ofe.v 37.1 KB
Newer Older
1
From iris.algebra Require Export base.
Robbert Krebbers's avatar
Robbert Krebbers committed
2

3
(** This files defines (a shallow embedding of) the category of OFEs:
4
5
6
7
8
9
10
11
    Complete ordered families of equivalences. This is a cartesian closed
    category, and mathematically speaking, the entire development lives
    in this category. However, we will generally prefer to work with raw
    Coq functions plus some registered Proper instances for non-expansiveness.
    This makes writing such functions much easier. It turns out that it many 
    cases, we do not even need non-expansiveness.
*)

Robbert Krebbers's avatar
Robbert Krebbers committed
12
13
(** Unbundeled version *)
Class Dist A := dist : nat  relation A.
14
Instance: Params (@dist) 3.
15
16
Notation "x ≡{ n }≡ y" := (dist n x y)
  (at level 70, n at next level, format "x  ≡{ n }≡  y").
17
Hint Extern 0 (_ {_} _) => reflexivity.
18
Hint Extern 0 (_ {_} _) => symmetry; assumption.
19
20
21

Tactic Notation "cofe_subst" ident(x) :=
  repeat match goal with
22
  | _ => progress simplify_eq/=
23
24
25
26
  | H:@dist ?A ?d ?n x _ |- _ => setoid_subst_aux (@dist A d n) x
  | H:@dist ?A ?d ?n _ x |- _ => symmetry in H;setoid_subst_aux (@dist A d n) x
  end.
Tactic Notation "cofe_subst" :=
27
  repeat match goal with
28
  | _ => progress simplify_eq/=
29
30
  | H:@dist ?A ?d ?n ?x _ |- _ => setoid_subst_aux (@dist A d n) x
  | H:@dist ?A ?d ?n _ ?x |- _ => symmetry in H;setoid_subst_aux (@dist A d n) x
31
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
32

33
Record OfeMixin A `{Equiv A, Dist A} := {
34
  mixin_equiv_dist x y : x  y   n, x {n} y;
35
  mixin_dist_equivalence n : Equivalence (dist n);
36
  mixin_dist_S n x y : x {S n} y  x {n} y
Robbert Krebbers's avatar
Robbert Krebbers committed
37
38
39
}.

(** Bundeled version *)
40
41
42
43
44
Structure ofeT := OfeT' {
  ofe_car :> Type;
  ofe_equiv : Equiv ofe_car;
  ofe_dist : Dist ofe_car;
  ofe_mixin : OfeMixin ofe_car;
45
  _ : Type
Robbert Krebbers's avatar
Robbert Krebbers committed
46
}.
47
48
49
50
51
52
53
54
55
Arguments OfeT' _ {_ _} _ _.
Notation OfeT A m := (OfeT' A m A).
Add Printing Constructor ofeT.
Hint Extern 0 (Equiv _) => eapply (@ofe_equiv _) : typeclass_instances.
Hint Extern 0 (Dist _) => eapply (@ofe_dist _) : typeclass_instances.
Arguments ofe_car : simpl never.
Arguments ofe_equiv : simpl never.
Arguments ofe_dist : simpl never.
Arguments ofe_mixin : simpl never.
56
57

(** Lifting properties from the mixin *)
58
59
Section ofe_mixin.
  Context {A : ofeT}.
60
  Implicit Types x y : A.
61
  Lemma equiv_dist x y : x  y   n, x {n} y.
62
  Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed.
63
  Global Instance dist_equivalence n : Equivalence (@dist A _ n).
64
  Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed.
65
  Lemma dist_S n x y : x {S n} y  x {n} y.
66
67
  Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed.
End ofe_mixin.
68

Robbert Krebbers's avatar
Robbert Krebbers committed
69
70
Hint Extern 1 (_ {_} _) => apply equiv_dist; assumption.

