auth.v 10.5 KB
Newer Older
1 2
From iris.algebra Require Export excl.
From iris.algebra Require Import upred.
3
Local Arguments valid _ _ !_ /.
Robbert Krebbers's avatar
Robbert Krebbers committed
4
Local Arguments validN _ _ _ !_ /.
Robbert Krebbers's avatar
Robbert Krebbers committed
5

Robbert Krebbers's avatar
Robbert Krebbers committed
6
Record auth (A : Type) := Auth { authoritative : option (excl A); own : A }.
7
Add Printing Constructor auth.
Robbert Krebbers's avatar
Robbert Krebbers committed
8
Arguments Auth {_} _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
9
Arguments authoritative {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
10
Arguments own {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
11 12
Notation "◯ a" := (Auth None a) (at level 20).
Notation "● a" := (Auth (Excl' a) ) (at level 20).
Robbert Krebbers's avatar
Robbert Krebbers committed
13

Robbert Krebbers's avatar
Robbert Krebbers committed
14
(* COFE *)
15 16
Section cofe.
Context {A : cofeT}.
Robbert Krebbers's avatar
Robbert Krebbers committed
17
Implicit Types a : option (excl A).
18
Implicit Types b : A.
19
Implicit Types x y : auth A.
20 21

Instance auth_equiv : Equiv (auth A) := λ x y,
Robbert Krebbers's avatar
Robbert Krebbers committed
22
  authoritative x  authoritative y  own x  own y.
23
Instance auth_dist : Dist (auth A) := λ n x y,
24
  authoritative x {n} authoritative y  own x {n} own y.
Robbert Krebbers's avatar
Robbert Krebbers committed
25

26
Global Instance Auth_ne : Proper (dist n ==> dist n ==> dist n) (@Auth A).
Robbert Krebbers's avatar
Robbert Krebbers committed
27
Proof. by split. Qed.
28 29
Global Instance Auth_proper : Proper (() ==> () ==> ()) (@Auth A).
Proof. by split. Qed.
30
Global Instance authoritative_ne: Proper (dist n ==> dist n) (@authoritative A).
Robbert Krebbers's avatar
Robbert Krebbers committed
31
Proof. by destruct 1. Qed.
32 33
Global Instance authoritative_proper : Proper (() ==> ()) (@authoritative A).
Proof. by destruct 1. Qed.
34
Global Instance own_ne : Proper (dist n ==> dist n) (@own A).
Robbert Krebbers's avatar
Robbert Krebbers committed
35
Proof. by destruct 1. Qed.
36 37
Global Instance own_proper : Proper (() ==> ()) (@own A).
Proof. by destruct 1. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
38

39
Instance auth_compl : Compl (auth A) := λ c,
Robbert Krebbers's avatar
Robbert Krebbers committed
40
  Auth (compl (chain_map authoritative c)) (compl (chain_map own c)).
41
Definition auth_cofe_mixin : CofeMixin (auth A).
Robbert Krebbers's avatar
Robbert Krebbers committed
42 43
Proof.
  split.
44
  - intros x y; unfold dist, auth_dist, equiv, auth_equiv.
Robbert Krebbers's avatar
Robbert Krebbers committed
45
    rewrite !equiv_dist; naive_solver.
46
  - intros n; split.
Robbert Krebbers's avatar
Robbert Krebbers committed
47 48
    + by intros ?; split.
    + by intros ?? [??]; split; symmetry.
49
    + intros ??? [??] [??]; split; etrans; eauto.
50
  - by intros ? [??] [??] [??]; split; apply dist_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
51 52
  - intros n c; split. apply (conv_compl n (chain_map authoritative c)).
    apply (conv_compl n (chain_map own c)).
Robbert Krebbers's avatar
Robbert Krebbers committed
53
Qed.
54
Canonical Structure authC := CofeT (auth A) auth_cofe_mixin.
55 56 57 58 59 60

Global Instance Auth_timeless a b :
  Timeless a  Timeless b  Timeless (Auth a b).
Proof. by intros ?? [??] [??]; split; apply: timeless. Qed.
Global Instance auth_discrete : Discrete A  Discrete authC.
Proof. intros ? [??]; apply _. Qed.
61
Global Instance auth_leibniz : LeibnizEquiv A  LeibnizEquiv (auth A).
62
Proof. by intros ? [??] [??] [??]; f_equal/=; apply leibniz_equiv. Qed.
63 64 65
End cofe.

