cmra.v 48 KB
Newer Older
1
From iris.algebra Require Export cofe.
2

Robbert Krebbers's avatar
Robbert Krebbers committed
3
4
Class PCore (A : Type) := pcore : A  option A.
Instance: Params (@pcore) 2.
5
6
7
8
9
10

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.

11
12
13
14
15
(* The inclusion quantifies over [A], not [option A].  This means we do not get
   reflexivity.  However, if we used [option A], the following would no longer
   hold:
     x ≼ y ↔ x.1 ≼ y.1 ∧ x.2 ≼ y.2
*)
16
17
18
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.
19
Hint Extern 0 (_  _) => reflexivity.
20
21
Instance: Params (@included) 3.

Robbert Krebbers's avatar
Robbert Krebbers committed
22
23
Class ValidN (A : Type) := validN : nat  A  Prop.
Instance: Params (@validN) 3.
24
Notation "✓{ n } x" := (validN n x)
25
  (at level 20, n at next level, format "✓{ n }  x").
Robbert Krebbers's avatar
Robbert Krebbers committed
26

27
28
Class Valid (A : Type) := valid : A  Prop.
Instance: Params (@valid) 2.
29
Notation "✓ x" := (valid x) (at level 20) : C_scope.
30

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
Notation "x ≼{ n } y" := (includedN n x y)
33
  (at level 70, n at next level, format "x  ≼{ n }  y") : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
34
Instance: Params (@includedN) 4.
35
Hint Extern 0 (_ {_} _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
36

Robbert Krebbers's avatar
Robbert Krebbers committed
37
Record CMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A} := {
Robbert Krebbers's avatar
Robbert Krebbers committed
38
  (* setoids *)
39
  mixin_cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x);
Robbert Krebbers's avatar
Robbert Krebbers committed
40
41
  mixin_cmra_pcore_ne n x y cx :
    x {n} y  pcore x = Some cx   cy, pcore y = Some cy  cx {n} cy;
42
  mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n);
Robbert Krebbers's avatar
Robbert Krebbers committed
43
  (* valid *)
44
  mixin_cmra_valid_validN x :  x   n, {n} x;
45
  mixin_cmra_validN_S n x : {S n} x  {n} x;
Robbert Krebbers's avatar
Robbert Krebbers committed
46
  (* monoid *)
47
48
  mixin_cmra_assoc : Assoc () ();
  mixin_cmra_comm : Comm () ();
Robbert Krebbers's avatar
Robbert Krebbers committed
49
50
  mixin_cmra_pcore_l x cx : pcore x = Some cx  cx  x  x;
  mixin_cmra_pcore_idemp x cx : pcore x = Some cx  pcore cx  Some cx;
51
  mixin_cmra_pcore_mono x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
52
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy;
53
  mixin_cmra_validN_op_l n x y : {n} (x  y)  {n} x;
54
55
  mixin_cmra_extend n x y1 y2 :
    {n} x  x {n} y1  y2 
56
     z1 z2, x  z1  z2  z1 {n} y1  z2 {n} y2
Robbert Krebbers's avatar
Robbert Krebbers committed
57
}.
Robbert Krebbers's avatar
Robbert Krebbers committed
58

Robbert Krebbers's avatar
Robbert Krebbers committed
59
(** Bundeled version *)
60
Structure cmraT := CMRAT' {
Robbert Krebbers's avatar
Robbert Krebbers committed
61
62
63
64
  cmra_car :> Type;
  cmra_equiv : Equiv cmra_car;
  cmra_dist : Dist cmra_car;
  cmra_compl : Compl cmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
65
  cmra_pcore : PCore cmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
66
  cmra_op : Op cmra_car;
67
  cmra_valid : Valid cmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
68
  cmra_validN : ValidN cmra_car;
69
  cmra_cofe_mixin : CofeMixin cmra_car;
70
  cmra_mixin : CMRAMixin cmra_car;
71
  _ : Type
Robbert Krebbers's avatar
Robbert Krebbers committed
72
}.
73
74
Arguments CMRAT' _ {_ _ _ _ _ _ _} _ _ _.
Notation CMRAT A m m' := (CMRAT' A m m' A).
75
76
77
78
Arguments cmra_car : simpl never.
Arguments cmra_equiv : simpl never.
Arguments cmra_dist : simpl never.
Arguments cmra_compl : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
79
Arguments cmra_pcore : simpl never.
80
Arguments cmra_op : simpl never.
81
Arguments cmra_valid : simpl never.
82
83
84
Arguments cmra_validN : simpl never.
Arguments cmra_cofe_mixin : simpl never.
Arguments cmra_mixin : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
85
Add Printing Constructor cmraT.
86
87
88
89
Hint Extern 0 (PCore _) => eapply (@cmra_pcore _) : typeclass_instances.
Hint Extern 0 (Op _) => eapply (@cmra_op _) : typeclass_instances.
Hint Extern 0 (Valid _) => eapply (@cmra_valid _) : typeclass_instances.
Hint Extern 0 (ValidN _) => eapply (@cmra_validN _) : typeclass_instances.
90
Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT A (cmra_cofe_mixin A).
Robbert Krebbers's avatar
Robbert Krebbers committed
91
92
Canonical Structure cmra_cofeC.

