> impl) (@validN A _ n).
  Proof. apply (mixin_cmra_validN_ne _ (cmra_mixin A)). Qed.
99
100
101
  Global Instance cmra_minus_ne n :
    Proper (dist n ==> dist n ==> dist n) (@minus A _).
  Proof. apply (mixin_cmra_minus_ne _ (cmra_mixin A)). Qed.
102
103
104
105
106
107
108
109
110
111
112
113
114
115
  Lemma cmra_validN_S n x : {S n} x  {n} x.
  Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed.
  Global Instance cmra_associative : Associative () (@op A _).
  Proof. apply (mixin_cmra_associative _ (cmra_mixin A)). Qed.
  Global Instance cmra_commutative : Commutative () (@op A _).
  Proof. apply (mixin_cmra_commutative _ (cmra_mixin A)). Qed.
  Lemma cmra_unit_l x : unit x  x  x.
  Proof. apply (mixin_cmra_unit_l _ (cmra_mixin A)). Qed.
  Lemma cmra_unit_idempotent x : unit (unit x)  unit x.
  Proof. apply (mixin_cmra_unit_idempotent _ (cmra_mixin A)). Qed.
  Lemma cmra_unit_preservingN n x y : x {n} y  unit x {n} unit y.
  Proof. apply (mixin_cmra_unit_preservingN _ (cmra_mixin A)). Qed.
  Lemma cmra_validN_op_l n x y : {n} (x  y)  {n} x.
  Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed.
116
  Lemma cmra_op_minus n x y : x {n} y  x  y  x {n} y.
117
118
  Proof. apply (mixin_cmra_op_minus _ (cmra_mixin A)). Qed.
  Lemma cmra_extend_op n x y1 y2 :
119
120
    {n} x  x {n} y1  y2 
    { z | x  z.1  z.2  z.1 {n} y1  z.2 {n} y2 }.
121
122
123
  Proof. apply (cmra_extend_mixin A). Qed.
End cmra_mixin.

124
125
126
127
128
129
130
131
(** * CMRAs with a global identity element *)
(** We use the notation  because for most instances (maps, sets, etc) the
`empty' element is the global identity. *)
Class CMRAIdentity (A : cmraT) `{Empty A} : Prop := {
  cmra_empty_valid :  ;
  cmra_empty_left_id :> LeftId ()  ();
  cmra_empty_timeless :> Timeless 
}.
132
Instance cmra_identity_inhabited `{CMRAIdentity A} : Inhabited A := populate .
133

Robbert Krebbers's avatar
Robbert Krebbers committed
134
(** * Morphisms *)
135
136
Class CMRAMonotone {A B : cmraT} (f : A  B) := {
  includedN_preserving n x y : x {n} y  f x {n} f y;
137
  validN_preserving n x : {n} x  {n} f x
138
139
}.

140
(** * Local updates *)
141
142
143
Class LocalUpdate {A : cmraT} (Lv : A  Prop) (L : A  A) := {
  local_update_ne n :> Proper (dist n ==> dist n) L;
  local_updateN n x y : Lv x  {n} (x  y)  L (x  y) {n} L x  y
144
145
146
}.
Arguments local_updateN {_ _} _ {_} _ _ _ _ _.

147
(** * Frame preserving updates *)
148
Definition cmra_updateP {A : cmraT} (x : A) (P : A  Prop) :=  z n,
149
  {n} (x  z)   y, P y  {n} (y  z).
150
Instance: Params (@cmra_updateP) 1.
151
Infix "~~>:" := cmra_updateP (at level 70).
152
Definition cmra_update {A : cmraT} (x y : A) :=  z n,
153
  {n} (x  z)  {n} (y  z).
154
Infix "~~>" := cmra_update (at level 70).
155
Instance: Params (@cmra_update) 1.
Robbert Krebbers's avatar
Robbert Krebbers committed
156

Robbert Krebbers's avatar
Robbert Krebbers committed
157
(** * Properties **)
Robbert Krebbers's avatar
Robbert Krebbers committed
158
Section cmra.
159
Context {A : cmraT}.
Robbert Krebbers's avatar
Robbert Krebbers committed
160
Implicit Types x y z : A.
161
Implicit Types xs ys zs : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
162

