auth.v 10.2 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
Lemma auth_cmra_mixin : CMRAMixin (auth A).
Robbert Krebbers's avatar
Robbert Krebbers committed
104
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
105 106
  apply cmra_total_mixin.
  - eauto.
107 108
  - 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
109 110 111
  - 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;
112
      naive_solver eauto using O.
Robbert Krebbers's avatar
Robbert Krebbers committed
113
  - intros n [[[]|] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
114 115
  - by split; simpl; rewrite assoc.
  - by split; simpl; rewrite comm.
Ralf Jung's avatar
Ralf Jung committed
116 117
  - by split; simpl; rewrite ?cmra_core_l.
  - by split; simpl; rewrite ?cmra_core_idemp.
Robbert Krebbers's avatar
Robbert Krebbers committed
118
  - intros ??; rewrite! auth_included; intros [??].
Ralf Jung's avatar
Ralf Jung committed
119
    by split; simpl; apply cmra_core_preserving.
120
  - assert ( n (a b1 b2 : A), b1  b2 {n} a  b1 {n} a).
121
    { intros n a b1 b2 <-; apply cmra_includedN_l. }
Robbert Krebbers's avatar
Robbert Krebbers committed
122
   intros n [[[a1|]|] b1] [[[a2|]|] b2];
123
     naive_solver eauto using cmra_validN_op_l, cmra_validN_includedN.
124 125 126 127 128 129
  - 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
130
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
131 132
Canonical Structure authR := CMRAT (auth A) auth_cofe_mixin auth_cmra_mixin.

133
Global Instance auth_cmra_discrete : CMRADiscrete A  CMRADiscrete authR.
134 135
Proof.
  split; first apply _.
Robbert Krebbers's avatar
Robbert Krebbers committed
136
  intros [[[?|]|] ?]; rewrite /= /cmra_valid /cmra_validN /=; auto.
137 138 139 140
  - setoid_rewrite <-cmra_discrete_included_iff.
    rewrite -cmra_discrete_valid_iff. tauto.
  - by rewrite -cmra_discrete_valid_iff.
Qed.
141

142 143 144 145 146 147 148
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
149
  - do 2 constructor; simpl; apply (persistent_core _).
150 151 152 153
Qed.
Canonical Structure authUR :=
  UCMRAT (auth A) auth_cofe_mixin auth_cmra_mixin auth_ucmra_mixin.

154 155
(** Internalized properties *)
Lemma auth_equivI {M} (x y : auth A) :
156
  (x  y)  (authoritative x  authoritative y  own x  own y : uPred M).
157
Proof. by uPred.unseal. Qed.
158
Lemma auth_validI {M} (x : auth A) :
159
  ( x)  (match authoritative x with
Robbert Krebbers's avatar
Robbert Krebbers committed
160 161 162
             | Excl' a => ( b, a  own x  b)   a
             | None =>  own x
             | ExclBot' => False
163
             end : uPred M).
Robbert Krebbers's avatar
Robbert Krebbers committed
164
Proof. uPred.unseal. by destruct x as [[[]|]]. Qed.
165

166
Lemma auth_frag_op a b :  (a  b)   a   b.
Robbert Krebbers's avatar
Robbert Krebbers committed
167
Proof. done. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
168
Lemma auth_both_op a b : Auth (Excl' a) b   a   b.
169
Proof. by rewrite /op /auth_op /= left_id. Qed.
170 171

Lemma auth_update a a' b b' :
Ralf Jung's avatar
Ralf Jung committed
172
  ( n af, {n} a  a {n} a'  af  b {n} b'  af  {n} b) 
173
   a   a' ~~>  b   b'.
174
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
175 176
  intros Hab; apply cmra_total_update.
  move=> n [[[?|]|] bf1] // =>-[[bf2 Ha] ?]; do 2 red; simpl in *.
177
  destruct (Hab n (bf1  bf2)) as [Ha' ?]; auto.
178 179
  { by rewrite Ha left_id assoc. }
  split; [by rewrite Ha' left_id assoc; apply cmra_includedN_l|done].
180
Qed.
181