93
94
95
96
97
98
(** 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.
Robbert Krebbers's avatar
Robbert Krebbers committed
99
100
101
  Lemma cmra_pcore_ne n x y cx :
    x {n} y  pcore x = Some cx   cy, pcore y = Some cy  cx {n} cy.
  Proof. apply (mixin_cmra_pcore_ne _ (cmra_mixin A)). Qed.
102
103
  Global Instance cmra_validN_ne n : Proper (dist n ==> impl) (@validN A _ n).
  Proof. apply (mixin_cmra_validN_ne _ (cmra_mixin A)). Qed.
104
105
  Lemma cmra_valid_validN x :  x   n, {n} x.
  Proof. apply (mixin_cmra_valid_validN _ (cmra_mixin A)). Qed.
106
107
  Lemma cmra_validN_S n x : {S n} x  {n} x.
  Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed.
108
109
110
111
  Global Instance cmra_assoc : Assoc () (@op A _).
  Proof. apply (mixin_cmra_assoc _ (cmra_mixin A)). Qed.
  Global Instance cmra_comm : Comm () (@op A _).
  Proof. apply (mixin_cmra_comm _ (cmra_mixin A)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
112
113
114
115
  Lemma cmra_pcore_l x cx : pcore x = Some cx  cx  x  x.
  Proof. apply (mixin_cmra_pcore_l _ (cmra_mixin A)). Qed.
  Lemma cmra_pcore_idemp x cx : pcore x = Some cx  pcore cx  Some cx.
  Proof. apply (mixin_cmra_pcore_idemp _ (cmra_mixin A)). Qed.
116
  Lemma cmra_pcore_mono x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
117
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy.
118
  Proof. apply (mixin_cmra_pcore_mono _ (cmra_mixin A)). Qed.
119
120
  Lemma cmra_validN_op_l n x y : {n} (x  y)  {n} x.
  Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed.
121
  Lemma cmra_extend n x y1 y2 :
122
    {n} x  x {n} y1  y2 
123
     z1 z2, x  z1  z2  z1 {n} y1  z2 {n} y2.
124
  Proof. apply (mixin_cmra_extend _ (cmra_mixin A)). Qed.
125
126
End cmra_mixin.

Robbert Krebbers's avatar
Robbert Krebbers committed
127
128
129
130
131
132
133
134
Definition opM {A : cmraT} (x : A) (my : option A) :=
  match my with Some y => x  y | None => x end.
Infix "⋅?" := opM (at level 50, left associativity) : C_scope.

(** * Persistent elements *)
Class Persistent {A : cmraT} (x : A) := persistent : pcore x  Some x.
Arguments persistent {_} _ {_}.

135
(** * Exclusive elements (i.e., elements that cannot have a frame). *)
136
137
Class Exclusive {A : cmraT} (x : A) := exclusive0_l y : {0} (x  y)  False.
Arguments exclusive0_l {_} _ {_} _ _.
138

Robbert Krebbers's avatar
Robbert Krebbers committed
139
140
141
142
143
144
145
146
147
148
149
(** * CMRAs whose core is total *)
(** The function [core] may return a dummy when used on CMRAs without total
core. *)
Class CMRATotal (A : cmraT) := cmra_total (x : A) : is_Some (pcore x).

Class Core (A : Type) := core : A  A.
Instance: Params (@core) 2.

Instance core' `{PCore A} : Core A := λ x, from_option id x (pcore x).
Arguments core' _ _ _ /.

Ralf Jung's avatar
Ralf Jung committed
150
(** * CMRAs with a unit element *)
151
(** We use the notation ∅ because for most instances (maps, sets, etc) the
Ralf Jung's avatar
Ralf Jung committed
152
`empty' element is the unit. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
153
Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := {
154
155
  mixin_ucmra_unit_valid :  ;
  mixin_ucmra_unit_left_id : LeftId ()  ();
Robbert Krebbers's avatar
Robbert Krebbers committed
156
  mixin_ucmra_pcore_unit : pcore   Some 
157
}.
158

159
Structure ucmraT := UCMRAT' {
160
161
162
163
  ucmra_car :> Type;
  ucmra_equiv : Equiv ucmra_car;
  ucmra_dist : Dist ucmra_car;
  ucmra_compl : Compl ucmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
164
  ucmra_pcore : PCore ucmra_car;
165
166
167
168
169
170
  ucmra_op : Op ucmra_car;
  ucmra_valid : Valid ucmra_car;
  ucmra_validN : ValidN ucmra_car;
  ucmra_empty : Empty ucmra_car;
  ucmra_cofe_mixin : CofeMixin ucmra_car;
  ucmra_cmra_mixin : CMRAMixin ucmra_car;
171
  ucmra_mixin : UCMRAMixin ucmra_car;
172
  _ : Type;
173
}.
174
175
Arguments UCMRAT' _ {_ _ _ _ _ _ _ _} _ _ _ _.
Notation UCMRAT A m m' m'' := (UCMRAT' A m m' m'' A).
176
177
178
179
Arguments ucmra_car : simpl never.
Arguments ucmra_equiv : simpl never.
Arguments ucmra_dist : simpl never.
Arguments ucmra_compl : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
180
Arguments ucmra_pcore : simpl never.
181
182
183
184
185
186
187
Arguments ucmra_op : simpl never.
Arguments ucmra_valid : simpl never.
Arguments ucmra_validN : simpl never.
Arguments ucmra_cofe_mixin : simpl never.
Arguments ucmra_cmra_mixin : simpl never.
Arguments ucmra_mixin : simpl never.
Add Printing Constructor ucmraT.
188
Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances.
189
190
191
192
193
194
195
196
197
198
199
200
201
202
Coercion ucmra_cofeC (A : ucmraT) : cofeT := CofeT A (ucmra_cofe_mixin A).
Canonical Structure ucmra_cofeC.
Coercion ucmra_cmraR (A : ucmraT) : cmraT :=
  CMRAT A (ucmra_cofe_mixin A) (ucmra_cmra_mixin A).
Canonical Structure ucmra_cmraR.