163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
(** ** Setoids *)
Global Instance cmra_unit_proper : Proper (() ==> ()) (@unit A _).
Proof. apply (ne_proper _). Qed.
Global Instance cmra_op_ne' n : Proper (dist n ==> dist n ==> dist n) (@op A _).
Proof.
  intros x1 x2 Hx y1 y2 Hy.
  by rewrite Hy (commutative _ x1) Hx (commutative _ y2).
Qed.
Global Instance ra_op_proper' : Proper (() ==> () ==> ()) (@op A _).
Proof. apply (ne_proper_2 _). Qed.
Global Instance cmra_validN_ne' : Proper (dist n ==> iff) (@validN A _ n) | 1.
Proof. by split; apply cmra_validN_ne. Qed.
Global Instance cmra_validN_proper : Proper (() ==> iff) (@validN A _ n) | 1.
Proof. by intros n x1 x2 Hx; apply cmra_validN_ne', equiv_dist. Qed.
Global Instance cmra_minus_proper : Proper (() ==> () ==> ()) (@minus A _).
Proof. apply (ne_proper_2 _). Qed.

Global Instance cmra_valid_proper : Proper (() ==> iff) (@valid A _).
Proof. by intros x y Hxy; split; intros ? n; [rewrite -Hxy|rewrite Hxy]. Qed.
Global Instance cmra_includedN_ne n :
  Proper (dist n ==> dist n ==> iff) (@includedN A _ _ n) | 1.
Proof.
  intros x x' Hx y y' Hy.
  by split; intros [z ?]; exists z; [rewrite -Hx -Hy|rewrite Hx Hy].
Qed.
Global Instance cmra_includedN_proper n :
  Proper (() ==> () ==> iff) (@includedN A _ _ n) | 1.
Proof.
  intros x x' Hx y y' Hy; revert Hx Hy; rewrite !equiv_dist=> Hx Hy.
  by rewrite (Hx n) (Hy n).
Qed.
Global Instance cmra_included_proper :
  Proper (() ==> () ==> iff) (@included A _ _) | 1.
Proof.
  intros x x' Hx y y' Hy.
  by split; intros [z ?]; exists z; [rewrite -Hx -Hy|rewrite Hx Hy].
Qed.
200
201
202
203
204
205
206
207
208
209
210
Global Instance cmra_update_proper :
  Proper (() ==> () ==> iff) (@cmra_update A).
Proof.
  intros x1 x2 Hx y1 y2 Hy; split=>? z n; [rewrite -Hx -Hy|rewrite Hx Hy]; auto.
Qed.
Global Instance cmra_updateP_proper :
  Proper (() ==> pointwise_relation _ iff ==> iff) (@cmra_updateP A).
Proof.
  intros x1 x2 Hx P1 P2 HP; split=>Hup z n;
    [rewrite -Hx; setoid_rewrite <-HP|rewrite Hx; setoid_rewrite HP]; auto.
Qed.
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228

(** ** Validity *)
Lemma cmra_valid_validN x :  x   n, {n} x.
Proof. done. Qed.
Lemma cmra_validN_le x n n' : {n} x  n'  n  {n'} x.
Proof. induction 2; eauto using cmra_validN_S. Qed.
Lemma cmra_valid_op_l x y :  (x  y)   x.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_l. Qed.
Lemma cmra_validN_op_r x y n : {n} (x  y)  {n} y.
Proof. rewrite (commutative _ x); apply cmra_validN_op_l. Qed.
Lemma cmra_valid_op_r x y :  (x  y)   y.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed.

(** ** Units *)
Lemma cmra_unit_r x : x  unit x  x.
Proof. by rewrite (commutative _ x) cmra_unit_l. Qed.
Lemma cmra_unit_unit x : unit x  unit x  unit x.
Proof. by rewrite -{2}(cmra_unit_idempotent x) cmra_unit_r. Qed.
229
Lemma cmra_unit_validN x n : {n} x  {n} unit x.
230
Proof. rewrite -{1}(cmra_unit_l x); apply cmra_validN_op_l. Qed.
231
Lemma cmra_unit_valid x :  x   unit x.
232
233
234
Proof. rewrite -{1}(cmra_unit_l x); apply cmra_valid_op_l. Qed.