Arguments authC : clear implicits.
Robbert Krebbers's avatar
Robbert Krebbers committed
66 67

(* CMRA *)
68
Section cmra.
69
Context {A : ucmraT}.
70 71
Implicit Types a b : A.
Implicit Types x y : auth A.
72

73 74
Instance auth_valid : Valid (auth A) := λ x,
  match authoritative x with
Robbert Krebbers's avatar
Robbert Krebbers committed
75 76 77
  | Excl' a => ( n, own x {n} a)   a
  | None =>  own x
  | ExclBot' => False
78 79
  end.
Global Arguments auth_valid !_ /.
80
Instance auth_validN : ValidN (auth A) := λ n x,
Robbert Krebbers's avatar
Robbert Krebbers committed
81
  match authoritative x with
Robbert Krebbers's avatar
Robbert Krebbers committed
82 83 84
  | Excl' a => own x {n} a  {n} a
  | None => {n} own x
  | ExclBot' => False
Robbert Krebbers's avatar
Robbert Krebbers committed
85
  end.
86
Global Arguments auth_validN _ !_ /.
Robbert Krebbers's avatar
Robbert Krebbers committed
87 88
Instance auth_pcore : PCore (auth A) := λ x,
  Some (Auth (core (authoritative x)) (core (own x))).
89
Instance auth_op : Op (auth A) := λ x y,
Robbert Krebbers's avatar
Robbert Krebbers committed
90
  Auth (authoritative x  authoritative y) (own x  own y).
91

92
Lemma auth_included (x y : auth A) :
Robbert Krebbers's avatar
Robbert Krebbers committed
93 94 95 96 97
  x  y  authoritative x  authoritative y  own x  own y.
Proof.
  split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|].
  intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto.
Qed.
98
Lemma authoritative_validN n (x : auth A) : {n} x  {n} authoritative x.
Robbert Krebbers's avatar
Robbert Krebbers committed
99
Proof. by destruct x as [[[]|]]. Qed.
100
Lemma own_validN n (x : auth A) : {n} x  {n} own x.
Robbert Krebbers's avatar
Robbert Krebbers committed
101
Proof. destruct x as [[[]|]]; naive_solver eauto using cmra_validN_includedN. Qed.
102

103 104 105 106 107 108 109 110 111 112 113
Lemma auth_valid_discrete `{CMRADiscrete A} x :
   x  match authoritative x with
        | Excl' a => own x  a   a
        | None =>  own x
        | ExclBot' => False
        end.
Proof.
  destruct x as [[[?|]|] ?]; simpl; try done.
  setoid_rewrite <-cmra_discrete_included_iff; naive_solver eauto using 0.
Qed.

114
Lemma auth_cmra_mixin : CMRAMixin (auth A).
Robbert Krebbers's avatar
Robbert Krebbers committed
115
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
116 117
  apply cmra_total_mixin.
  - eauto.
118 119
  - by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
  - by intros n y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
Robbert Krebbers's avatar
Robbert Krebbers committed
120 121 122
  - intros n [x a] [y b] [Hx Ha]; simpl in *.
    destruct Hx as [?? Hx|]; first destruct Hx; intros ?; cofe_subst; auto.
  - intros [[[?|]|] ?]; rewrite /= ?cmra_included_includedN ?cmra_valid_validN;
123
      naive_solver eauto using O.
Robbert Krebbers's avatar
Robbert Krebbers committed
124
  - intros n [[[]|] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
125 126
  - by split; simpl; rewrite assoc.
  - by split; simpl; rewrite comm.
Ralf Jung's avatar
Ralf Jung committed
127 128
  - by split; simpl; rewrite ?cmra_core_l.
  - by split; simpl; rewrite ?cmra_core_idemp.
Robbert Krebbers's avatar
Robbert Krebbers committed
129
  - intros ??; rewrite! auth_included; intros [??].
Ralf Jung's avatar
Ralf Jung committed
130
    by split; simpl; apply cmra_core_preserving.
131
  - assert ( n (a b1 b2 : A), b1  b2 {n} a  b1 {n} a).
132
    { intros n a b1 b2 <-; apply cmra_includedN_l. }
Robbert Krebbers's avatar
Robbert Krebbers committed
133
   intros n [[[a1|]|] b1] [[[a2|]|] b2];
134
     naive_solver eauto using cmra_validN_op_l, cmra_validN_includedN.
135 136 137 138 139 140
  - intros n x y1 y2 ? [??]; simpl in *.
    destruct (cmra_extend n (authoritative x) (authoritative y1)
      (authoritative y2)) as (ea&?&?&?); auto using authoritative_validN.
    destruct (cmra_extend n (own x) (own y1) (own y2))
      as (b&?&?&?); auto using own_validN.
    by exists (Auth (ea.1) (b.1), Auth (ea.2) (b.2)).
Robbert Krebbers's avatar
Robbert Krebbers committed
141
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
142 143
Canonical Structure authR := CMRAT (auth A) auth_cofe_mixin auth_cmra_mixin.