(** Lifting properties from the mixin *)
Section ucmra_mixin.
  Context {A : ucmraT}.
  Implicit Types x y : A.
  Lemma ucmra_unit_valid :  ( : A).
  Proof. apply (mixin_ucmra_unit_valid _ (ucmra_mixin A)). Qed.
  Global Instance ucmra_unit_left_id : LeftId ()  (@op A _).
  Proof. apply (mixin_ucmra_unit_left_id _ (ucmra_mixin A)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
203
204
  Lemma ucmra_pcore_unit : pcore (:A)  Some .
  Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed.
205
End ucmra_mixin.
206

207
(** * Discrete CMRAs *)
208
Class CMRADiscrete (A : cmraT) := {
209
210
211
212
  cmra_discrete :> Discrete A;
  cmra_discrete_valid (x : A) : {0} x   x
}.

Robbert Krebbers's avatar
Robbert Krebbers committed
213
(** * Morphisms *)
214
Class CMRAMonotone {A B : cmraT} (f : A  B) := {
Robbert Krebbers's avatar
Robbert Krebbers committed
215
216
  cmra_monotone_ne n :> Proper (dist n ==> dist n) f;
  validN_preserving n x : {n} x  {n} f x;
217
  cmra_monotone x y : x  y  f x  f y
218
}.
219
Arguments validN_preserving {_ _} _ {_} _ _ _.
220
Arguments cmra_monotone {_ _} _ {_} _ _ _.
221

Robbert Krebbers's avatar
Robbert Krebbers committed
222
(** * Properties **)
Robbert Krebbers's avatar
Robbert Krebbers committed
223
Section cmra.
224
Context {A : cmraT}.
Robbert Krebbers's avatar
Robbert Krebbers committed
225
Implicit Types x y z : A.
226
Implicit Types xs ys zs : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
227

228
(** ** Setoids *)
Robbert Krebbers's avatar
Robbert Krebbers committed
229
230
231
232
233
234
235
236
237
Global Instance cmra_pcore_ne' n : Proper (dist n ==> dist n) (@pcore A _).
Proof.
  intros x y Hxy. destruct (pcore x) as [cx|] eqn:?.
  { destruct (cmra_pcore_ne n x y cx) as (cy&->&->); auto. }
  destruct (pcore y) as [cy|] eqn:?; auto.
  destruct (cmra_pcore_ne n y x cy) as (cx&?&->); simplify_eq/=; auto.
Qed.
Lemma cmra_pcore_proper x y cx :
  x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy.
238
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
239
240
241
  intros. destruct (cmra_pcore_ne 0 x y cx) as (cy&?&?); auto.
  exists cy; split; [done|apply equiv_dist=> n].
  destruct (cmra_pcore_ne n x y cx) as (cy'&?&?); naive_solver.
242
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
243
244
245
246
Global Instance cmra_pcore_proper' : Proper (() ==> ()) (@pcore 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 (comm _ x1) Hx (comm _ y2). Qed.
247
Global Instance cmra_op_proper' : Proper (() ==> () ==> ()) (@op A _).
248
249
250
251
252
253
254
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_valid_proper : Proper (() ==> iff) (@valid A _).
255
256
257
258
Proof.
  intros x y Hxy; rewrite !cmra_valid_validN.
  by split=> ? n; [rewrite -Hxy|rewrite Hxy].
Qed.
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
277
278
279
280
Global Instance cmra_opM_ne n : Proper (dist n ==> dist n ==> dist n) (@opM A).
Proof. destruct 2; by cofe_subst. Qed.
Global Instance cmra_opM_proper : Proper (() ==> () ==> ()) (@opM A).
Proof. destruct 2; by setoid_subst. Qed.
281

Robbert Krebbers's avatar
Robbert Krebbers committed
282
283
284
285
(** ** Op *)
Lemma cmra_opM_assoc x y mz : (x  y) ? mz  x  (y ? mz).
Proof. destruct mz; by rewrite /= -?assoc. Qed.

286
(** ** Validity *)
Robbert Krebbers's avatar
Robbert Krebbers committed
287
Lemma cmra_validN_le n n' x : {n} x  n'  n  {n'} x.
288
289
290
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
291
Lemma cmra_validN_op_r n x y : {n} (x  y)  {n} y.
292
Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed.
293
294
295
Lemma cmra_valid_op_r x y :  (x  y)   y.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed.

Ralf Jung's avatar
Ralf Jung committed
296
(** ** Core *)
Robbert Krebbers's avatar
Robbert Krebbers committed
297
298
299
300
301
302
303
304
Lemma cmra_pcore_l' x cx : pcore x  Some cx  cx  x  x.
Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_l. Qed.
Lemma cmra_pcore_r x cx : pcore x = Some cx  x  cx  x.
Proof. intros. rewrite comm. by apply cmra_pcore_l. Qed. 
Lemma cmra_pcore_r' x cx : pcore x  Some cx  x  cx  x.
Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_r. Qed. 
Lemma cmra_pcore_idemp' x cx : pcore x  Some cx  pcore cx  Some cx.
Proof. intros (cx'&?&->)%equiv_Some_inv_r'. eauto using cmra_pcore_idemp. Qed. 
305
306
307
308
Lemma cmra_pcore_dup x cx : pcore x = Some cx  cx  cx  cx.
Proof. intros; symmetry; eauto using cmra_pcore_r', cmra_pcore_idemp. Qed.
Lemma cmra_pcore_dup' x cx : pcore x  Some cx  cx  cx  cx.
Proof. intros; symmetry; eauto using cmra_pcore_r', cmra_pcore_idemp'. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
309
310
311
312
313
314
315
316
Lemma cmra_pcore_validN n x cx : {n} x  pcore x = Some cx  {n} cx.
Proof.
  intros Hvx Hx%cmra_pcore_l. move: Hvx; rewrite -Hx. apply cmra_validN_op_l.
Qed.
Lemma cmra_pcore_valid x cx :  x  pcore x = Some cx   cx.
Proof.
  intros Hv Hx%cmra_pcore_l. move: Hv; rewrite -Hx. apply cmra_valid_op_l.
Qed.
317

318
319
320
321
(** ** Persistent elements *)
Lemma persistent_dup x `{!Persistent x} : x  x  x.
Proof. by apply cmra_pcore_dup' with x. Qed.