(** ** Order *)
Robbert Krebbers's avatar
Robbert Krebbers committed
235
236
237
238
239
240
Lemma cmra_included_includedN x y : x  y   n, x {n} y.
Proof.
  split; [by intros [z Hz] n; exists z; rewrite Hz|].
  intros Hxy; exists (y  x); apply equiv_dist; intros n.
  symmetry; apply cmra_op_minus, Hxy.
Qed.
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
Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
Proof.
  split.
  * by intros x; exists (unit x); rewrite cmra_unit_r.
  * intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2).
    by rewrite (associative _) -Hy -Hz.
Qed.
Global Instance cmra_included_preorder: PreOrder (@included A _ _).
Proof.
  split; red; intros until 0; rewrite !cmra_included_includedN; first done.
  intros; etransitivity; eauto.
Qed.
Lemma cmra_validN_includedN x y n : {n} y  x {n} y  {n} x.
Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed.
Lemma cmra_validN_included x y n : {n} y  x  y  {n} x.
Proof. rewrite cmra_included_includedN; eauto using cmra_validN_includedN. Qed.

Lemma cmra_includedN_S x y n : x {S n} y  x {n} y.
Proof. by intros [z Hz]; exists z; apply dist_S. Qed.
Lemma cmra_includedN_le x y n n' : x {n} y  n'  n  x {n'} y.
Proof. induction 2; auto using cmra_includedN_S. Qed.

Lemma cmra_includedN_l n x y : x {n} x  y.
Proof. by exists y. Qed.
Lemma cmra_included_l x y : x  x  y.
Proof. by exists y. Qed.
Lemma cmra_includedN_r n x y : y {n} x  y.
Proof. rewrite (commutative op); apply cmra_includedN_l. Qed.
Lemma cmra_included_r x y : y  x  y.
Proof. rewrite (commutative op); apply cmra_included_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
271

272
273
274
275
Lemma cmra_unit_preserving x y : x  y  unit x  unit y.
Proof. rewrite !cmra_included_includedN; eauto using cmra_unit_preservingN. Qed.
Lemma cmra_included_unit x : unit x  x.
Proof. by exists x; rewrite cmra_unit_l. Qed.
276
277
Lemma cmra_preservingN_l n x y z : x {n} y  z  x {n} z  y.
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (associative op). Qed.
278
279
Lemma cmra_preserving_l x y z : x  y  z  x  z  y.
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (associative op). Qed.
280
281
Lemma cmra_preservingN_r n x y z : x {n} y  x  z {n} y  z.
Proof. by intros; rewrite -!(commutative _ z); apply cmra_preservingN_l. Qed.
282
283
284
285
Lemma cmra_preserving_r x y z : x  y  x  z  y  z.
Proof. by intros; rewrite -!(commutative _ z); apply cmra_preserving_l. Qed.

Lemma cmra_included_dist_l x1 x2 x1' n :
286
  x1  x2  x1' {n} x1   x2', x1'  x2'  x2' {n} x2.
Robbert Krebbers's avatar
Robbert Krebbers committed
287
Proof.
288
289
  intros [z Hx2] Hx1; exists (x1'  z); split; auto using cmra_included_l.
  by rewrite Hx1 Hx2.
Robbert Krebbers's avatar
Robbert Krebbers committed
290
Qed.
291
292
293

(** ** Minus *)
Lemma cmra_op_minus' x y : x  y  x  y  x  y.
Robbert Krebbers's avatar
Robbert Krebbers committed
294
Proof.
295
  rewrite cmra_included_includedN equiv_dist; eauto using cmra_op_minus.
Robbert Krebbers's avatar
Robbert Krebbers committed
296
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
297