144
Global Instance auth_cmra_discrete : CMRADiscrete A  CMRADiscrete authR.
145 146
Proof.
  split; first apply _.
Robbert Krebbers's avatar
Robbert Krebbers committed
147
  intros [[[?|]|] ?]; rewrite /= /cmra_valid /cmra_validN /=; auto.
148 149 150 151
  - setoid_rewrite <-cmra_discrete_included_iff.
    rewrite -cmra_discrete_valid_iff. tauto.
  - by rewrite -cmra_discrete_valid_iff.
Qed.
152

153 154 155 156 157 158 159
Instance auth_empty : Empty (auth A) := Auth  .
Lemma auth_ucmra_mixin : UCMRAMixin (auth A).
Proof.
  split; simpl.
  - apply (@ucmra_unit_valid A).
  - by intros x; constructor; rewrite /= left_id.
  - apply _.
Robbert Krebbers's avatar
Robbert Krebbers committed
160
  - do 2 constructor; simpl; apply (persistent_core _).
161 162 163 164
Qed.
Canonical Structure authUR :=
  UCMRAT (auth A) auth_cofe_mixin auth_cmra_mixin auth_ucmra_mixin.

165 166
(** Internalized properties *)
Lemma auth_equivI {M} (x y : auth A) :
167
  x  y  (authoritative x  authoritative y  own x  own y : uPred M).
168
Proof. by uPred.unseal. Qed.
169
Lemma auth_validI {M} (x : auth A) :
170 171 172 173 174
   x  (match authoritative x with
          | Excl' a => ( b, a  own x  b)   a
          | None =>  own x
          | ExclBot' => False
          end : uPred M).
Robbert Krebbers's avatar
Robbert Krebbers committed
175
Proof. uPred.unseal. by destruct x as [[[]|]]. Qed.
176

177
Lemma auth_frag_op a b :  (a  b)   a   b.
Robbert Krebbers's avatar
Robbert Krebbers committed
178
Proof. done. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
179
Lemma auth_both_op a b : Auth (Excl' a) b   a   b.
180
Proof. by rewrite /op /auth_op /= left_id. Qed.
181 182

Lemma auth_update a a' b b' :
Ralf Jung's avatar
Ralf Jung committed
183
  ( n af, {n} a  a {n} a'  af  b {n} b'  af  {n} b) 
184
   a   a' ~~>  b   b'.
185
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
186 187
  intros Hab; apply cmra_total_update.
  move=> n [[[?|]|] bf1] // =>-[[bf2 Ha] ?]; do 2 red; simpl in *.
188
  destruct (Hab n (bf1  bf2)) as [Ha' ?]; auto.
189 190
  { by rewrite Ha left_id assoc. }
  split; [by rewrite Ha' left_id assoc; apply cmra_includedN_l|done].
191
Qed.
192

193
Lemma auth_local_update L `{!LocalUpdate Lv L} a a' :
194
  Lv a   L a' 
195
   a'   a ~~>  L a'   L a.
196
Proof.
197
  intros. apply auth_update=>n af ? EQ; split; last by apply cmra_valid_validN.
198
  by rewrite EQ (local_updateN L) // -EQ.
199
Qed.
200 201 202 203

Lemma auth_update_op_l a a' b :
   (b  a)   a   a' ~~>  (b  a)   (b  a').
Proof. by intros; apply (auth_local_update _). Qed.
204
Lemma auth_update_op_r a a' b :
205
   (a  b)   a   a' ~~>  (a  b)   (a'  b).
206
Proof. rewrite -!(comm _ b); apply auth_update_op_l. Qed.
207

Ralf Jung's avatar
Ralf Jung committed
208
(* This does not seem to follow from auth_local_update.
209
   The trouble is that given ✓ (L a ⋅ a'), Lv a
Ralf Jung's avatar
Ralf Jung committed
210 211
   we need ✓ (a ⋅ a'). I think this should hold for every local update,
   but adding an extra axiom to local updates just for this is silly. *)
212 213 214
Lemma auth_local_update_l L `{!LocalUpdate Lv L} a a' :
  Lv a   (L a  a') 
   (a  a')   a ~~>  (L a  a')   L a.
Ralf Jung's avatar
Ralf Jung committed
215
Proof.
216
  intros. apply auth_update=>n af ? EQ; split; last by apply cmra_valid_validN.
217
  by rewrite -(local_updateN L) // EQ -(local_updateN L) // -EQ.
Ralf Jung's avatar
Ralf Jung committed
218
Qed.
219 220
End cmra.