322
(** ** Exclusive elements *)
323
Lemma exclusiveN_l n x `{!Exclusive x} y : {n} (x  y)  False.
324
Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed.
325
326
327
328
329
330
Lemma exclusiveN_r n x `{!Exclusive x} y : {n} (y  x)  False.
Proof. rewrite comm. by apply exclusiveN_l. Qed.
Lemma exclusive_l x `{!Exclusive x} y :  (x  y)  False.
Proof. by move /cmra_valid_validN /(_ 0) /exclusive0_l. Qed.
Lemma exclusive_r x `{!Exclusive x} y :  (y  x)  False.
Proof. rewrite comm. by apply exclusive_l. Qed.
331
Lemma exclusiveN_opM n x `{!Exclusive x} my : {n} (x ? my)  my = None.
332
Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed.
333

334
(** ** Order *)
Robbert Krebbers's avatar
Robbert Krebbers committed
335
336
Lemma cmra_included_includedN n x y : x  y  x {n} y.
Proof. intros [z ->]. by exists z. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
337
Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n).
338
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
339
  intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2). by rewrite assoc -Hy -Hz.
340
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
341
Global Instance cmra_included_trans: Transitive (@included A _ _).
342
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
343
  intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2). by rewrite assoc -Hy -Hz.
344
Qed.
345
346
Lemma cmra_valid_included x y :  y  x  y   x.
Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_valid_op_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
347
Lemma cmra_validN_includedN n x y : {n} y  x {n} y  {n} x.
348
Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
349
Lemma cmra_validN_included n x y : {n} y  x  y  {n} x.
Robbert Krebbers's avatar
Robbert Krebbers committed
350
Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed.
351

Robbert Krebbers's avatar
Robbert Krebbers committed
352
Lemma cmra_includedN_S n x y : x {S n} y  x {n} y.
353
Proof. by intros [z Hz]; exists z; apply dist_S. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
354
Lemma cmra_includedN_le n n' x y : x {n} y  n'  n  x {n'} y.
355
356
357
358
359
360
361
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.
362
Proof. rewrite (comm op); apply cmra_includedN_l. Qed.
363
Lemma cmra_included_r x y : y  x  y.
364
Proof. rewrite (comm op); apply cmra_included_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
365

366
Lemma cmra_pcore_mono' x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
367
368
369
  x  y  pcore x  Some cx   cy, pcore y = Some cy  cx  cy.
Proof.
  intros ? (cx'&?&Hcx)%equiv_Some_inv_r'.
370
  destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
371
372
  exists cy; by rewrite Hcx.
Qed.
373
Lemma cmra_pcore_monoN' n x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
374
  x {n} y  pcore x {n} Some cx   cy, pcore y = Some cy  cx {n} cy.
Robbert Krebbers's avatar
Robbert Krebbers committed
375
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
376
  intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'.
377
  destruct (cmra_pcore_mono x (x  z) cx')
Robbert Krebbers's avatar
Robbert Krebbers committed
378
379
380
381
382
    as (cy&Hxy&?); auto using cmra_included_l.
  assert (pcore y {n} Some cy) as (cy'&?&Hcy')%dist_Some_inv_r'.
  { by rewrite Hy Hxy. }
  exists cy'; split; first done.
  rewrite Hcx -Hcy'; auto using cmra_included_includedN.
Robbert Krebbers's avatar
Robbert Krebbers committed
383
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
384
385
Lemma cmra_included_pcore x cx : pcore x = Some cx  cx  x.
Proof. exists x. by rewrite cmra_pcore_l. Qed.
386

387
Lemma cmra_monoN_l n x y z : x {n} y  z  x {n} z  y.
388
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
389
Lemma cmra_mono_l x y z : x  y  z  x  z  y.
390
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
391
392
393
394
Lemma cmra_monoN_r n x y z : x {n} y  x  z {n} y  z.
Proof. by intros; rewrite -!(comm _ z); apply cmra_monoN_l. Qed.
Lemma cmra_mono_r x y z : x  y  x  z  y  z.
Proof. by intros; rewrite -!(comm _ z); apply cmra_mono_l. Qed.
395
396
397
398
Lemma cmra_monoN n x1 x2 y1 y2 : x1 {n} y1  x2 {n} y2  x1  x2 {n} y1  y2.
Proof. intros; etrans; eauto using cmra_monoN_l, cmra_monoN_r. Qed.
Lemma cmra_mono x1 x2 y1 y2 : x1  y1  x2  y2  x1  x2  y1  y2.
Proof. intros; etrans; eauto using cmra_mono_l, cmra_mono_r. Qed.
399