Robbert Krebbers's avatar
Robbert Krebbers committed
298
(** ** Timeless *)
299
Lemma cmra_timeless_included_l x y : Timeless x  {0} y  x {0} y  x  y.
Robbert Krebbers's avatar
Robbert Krebbers committed
300
301
Proof.
  intros ?? [x' ?].
302
  destruct (cmra_extend_op 0 y x x') as ([z z']&Hy&Hz&Hz'); auto; simpl in *.
Robbert Krebbers's avatar
Robbert Krebbers committed
303
  by exists z'; rewrite Hy (timeless x z).
Robbert Krebbers's avatar
Robbert Krebbers committed
304
Qed.
305
Lemma cmra_timeless_included_r n x y : Timeless y  x {0} y  x {n} y.
Robbert Krebbers's avatar
Robbert Krebbers committed
306
Proof. intros ? [x' ?]. exists x'. by apply equiv_dist, (timeless y). Qed.
307
Lemma cmra_op_timeless x1 x2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
308
   (x1  x2)  Timeless x1  Timeless x2  Timeless (x1  x2).
Robbert Krebbers's avatar
Robbert Krebbers committed
309
310
Proof.
  intros ??? z Hz.
311
  destruct (cmra_extend_op 0 z x1 x2) as ([y1 y2]&Hz'&?&?); auto; simpl in *.
312
  { by rewrite -?Hz. }
Robbert Krebbers's avatar
Robbert Krebbers committed
313
  by rewrite Hz' (timeless x1 y1) // (timeless x2 y2).
Robbert Krebbers's avatar
Robbert Krebbers committed
314
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
315

316
317
318
319
320
321
322
323
324
325
326
327
(** ** RAs with an empty element *)
Section identity.
  Context `{Empty A, !CMRAIdentity A}.
  Lemma cmra_empty_leastN  n x :  {n} x.
  Proof. by exists x; rewrite left_id. Qed.
  Lemma cmra_empty_least x :   x.
  Proof. by exists x; rewrite left_id. Qed.
  Global Instance cmra_empty_right_id : RightId ()  ().
  Proof. by intros x; rewrite (commutative op) left_id. Qed.
  Lemma cmra_unit_empty : unit   .
  Proof. by rewrite -{2}(cmra_unit_l ) right_id. Qed.
End identity.
Robbert Krebbers's avatar
Robbert Krebbers committed
328

329
(** ** Local updates *)
330
331
Global Instance local_update_proper Lv (L : A  A) :
  LocalUpdate Lv L  Proper (() ==> ()) L.
332
333
Proof. intros; apply (ne_proper _). Qed.

334
335
336
Lemma local_update L `{!LocalUpdate Lv L} x y :
  Lv x   (x  y)  L (x  y)  L x  y.
Proof. by rewrite equiv_dist=>?? n; apply (local_updateN L). Qed.
337
338
339
340

Global Instance local_update_op x : LocalUpdate (λ _, True) (op x).
Proof. split. apply _. by intros n y1 y2 _ _; rewrite associative. Qed.

341
(** ** Updates *)
342
Global Instance cmra_update_preorder : PreOrder (@cmra_update A).
Robbert Krebbers's avatar
Robbert Krebbers committed
343
Proof. split. by intros x y. intros x y y' ?? z ?; naive_solver. Qed.
344
Lemma cmra_update_updateP x y : x ~~> y  x ~~>: (y =).
Robbert Krebbers's avatar
Robbert Krebbers committed
345
346
347
348
349
Proof.
  split.
  * by intros Hx z ?; exists y; split; [done|apply (Hx z)].
  * by intros Hx z n ?; destruct (Hx z n) as (?&<-&?).
Qed.
350
Lemma cmra_updateP_id (P : A  Prop) x : P x  x ~~>: P.
351
Proof. by intros ? z n ?; exists x. Qed.
352
Lemma cmra_updateP_compose (P Q : A  Prop) x :
353
  x ~~>: P  ( y, P y  y ~~>: Q)  x ~~>: Q.
354
355
356
Proof.
  intros Hx Hy z n ?. destruct (Hx z n) as (y&?&?); auto. by apply (Hy y).
Qed.
357
358
359
360
361
Lemma cmra_updateP_compose_l (Q : A  Prop) x y : x ~~> y  y ~~>: Q  x ~~>: Q.
Proof.
  rewrite cmra_update_updateP.
  intros; apply cmra_updateP_compose with (y =); intros; subst; auto.
Qed.
362
Lemma cmra_updateP_weaken (P Q : A  Prop) x : x ~~>: P  ( y, P y  Q y)  x ~~>: Q.
363
Proof. eauto using cmra_updateP_compose, cmra_updateP_id. Qed.
364