221
Arguments authR : clear implicits.
222
Arguments authUR : clear implicits.
Robbert Krebbers's avatar
Robbert Krebbers committed
223 224

(* Functor *)
225
Definition auth_map {A B} (f : A  B) (x : auth A) : auth B :=
Robbert Krebbers's avatar
Robbert Krebbers committed
226
  Auth (excl_map f <$> authoritative x) (f (own x)).
227
Lemma auth_map_id {A} (x : auth A) : auth_map id x = x.
Robbert Krebbers's avatar
Robbert Krebbers committed
228
Proof. by destruct x as [[[]|]]. Qed.
229 230
Lemma auth_map_compose {A B C} (f : A  B) (g : B  C) (x : auth A) :
  auth_map (g  f) x = auth_map g (auth_map f x).
Robbert Krebbers's avatar
Robbert Krebbers committed
231
Proof. by destruct x as [[[]|]]. Qed.
232 233
Lemma auth_map_ext {A B : cofeT} (f g : A  B) x :
  ( x, f x  g x)  auth_map f x  auth_map g x.
Robbert Krebbers's avatar
Robbert Krebbers committed
234 235 236 237 238
Proof.
  constructor; simpl; auto.
  apply option_fmap_setoid_ext=> a; by apply excl_map_ext.
Qed.
Instance auth_map_ne {A B : cofeT} n :
239
  Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@auth_map A B).
Robbert Krebbers's avatar
Robbert Krebbers committed
240
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
241 242
  intros f g Hf [??] [??] [??]; split; simpl in *; [|by apply Hf].
  apply option_fmap_ne; [|done]=> x y ?; by apply excl_map_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
243
Qed.
244
Instance auth_map_cmra_monotone {A B : ucmraT} (f : A  B) :
245
  CMRAMonotone f  CMRAMonotone (auth_map f).
Robbert Krebbers's avatar
Robbert Krebbers committed
246
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
247
  split; try apply _.
Robbert Krebbers's avatar
Robbert Krebbers committed
248
  - intros n [[[a|]|] b]; rewrite /= /cmra_validN /=; try
Robbert Krebbers's avatar
Robbert Krebbers committed
249 250 251
      naive_solver eauto using includedN_preserving, validN_preserving.
  - by intros [x a] [y b]; rewrite !auth_included /=;
      intros [??]; split; simpl; apply: included_preserving.
Robbert Krebbers's avatar
Robbert Krebbers committed
252
Qed.
253
Definition authC_map {A B} (f : A -n> B) : authC A -n> authC B :=
254
  CofeMor (auth_map f).
255
Lemma authC_map_ne A B n : Proper (dist n ==> dist n) (@authC_map A B).
Robbert Krebbers's avatar
Robbert Krebbers committed
256
Proof. intros f f' Hf [[[a|]|] b]; repeat constructor; apply Hf. Qed.
Ralf Jung's avatar
Ralf Jung committed
257

258 259 260
Program Definition authURF (F : urFunctor) : urFunctor := {|
  urFunctor_car A B := authUR (urFunctor_car F A B);
  urFunctor_map A1 A2 B1 B2 fg := authC_map (urFunctor_map F fg)
Ralf Jung's avatar
Ralf Jung committed
261
|}.
262
Next Obligation.
263
  by intros F A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_ne.
264
Qed.
Ralf Jung's avatar
Ralf Jung committed
265
Next Obligation.
266
  intros F A B x. rewrite /= -{2}(auth_map_id x).
267
  apply auth_map_ext=>y; apply urFunctor_id.
Ralf Jung's avatar
Ralf Jung committed
268 269
Qed.
Next Obligation.
270
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -auth_map_compose.
271
  apply auth_map_ext=>y; apply urFunctor_compose.
Ralf Jung's avatar
Ralf Jung committed
272
Qed.
273

274 275
Instance authURF_contractive F :
  urFunctorContractive F  urFunctorContractive (authURF F).
276
Proof.
277
  by intros ? A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_contractive.
278
Qed.