400
401
402
403
404
405
406
Global Instance cmra_monoN' n :
  Proper (includedN n ==> includedN n ==> includedN n) (@op A _).
Proof. intros x1 x2 Hx y1 y2 Hy. by apply cmra_monoN. Qed.
Global Instance cmra_mono' :
  Proper (included ==> included ==> included) (@op A _).
Proof. intros x1 x2 Hx y1 y2 Hy. by apply cmra_mono. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
407
Lemma cmra_included_dist_l n x1 x2 x1' :
408
  x1  x2  x1' {n} x1   x2', x1'  x2'  x2' {n} x2.
Robbert Krebbers's avatar
Robbert Krebbers committed
409
Proof.
410
411
  intros [z Hx2] Hx1; exists (x1'  z); split; auto using cmra_included_l.
  by rewrite Hx1 Hx2.
Robbert Krebbers's avatar
Robbert Krebbers committed
412
Qed.
413

Robbert Krebbers's avatar
Robbert Krebbers committed
414
415
416
417
418
419
420
421
422
423
424
425
(** ** Total core *)
Section total_core.
  Context `{CMRATotal A}.

  Lemma cmra_core_l x : core x  x  x.
  Proof.
    destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= Hcx cmra_pcore_l.
  Qed.
  Lemma cmra_core_idemp x : core (core x)  core x.
  Proof.
    destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= Hcx cmra_pcore_idemp.
  Qed.
426
  Lemma cmra_core_mono x y : x  y  core x  core y.
Robbert Krebbers's avatar
Robbert Krebbers committed
427
428
  Proof.
    intros; destruct (cmra_total x) as [cx Hcx].
429
    destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
430
431
432
433
434
435
436
437
438
439
440
441
442
    by rewrite /core /= Hcx Hcy.
  Qed.

  Global Instance cmra_core_ne n : Proper (dist n ==> dist n) (@core A _).
  Proof.
    intros x y Hxy. destruct (cmra_total x) as [cx Hcx].
    by rewrite /core /= -Hxy Hcx.
  Qed.
  Global Instance cmra_core_proper : Proper (() ==> ()) (@core A _).
  Proof. apply (ne_proper _). Qed.

  Lemma cmra_core_r x : x  core x  x.
  Proof. by rewrite (comm _ x) cmra_core_l. Qed.
443
444
  Lemma cmra_core_dup x : core x  core x  core x.
  Proof. by rewrite -{3}(cmra_core_idemp x) cmra_core_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
  Lemma cmra_core_validN n x : {n} x  {n} core x.
  Proof. rewrite -{1}(cmra_core_l x); apply cmra_validN_op_l. Qed.
  Lemma cmra_core_valid x :  x   core x.
  Proof. rewrite -{1}(cmra_core_l x); apply cmra_valid_op_l. Qed.

  Lemma persistent_total x : Persistent x  core x  x.
  Proof.
    split; [intros; by rewrite /core /= (persistent x)|].
    rewrite /Persistent /core /=.
    destruct (cmra_total x) as [? ->]. by constructor.
  Qed.
  Lemma persistent_core x `{!Persistent x} : core x  x.
  Proof. by apply persistent_total. Qed.

  Global Instance cmra_core_persistent x : Persistent (core x).
  Proof.
    destruct (cmra_total x) as [cx Hcx].
    rewrite /Persistent /core /= Hcx /=. eauto using cmra_pcore_idemp.
  Qed.

  Lemma cmra_included_core x : core x  x.
  Proof. by exists x; rewrite cmra_core_l. Qed.
  Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
  Proof.
    split; [|apply _]. by intros x; exists (core x); rewrite cmra_core_r.
  Qed.
  Global Instance cmra_included_preorder : PreOrder (@included A _ _).
  Proof.
    split; [|apply _]. by intros x; exists (core x); rewrite cmra_core_r.
  Qed.
475
  Lemma cmra_core_monoN n x y : x {n} y  core x {n} core y.
Robbert Krebbers's avatar
Robbert Krebbers committed
476
477
  Proof.
    intros [z ->].
478
    apply cmra_included_includedN, cmra_core_mono, cmra_included_l.
Robbert Krebbers's avatar
Robbert Krebbers committed
479
480
481
  Qed.
End total_core.

Robbert Krebbers's avatar
Robbert Krebbers committed
482
(** ** Timeless *)
483
Lemma cmra_timeless_included_l x y : Timeless x  {0} y  x {0} y  x  y.
Robbert Krebbers's avatar
Robbert Krebbers committed
484
485
Proof.
  intros ?? [x' ?].
486
  destruct (cmra_extend 0 y x x') as (z&z'&Hy&Hz&Hz'); auto; simpl in *.
Robbert Krebbers's avatar
Robbert Krebbers committed
487
  by exists z'; rewrite Hy (timeless x z).
Robbert Krebbers's avatar
Robbert Krebbers committed
488
Qed.
489
490
Lemma cmra_timeless_included_r x y : Timeless y  x {0} y  x  y.
Proof. intros ? [x' ?]. exists x'. by apply (timeless y). Qed.
491
Lemma cmra_op_timeless x1 x2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
492
   (x1  x2)  Timeless x1  Timeless x2  Timeless (x1  x2).
Robbert Krebbers's avatar
Robbert Krebbers committed
493
494
Proof.
  intros ??? z Hz.
495
  destruct (cmra_extend 0 z x1 x2) as (y1&y2&Hz'&?&?); auto; simpl in *.
496
  { rewrite -?Hz. by apply cmra_valid_validN. }
Robbert Krebbers's avatar
Robbert Krebbers committed
497
  by rewrite Hz' (timeless x1 y1) // (timeless x2 y2).
Robbert Krebbers's avatar
Robbert Krebbers committed
498
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
499

500
501
502
503
504
505
506
507
(** ** Discrete *)
Lemma cmra_discrete_valid_iff `{CMRADiscrete A} n x :  x  {n} x.
Proof.
  split; first by rewrite cmra_valid_validN.
  eauto using cmra_discrete_valid, cmra_validN_le with lia.
Qed.
Lemma cmra_discrete_included_iff `{Discrete A} n x y : x  y  x {n} y.
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
508
  split; first by apply cmra_included_includedN.
509
510
  intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
511
512
End cmra.