365
Lemma cmra_updateP_op (P1 P2 Q : A  Prop) x1 x2 :
366
  x1 ~~>: P1  x2 ~~>: P2  ( y1 y2, P1 y1  P2 y2  Q (y1  y2))  x1  x2 ~~>: Q.
367
368
369
370
371
372
373
Proof.
  intros Hx1 Hx2 Hy z n ?.
  destruct (Hx1 (x2  z) n) as (y1&?&?); first by rewrite associative.
  destruct (Hx2 (y1  z) n) as (y2&?&?);
    first by rewrite associative (commutative _ x2) -associative.
  exists (y1  y2); split; last rewrite (commutative _ y1) -associative; auto.
Qed.
374
Lemma cmra_updateP_op' (P1 P2 : A  Prop) x1 x2 :
375
  x1 ~~>: P1  x2 ~~>: P2  x1  x2 ~~>: λ y,  y1 y2, y = y1  y2  P1 y1  P2 y2.
376
Proof. eauto 10 using cmra_updateP_op. Qed.
377
Lemma cmra_update_op x1 x2 y1 y2 : x1 ~~> y1  x2 ~~> y2  x1  x2 ~~> y1  y2.
378
Proof.
379
  rewrite !cmra_update_updateP; eauto using cmra_updateP_op with congruence.
380
Qed.
381
382
383
384
385
386
387
388

Section identity_updates.
  Context `{Empty A, !CMRAIdentity A}.
  Lemma cmra_update_empty x : x ~~> .
  Proof. intros z n; rewrite left_id; apply cmra_validN_op_r. Qed.
  Lemma cmra_update_empty_alt y :  ~~> y   x, x ~~> y.
  Proof. split; [intros; transitivity |]; auto using cmra_update_empty. Qed.
End identity_updates.
Robbert Krebbers's avatar
Robbert Krebbers committed
389
390
End cmra.

391
(** * Properties about monotone functions *)
392
Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A).
393
Proof. by split. Qed.
394
395
Instance cmra_monotone_compose {A B C : cmraT} (f : A  B) (g : B  C) :
  CMRAMonotone f  CMRAMonotone g  CMRAMonotone (g  f).
Robbert Krebbers's avatar
Robbert Krebbers committed
396
397
Proof.
  split.
398
399
  * move=> n x y Hxy /=. by apply includedN_preserving, includedN_preserving.
  * move=> n x Hx /=. by apply validN_preserving, validN_preserving.
Robbert Krebbers's avatar
Robbert Krebbers committed
400
Qed.
401

402
403
404
405
406
407
Section cmra_monotone.
  Context {A B : cmraT} (f : A  B) `{!CMRAMonotone f}.
  Lemma included_preserving x y : x  y  f x  f y.
  Proof.
    rewrite !cmra_included_includedN; eauto using includedN_preserving.
  Qed.
408
  Lemma valid_preserving x :  x   f x.
409
410
411
  Proof. rewrite !cmra_valid_validN; eauto using validN_preserving. Qed.
End cmra_monotone.

412

413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
(** * Transporting a CMRA equality *)
Definition cmra_transport {A B : cmraT} (H : A = B) (x : A) : B :=
  eq_rect A id x _ H.

Section cmra_transport.
  Context {A B : cmraT} (H : A = B).
  Notation T := (cmra_transport H).
  Global Instance cmra_transport_ne n : Proper (dist n ==> dist n) T.
  Proof. by intros ???; destruct H. Qed.
  Global Instance cmra_transport_proper : Proper (() ==> ()) T.
  Proof. by intros ???; destruct H. Qed.
  Lemma cmra_transport_op x y : T (x  y) = T x  T y.
  Proof. by destruct H. Qed.
  Lemma cmra_transport_unit x : T (unit x) = unit (T x).
  Proof. by destruct H. Qed.
428
  Lemma cmra_transport_validN n x : {n} T x  {n} x.
429
  Proof. by destruct H. Qed.
430
  Lemma cmra_transport_valid x :  T x   x.
431
432
433
434
435
436
437
438
439
440
441
  Proof. by destruct H. Qed.
  Global Instance cmra_transport_timeless x : Timeless x  Timeless (T x).
  Proof. by destruct H. Qed.
  Lemma cmra_transport_updateP (P : A  Prop) (Q : B  Prop) x :
    x ~~>: P  ( y, P y  Q (T y))  T x ~~>: Q.
  Proof. destruct H; eauto using cmra_updateP_weaken. Qed.
  Lemma cmra_transport_updateP' (P : A  Prop) x :
    x ~~>: P  T x ~~>: λ y,  y', y = cmra_transport H y'  P y'.
  Proof. eauto using cmra_transport_updateP. Qed.
End cmra_transport.