182
Lemma auth_local_update L `{!LocalUpdate Lv L} a a' :
183
  Lv a   L a' 
184
   a'   a ~~>  L a'   L a.
185
Proof.
186
  intros. apply auth_update=>n af ? EQ; split; last by apply cmra_valid_validN.
187
  by rewrite EQ (local_updateN L) // -EQ.
188
Qed.
189 190 191 192

Lemma auth_update_op_l a a' b :
   (b  a)   a   a' ~~>  (b  a)   (b  a').
Proof. by intros; apply (auth_local_update _). Qed.
193
Lemma auth_update_op_r a a' b :
194
   (a  b)   a   a' ~~>  (a  b)   (a'  b).
195
Proof. rewrite -!(comm _ b); apply auth_update_op_l. Qed.
196

Ralf Jung's avatar
Ralf Jung committed
197
(* This does not seem to follow from auth_local_update.
198
   The trouble is that given ✓ (L a ⋅ a'), Lv a
Ralf Jung's avatar
Ralf Jung committed
199 200
   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. *)
201 202 203
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
204
Proof.
205
  intros. apply auth_update=>n af ? EQ; split; last by apply cmra_valid_validN.
206
  by rewrite -(local_updateN L) // EQ -(local_updateN L) // -EQ.
Ralf Jung's avatar
Ralf Jung committed
207
Qed.
208 209
End cmra.

210
Arguments authR : clear implicits.
211
Arguments authUR : clear implicits.
Robbert Krebbers's avatar
Robbert Krebbers committed
212 213

(* Functor *)
214
Definition auth_map {A B} (f : A  B) (x : auth A) : auth B :=
Robbert Krebbers's avatar
Robbert Krebbers committed
215
  Auth (excl_map f <$> authoritative x) (f (own x)).
216
Lemma auth_map_id {A} (x : auth A) : auth_map id x = x.
Robbert Krebbers's avatar
Robbert Krebbers committed
217
Proof. by destruct x as [[[]|]]. Qed.
218 219
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
220
Proof. by destruct x as [[[]|]]. Qed.
221 222
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
223 224 225 226 227
Proof.
  constructor; simpl; auto.
  apply option_fmap_setoid_ext=> a; by apply excl_map_ext.
Qed.
Instance auth_map_ne {A B : cofeT} n :
228
  Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@auth_map A B).
Robbert Krebbers's avatar
Robbert Krebbers committed
229
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
230 231
  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
232
Qed.
233
Instance auth_map_cmra_monotone {A B : ucmraT} (f : A  B) :
234
  CMRAMonotone f  CMRAMonotone (auth_map f).
Robbert Krebbers's avatar
Robbert Krebbers committed
235
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
236
  split; try apply _.
Robbert Krebbers's avatar
Robbert Krebbers committed
237
  - intros n [[[a|]|] b]; rewrite /= /cmra_validN /=; try
Robbert Krebbers's avatar
Robbert Krebbers committed
238 239 240
      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
241
Qed.
242
Definition authC_map {A B} (f : A -n> B) : authC A -n> authC B :=
243
  CofeMor (auth_map f).
244
Lemma authC_map_ne A B n : Proper (dist n ==> dist n) (@authC_map A B).
Robbert Krebbers's avatar
Robbert Krebbers committed
245
Proof. intros f f' Hf [[[a|]|] b]; repeat constructor; apply Hf. Qed.
Ralf Jung's avatar
Ralf Jung committed
246

247 248 249
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
250
|}.
251
Next Obligation.
252
  by intros F A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_ne.
253
Qed.
Ralf Jung's avatar
Ralf Jung committed
254
Next Obligation.
255
  intros F A B x. rewrite /= -{2}(auth_map_id x).
256
  apply auth_map_ext=>y; apply urFunctor_id.
Ralf Jung's avatar
Ralf Jung committed
257 258
Qed.
Next Obligation.
259
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -auth_map_compose.
260
  apply auth_map_ext=>y; apply urFunctor_compose.
Ralf Jung's avatar
Ralf Jung committed
261
Qed.
262

263 264
Instance authURF_contractive F :
  urFunctorContractive F  urFunctorContractive (authURF F).
265
Proof.
266
  by intros ? A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_contractive.
267
Qed.