513
514
(** * Properties about CMRAs with a unit element **)
Section ucmra.
Robbert Krebbers's avatar
Robbert Krebbers committed
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
  Context {A : ucmraT}.
  Implicit Types x y z : A.

  Lemma ucmra_unit_validN n : {n} (:A).
  Proof. apply cmra_valid_validN, ucmra_unit_valid. Qed.
  Lemma ucmra_unit_leastN n x :  {n} x.
  Proof. by exists x; rewrite left_id. Qed.
  Lemma ucmra_unit_least x :   x.
  Proof. by exists x; rewrite left_id. Qed.
  Global Instance ucmra_unit_right_id : RightId ()  (@op A _).
  Proof. by intros x; rewrite (comm op) left_id. Qed.
  Global Instance ucmra_unit_persistent : Persistent (:A).
  Proof. apply ucmra_pcore_unit. Qed.

  Global Instance cmra_unit_total : CMRATotal A.
  Proof.
531
    intros x. destruct (cmra_pcore_mono'  x ) as (cx&->&?);
Robbert Krebbers's avatar
Robbert Krebbers committed
532
533
      eauto using ucmra_unit_least, (persistent ).
  Qed.
534
End ucmra.
Robbert Krebbers's avatar
Robbert Krebbers committed
535
536
Hint Immediate cmra_unit_total.

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
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594

(** * Properties about CMRAs with Leibniz equality *)
Section cmra_leibniz.
  Context {A : cmraT} `{!LeibnizEquiv A}.
  Implicit Types x y : A.

  Global Instance cmra_assoc_L : Assoc (=) (@op A _).
  Proof. intros x y z. unfold_leibniz. by rewrite assoc. Qed.
  Global Instance cmra_comm_L : Comm (=) (@op A _).
  Proof. intros x y. unfold_leibniz. by rewrite comm. Qed.

  Lemma cmra_pcore_l_L x cx : pcore x = Some cx  cx  x = x.
  Proof. unfold_leibniz. apply cmra_pcore_l'. Qed.
  Lemma cmra_pcore_idemp_L x cx : pcore x = Some cx  pcore cx = Some cx.
  Proof. unfold_leibniz. apply cmra_pcore_idemp'. Qed.

  Lemma cmra_opM_assoc_L x y mz : (x  y) ? mz = x  (y ? mz).
  Proof. unfold_leibniz. apply cmra_opM_assoc. Qed.

  (** ** Core *)
  Lemma cmra_pcore_r_L x cx : pcore x = Some cx  x  cx = x.
  Proof. unfold_leibniz. apply cmra_pcore_r'. Qed.
  Lemma cmra_pcore_dup_L x cx : pcore x = Some cx  cx = cx  cx.
  Proof. unfold_leibniz. apply cmra_pcore_dup'. Qed.

  (** ** Persistent elements *)
  Lemma persistent_dup_L x `{!Persistent x} : x  x  x.
  Proof. unfold_leibniz. by apply persistent_dup. Qed.

  (** ** Total core *)
  Section total_core.
    Context `{CMRATotal A}.

    Lemma cmra_core_r_L x : x  core x = x.
    Proof. unfold_leibniz. apply cmra_core_r. Qed.
    Lemma cmra_core_l_L x : core x  x = x.
    Proof. unfold_leibniz. apply cmra_core_l. Qed.
    Lemma cmra_core_idemp_L x : core (core x) = core x.
    Proof. unfold_leibniz. apply cmra_core_idemp. Qed.
    Lemma cmra_core_dup_L x : core x = core x  core x.
    Proof. unfold_leibniz. apply cmra_core_dup. Qed.
    Lemma persistent_total_L x : Persistent x  core x = x.
    Proof. unfold_leibniz. apply persistent_total. Qed.
    Lemma persistent_core_L x `{!Persistent x} : core x = x.
    Proof. by apply persistent_total_L. Qed.
  End total_core.
End cmra_leibniz.

Section ucmra_leibniz.
  Context {A : ucmraT} `{!LeibnizEquiv A}.
  Implicit Types x y z : A.

  Global Instance ucmra_unit_left_id_L : LeftId (=)  (@op A _).
  Proof. intros x. unfold_leibniz. by rewrite left_id. Qed.
  Global Instance ucmra_unit_right_id_L : RightId (=)  (@op A _).
  Proof. intros x. unfold_leibniz. by rewrite right_id. Qed.
End ucmra_leibniz.