442
443
444
445
446
447
(** * Instances *)
(** ** Discrete CMRA *)
Class RA A `{Equiv A, Unit A, Op A, Valid A, Minus A} := {
  (* setoids *)
  ra_op_ne (x : A) : Proper (() ==> ()) (op x);
  ra_unit_ne :> Proper (() ==> ()) unit;
448
  ra_validN_ne :> Proper (() ==> impl) valid;
449
450
451
452
453
454
455
456
457
458
459
  ra_minus_ne :> Proper (() ==> () ==> ()) minus;
  (* monoid *)
  ra_associative :> Associative () ();
  ra_commutative :> Commutative () ();
  ra_unit_l x : unit x  x  x;
  ra_unit_idempotent x : unit (unit x)  unit x;
  ra_unit_preserving x y : x  y  unit x  unit y;
  ra_valid_op_l x y :  (x  y)   x;
  ra_op_minus x y : x  y  x  y  x  y
}.

460
Section discrete.
461
462
463
  Context {A : cofeT} `{ x : A, Timeless x}.
  Context `{Unit A, Op A, Valid A, Minus A} (ra : RA A).

464
  Instance discrete_validN : ValidN A := λ n x,  x.
465
  Definition discrete_cmra_mixin : CMRAMixin A.
466
  Proof.
467
468
    by destruct ra; split; unfold Proper, respectful, includedN;
      try setoid_rewrite <-(timeless_iff _ _ _ _).
469
  Qed.
470
  Definition discrete_extend_mixin : CMRAExtendMixin A.
471
  Proof.
472
473
    intros n x y1 y2 ??; exists (y1,y2); split_ands; auto.
    apply (timeless _), dist_le with n; auto with lia.
474
  Qed.
475
  Definition discreteRA : cmraT :=
476
    CMRAT (cofe_mixin A) discrete_cmra_mixin discrete_extend_mixin.
477
  Lemma discrete_updateP (x : discreteRA) (P : A  Prop) :
478
    ( z,  (x  z)   y, P y   (y  z))  x ~~>: P.
479
  Proof. intros Hvalid z n; apply Hvalid. Qed.
480
  Lemma discrete_update (x y : discreteRA) :
481
    ( z,  (x  z)   (y  z))  x ~~> y.
482
  Proof. intros Hvalid z n; apply Hvalid. Qed.
483
484
End discrete.

485
486
487
488
489
490
491
492
493
494
495
496
497
498
(** ** CMRA for the unit type *)
Section unit.
  Instance unit_valid : Valid () := λ x, True.
  Instance unit_unit : Unit () := λ x, x.
  Instance unit_op : Op () := λ x y, ().
  Instance unit_minus : Minus () := λ x y, ().
  Global Instance unit_empty : Empty () := ().
  Definition unit_ra : RA ().
  Proof. by split. Qed.
  Canonical Structure unitRA : cmraT :=
    Eval cbv [unitC discreteRA cofe_car] in discreteRA unit_ra.
  Global Instance unit_cmra_identity : CMRAIdentity unitRA.
  Proof. by split; intros []. Qed.
End unit.
499

500
(** ** Product *)
501
502
503
504
505
Section prod.
  Context {A B : cmraT}.
  Instance prod_op : Op (A * B) := λ x y, (x.1  y.1, x.2  y.2).
  Global Instance prod_empty `{Empty A, Empty B} : Empty (A * B) := (, ).
  Instance prod_unit : Unit (A * B) := λ x, (unit (x.1), unit (x.2)).
506
  Instance prod_validN : ValidN (A * B) := λ n x, {n} x.1  {n} x.2.
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
  Instance prod_minus : Minus (A * B) := λ x y, (x.1  y.1, x.2  y.2).
  Lemma prod_included (x y : A * B) : x  y  x.1  y.1  x.2  y.2.
  Proof.
    split; [intros [z Hz]; split; [exists (z.1)|exists (z.2)]; apply Hz|].
    intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto.
  Qed.
  Lemma prod_includedN (x y : A * B) n : x {n} y  x.1 {n} y.1  x.2 {n} y.2.
  Proof.
    split; [intros [z Hz]; split; [exists (z.1)|exists (z.2)]; apply Hz|].
    intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto.
  Qed.
  Definition prod_cmra_mixin : CMRAMixin (A * B).
  Proof.
    split; try apply _.
    * by intros n x y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
    * by intros n y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
    * by intros n y1 y2 [Hy1 Hy2] [??]; split; rewrite /= -?Hy1 -?Hy2.
    * by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2];
        split; rewrite /= ?Hx1 ?Hx2 ?Hy1 ?Hy2.
526
    * by intros n x [??]; split; apply cmra_validN_S.
527
528
    * split; simpl; apply (associative _).
    * split; simpl; apply (commutative _).
529
530
    * split; simpl; apply cmra_unit_l.
    * split; simpl; apply cmra_unit_idempotent.
531
    * intros n x y; rewrite !prod_includedN.
532
533
      by intros [??]; split; apply cmra_unit_preservingN.
    * intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
534
535
536
537
538
539
540
541
542
543
544
545
    * intros x y n; rewrite prod_includedN; intros [??].
      by split; apply cmra_op_minus.
  Qed.
  Definition prod_cmra_extend_mixin : CMRAExtendMixin (A * B).
  Proof.
    intros n x y1 y2 [??] [??]; simpl in *.
    destruct (cmra_extend_op n (x.1) (y1.1) (y2.1)) as (z1&?&?&?); auto.
    destruct (cmra_extend_op n (x.2) (y1.2) (y2.2)) as (z2&?&?&?); auto.
    by exists ((z1.1,z2.1),(z1.2,z2.2)).
  Qed.
  Canonical Structure prodRA : cmraT :=
    CMRAT prod_cofe_mixin prod_cmra_mixin prod_cmra_extend_mixin.
546
547
548
549
550
551
552
553
  Global Instance prod_cmra_identity `{Empty A, Empty B} :
    CMRAIdentity A  CMRAIdentity B  CMRAIdentity prodRA.
  Proof.
    split.
    * split; apply cmra_empty_valid.
    * by split; rewrite /=left_id.
    * by intros ? [??]; split; apply (timeless _).
  Qed.
554
  Lemma prod_update x y : x.1 ~~> y.1  x.2 ~~> y.2  x ~~> y.
555
  Proof. intros ?? z n [??]; split; simpl in *; auto. Qed.
556
  Lemma prod_updateP P1 P2 (Q : A * B  Prop)  x :
557
    x.1 ~~>: P1  x.2 ~~>: P2  ( a b, P1 a  P2 b  Q (a,b))  x ~~>: Q.
558
559
560
561
562
  Proof.
    intros Hx1 Hx2 HP z n [??]; simpl in *.
    destruct (Hx1 (z.1) n) as (a&?&?), (Hx2 (z.2) n) as (b&?&?); auto.
    exists (a,b); repeat split; auto.
  Qed.
563
  Lemma prod_updateP' P1 P2 x :
564
    x.1 ~~>: P1  x.2 ~~>: P2  x ~~>: λ y, P1 (y.1)  P2 (y.2).
565
  Proof. eauto using prod_updateP. Qed.
566
567
568
569
570
End prod.
Arguments prodRA : clear implicits.

Instance prod_map_cmra_monotone {A A' B B' : cmraT} (f : A  A') (g : B  B') :
  CMRAMonotone f  CMRAMonotone g  CMRAMonotone (prod_map f g).
571
572
Proof.
  split.
573
  * intros n x y; rewrite !prod_includedN; intros [??]; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
574
    by split; apply includedN_preserving.
575
576
  * by intros n x [??]; split; simpl; apply validN_preserving.
Qed.