Robbert Krebbers's avatar
Robbert Krebbers committed
595
596
597
598
599
600
601
602
603
604
605
606
607
(** * Constructing a CMRA with total core *)
Section cmra_total.
  Context A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A}.
  Context (total :  x, is_Some (pcore x)).
  Context (op_ne :  n (x : A), Proper (dist n ==> dist n) (op x)).
  Context (core_ne :  n, Proper (dist n ==> dist n) (@core A _)).
  Context (validN_ne :  n, Proper (dist n ==> impl) (@validN A _ n)).
  Context (valid_validN :  (x : A),  x   n, {n} x).
  Context (validN_S :  n (x : A), {S n} x  {n} x).
  Context (op_assoc : Assoc () (@op A _)).
  Context (op_comm : Comm () (@op A _)).
  Context (core_l :  x : A, core x  x  x).
  Context (core_idemp :  x : A, core (core x)  core x).
608
  Context (core_mono :  x y : A, x  y  core x  core y).
Robbert Krebbers's avatar
Robbert Krebbers committed
609
610
611
  Context (validN_op_l :  n (x y : A), {n} (x  y)  {n} x).
  Context (extend :  n (x y1 y2 : A),
    {n} x  x {n} y1  y2 
612
     z1 z2, x  z1  z2  z1 {n} y1  z2 {n} y2).
Robbert Krebbers's avatar
Robbert Krebbers committed
613
614
615
616
617
618
619
620
  Lemma cmra_total_mixin : CMRAMixin A.
  Proof.
    split; auto.
    - intros n x y ? Hcx%core_ne Hx; move: Hcx. rewrite /core /= Hx /=.
      case (total y)=> [cy ->]; eauto.
    - intros x cx Hcx. move: (core_l x). by rewrite /core /= Hcx.
    - intros x cx Hcx. move: (core_idemp x). rewrite /core /= Hcx /=.
      case (total cx)=>[ccx ->]; by constructor.
621
    - intros x y cx Hxy%core_mono Hx. move: Hxy.
Robbert Krebbers's avatar
Robbert Krebbers committed
622
623
624
      rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto.
  Qed.
End cmra_total.
625

626
(** * Properties about monotone functions *)
627
Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A).
Robbert Krebbers's avatar
Robbert Krebbers committed
628
Proof. repeat split; by try apply _. Qed.
629
630
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
631
632
Proof.
  split.
Robbert Krebbers's avatar
Robbert Krebbers committed
633
  - apply _. 
634
  - move=> n x Hx /=. by apply validN_preserving, validN_preserving.
635
  - move=> x y Hxy /=. by apply cmra_monotone, cmra_monotone.
Robbert Krebbers's avatar
Robbert Krebbers committed
636
Qed.
637

638
639
Section cmra_monotone.
  Context {A B : cmraT} (f : A  B) `{!CMRAMonotone f}.
Robbert Krebbers's avatar
Robbert Krebbers committed
640
  Global Instance cmra_monotone_proper : Proper (() ==> ()) f := ne_proper _.
641
  Lemma cmra_monotoneN n x y : x {n} y  f x {n} f y.
642
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
643
    intros [z ->].
644
    apply cmra_included_includedN, (cmra_monotone f), cmra_included_l.
645
  Qed.
646
  Lemma valid_preserving x :  x   f x.
647
648
649
  Proof. rewrite !cmra_valid_validN; eauto using validN_preserving. Qed.
End cmra_monotone.

650
651
(** Functors *)
Structure rFunctor := RFunctor {
652
  rFunctor_car : cofeT  cofeT  cmraT;
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
  rFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  rFunctor_car A1 B1 -n> rFunctor_car A2 B2;
  rFunctor_ne A1 A2 B1 B2 n :
    Proper (dist n ==> dist n) (@rFunctor_map A1 A2 B1 B2);
  rFunctor_id {A B} (x : rFunctor_car A B) : rFunctor_map (cid,cid) x  x;
  rFunctor_compose {A1 A2 A3 B1 B2 B3}
      (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x :
    rFunctor_map (fg, g'f') x  rFunctor_map (g,g') (rFunctor_map (f,f') x);
  rFunctor_mono {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) :
    CMRAMonotone (rFunctor_map fg) 
}.
Existing Instances rFunctor_ne rFunctor_mono.
Instance: Params (@rFunctor_map) 5.

Class rFunctorContractive (F : rFunctor) :=
  rFunctor_contractive A1 A2 B1 B2 :> Contractive (@rFunctor_map F A1 A2 B1 B2).

Definition rFunctor_diag (F: rFunctor) (A: cofeT) : cmraT := rFunctor_car F A A.
Coercion rFunctor_diag : rFunctor >-> Funclass.

Program Definition constRF (B : cmraT) : rFunctor :=
  {| rFunctor_car A1 A2 := B; rFunctor_map A1 A2 B1 B2 f := cid |}.
Solve Obligations with done.

Instance constRF_contractive B : rFunctorContractive (constRF B).
Proof. rewrite /rFunctorContractive; apply _. Qed.

680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
Structure urFunctor := URFunctor {
  urFunctor_car : cofeT  cofeT  ucmraT;
  urFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  urFunctor_car A1 B1 -n> urFunctor_car A2 B2;
  urFunctor_ne A1 A2 B1 B2 n :
    Proper (dist n ==> dist n) (@urFunctor_map A1 A2 B1 B2);
  urFunctor_id {A B} (x : urFunctor_car A B) : urFunctor_map (cid,cid) x  x;
  urFunctor_compose {A1 A2 A3 B1 B2 B3}
      (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x :
    urFunctor_map (fg, g'f') x  urFunctor_map (g,g') (urFunctor_map (f,f') x);
  urFunctor_mono {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) :
    CMRAMonotone (urFunctor_map fg) 
}.
Existing Instances urFunctor_ne urFunctor_mono.
Instance: Params (@urFunctor_map) 5.

Class urFunctorContractive (F : urFunctor) :=
  urFunctor_contractive A1 A2 B1 B2 :> Contractive (@urFunctor_map F A1 A2 B1 B2).

Definition urFunctor_diag (F: urFunctor) (A: cofeT) : ucmraT := urFunctor_car F A A.
Coercion urFunctor_diag : urFunctor >-> Funclass.

Program Definition constURF (B : ucmraT) : urFunctor :=
  {| urFunctor_car A1 A2 := B; urFunctor_map A1 A2 B1 B2 f := cid |}.
Solve Obligations with done.

Instance constURF_contractive B : urFunctorContractive (constURF B).
Proof. rewrite /urFunctorContractive; apply _. Qed.

709
710
711
712
713
714
715
716
717
718
719
720
721
(** * 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.
Ralf Jung's avatar
Ralf Jung committed
722
  Lemma cmra_transport_core x : T (core x) = core (T x).
723
  Proof. by destruct H. Qed.
724
  Lemma cmra_transport_validN n x : {n} T x  {n} x.
725
  Proof. by destruct H. Qed.
726
  Lemma cmra_transport_valid x :  T x   x.
727
728
729
  Proof. by destruct H. Qed.
  Global Instance cmra_transport_timeless x : Timeless x  Timeless (T x).
  Proof. by destruct H. Qed.
730
731
  Global Instance cmra_transport_persistent x : Persistent x  Persistent (T x).
  Proof. by destruct H. Qed.
732
733
End cmra_transport.

734
735
(** * Instances *)
(** ** Discrete CMRA *)
Robbert Krebbers's avatar
Robbert Krebbers committed
736
Record RAMixin A `{Equiv A, PCore A, Op A, Valid A} := {
737
  (* setoids *)
Robbert Krebbers's avatar
Robbert Krebbers committed
738
739
740
741
  ra_op_proper (x : A) : Proper (() ==> ()) (op x);
  ra_core_proper x y cx :
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy;
  ra_validN_proper : Proper (() ==> impl) valid;
742
  (* monoid *)
743
744
  ra_assoc : Assoc () ();
  ra_comm : Comm () ();
Robbert Krebbers's avatar
Robbert Krebbers committed
745
746
  ra_pcore_l x cx : pcore x = Some cx  cx  x  x;
  ra_pcore_idemp x cx : pcore x = Some cx  pcore cx  Some cx;
747
  ra_pcore_mono x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
748
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy;
Robbert Krebbers's avatar
Robbert Krebbers committed
749
  ra_valid_op_l x y :  (x  y)   x
750
751
}.

752
Section discrete.
Robbert Krebbers's avatar
Robbert Krebbers committed
753
  Context `{Equiv A, PCore A, Op A, Valid A, @Equivalence A ()}.
754
755
  Context (ra_mix : RAMixin A).
  Existing Instances discrete_dist discrete_compl.
756

757
  Instance discrete_validN : ValidN A := λ n x,  x.
758
  Definition discrete_cmra_mixin : CMRAMixin A.
759
  Proof.
760
    destruct ra_mix; split; try done.
761
    - intros x; split; first done. by move=> /(_ 0).
762
    - intros n x y1 y2 ??; by exists y1, y2.
763
764
765
  Qed.
End discrete.

766
767
Notation discreteR A ra_mix :=
  (CMRAT A discrete_cofe_mixin (discrete_cmra_mixin ra_mix)).
768
769
Notation discreteUR A ra_mix ucmra_mix :=
  (UCMRAT A discrete_cofe_mixin (discrete_cmra_mixin ra_mix) ucmra_mix).
770

Robbert Krebbers's avatar
Robbert Krebbers committed
771
Global Instance discrete_cmra_discrete `{Equiv A, PCore A, Op A, Valid A,
772
773
774
  @Equivalence A ()} (ra_mix : RAMixin A) : CMRADiscrete (discreteR A ra_mix).
Proof. split. apply _. done. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
775
776
777
778
779
780
781
782
783
784
Section ra_total.
  Context A `{Equiv A, PCore A, Op A, Valid A}.
  Context (total :  x, is_Some (pcore x)).
  Context (op_proper :  (x : A), Proper (() ==> ()) (op x)).
  Context (core_proper: Proper (() ==> ()) (@core A _)).
  Context (valid_proper : Proper (() ==> impl) (@valid A _)).
  Context (op_assoc : Assoc () (@op A _)).
  Context (op_comm : Comm () (@op A _)).
  Context (core_l :  x : A, core x  x  x).
  Context (core_idemp :  x : A, core (core x)  core x).
785
  Context (core_mono :  x y : A, x  y  core x  core y).
Robbert Krebbers's avatar
Robbert Krebbers committed
786
787
788
789
790
791
792
793
794
  Context (valid_op_l :  x y : A,  (x  y)   x).
  Lemma ra_total_mixin : RAMixin A.
  Proof.
    split; auto.
    - intros x y ? Hcx%core_proper Hx; move: Hcx. rewrite /core /= Hx /=.
      case (total y)=> [cy ->]; eauto.
    - intros x cx Hcx. move: (core_l x). by rewrite /core /= Hcx.
    - intros x cx Hcx. move: (core_idemp x). rewrite /core /= Hcx /=.
      case (total cx)=>[ccx ->]; by constructor.
795
    - intros x y cx Hxy%core_mono Hx. move: Hxy.
Robbert Krebbers's avatar
Robbert Krebbers committed
796
797
798
799
      rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto.
  Qed.
End ra_total.

800
801
802
(** ** CMRA for the unit type *)
Section unit.
  Instance unit_valid : Valid () := λ x, True.
803
  Instance unit_validN : ValidN () := λ n x, True.
Robbert Krebbers's avatar
Robbert Krebbers committed
804
  Instance unit_pcore : PCore () := λ x, Some x.
805
  Instance unit_op : Op () := λ x y, ().
806
  Lemma unit_cmra_mixin : CMRAMixin ().
Robbert Krebbers's avatar
Robbert Krebbers committed
807
  Proof. apply discrete_cmra_mixin, ra_total_mixin; by eauto. Qed.