71
(** Discrete OFEs and Timeless elements *)
Ralf Jung's avatar
Ralf Jung committed
72
(* TODO: On paper, We called these "discrete elements". I think that makes
Ralf Jung's avatar
Ralf Jung committed
73
   more sense. *)
74
75
Class Timeless `{Equiv A, Dist A} (x : A) := timeless y : x {0} y  x  y.
Arguments timeless {_ _ _} _ {_} _ _.
76
77
78
79
80
81
82
83
84
85
Class Discrete (A : ofeT) := discrete_timeless (x : A) :> Timeless x.

(** OFEs with a completion *)
Record chain (A : ofeT) := {
  chain_car :> nat  A;
  chain_cauchy n i : n  i  chain_car i {n} chain_car n
}.
Arguments chain_car {_} _ _.
Arguments chain_cauchy {_} _ _ _ _.

86
87
88
89
90
Program Definition chain_map {A B : ofeT} (f : A  B)
    `{! n, Proper (dist n ==> dist n) f} (c : chain A) : chain B :=
  {| chain_car n := f (c n) |}.
Next Obligation. by intros A B f Hf c n i ?; apply Hf, chain_cauchy. Qed.

91
92
93
94
95
96
Notation Compl A := (chain A%type  A).
Class Cofe (A : ofeT) := {
  compl : Compl A;
  conv_compl n c : compl c {n} c n;
}.
Arguments compl : simpl never.
97

Robbert Krebbers's avatar
Robbert Krebbers committed
98
99
(** General properties *)
Section cofe.
100
  Context {A : ofeT}.
101
  Implicit Types x y : A.
Robbert Krebbers's avatar
Robbert Krebbers committed
102
103
104
  Global Instance cofe_equivalence : Equivalence (() : relation A).
  Proof.
    split.
105
106
    - by intros x; rewrite equiv_dist.
    - by intros x y; rewrite !equiv_dist.
107
    - by intros x y z; rewrite !equiv_dist; intros; trans y.
Robbert Krebbers's avatar
Robbert Krebbers committed
108
  Qed.
109
  Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
110
111
  Proof.
    intros x1 x2 ? y1 y2 ?; split; intros.
112
113
    - by trans x1; [|trans y1].
    - by trans x2; [|trans y2].
Robbert Krebbers's avatar
Robbert Krebbers committed
114
  Qed.
115
  Global Instance dist_proper n : Proper (() ==> () ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
116
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
117
    by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
118
119
120
  Qed.
  Global Instance dist_proper_2 n x : Proper (() ==> iff) (dist n x).
  Proof. by apply dist_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
121
  Lemma dist_le n n' x y : x {n} y  n'  n  x {n'} y.
Robbert Krebbers's avatar
Robbert Krebbers committed
122
  Proof. induction 2; eauto using dist_S. Qed.
123
124
  Lemma dist_le' n n' x y : n'  n  x {n} y  x {n'} y.
  Proof. intros; eauto using dist_le. Qed.
125
  Instance ne_proper {B : ofeT} (f : A  B)
Robbert Krebbers's avatar
Robbert Krebbers committed
126
127
    `{! n, Proper (dist n ==> dist n) f} : Proper (() ==> ()) f | 100.
  Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed.
128
  Instance ne_proper_2 {B C : ofeT} (f : A  B  C)
Robbert Krebbers's avatar
Robbert Krebbers committed
129
130
131
132
    `{! n, Proper (dist n ==> dist n ==> dist n) f} :
    Proper (() ==> () ==> ()) f | 100.
  Proof.
     unfold Proper, respectful; setoid_rewrite equiv_dist.
Robbert Krebbers's avatar
Robbert Krebbers committed
133
     by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
134
  Qed.
135

136
  Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c {n} c (S n).
137
138
139
140
  Proof.
    transitivity (c n); first by apply conv_compl. symmetry.
    apply chain_cauchy. omega.
  Qed.
141
142
  Lemma timeless_iff n (x : A) `{!Timeless x} y : x  y  x {n} y.
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
143
    split; intros; auto. apply (timeless _), dist_le with n; auto with lia.
144
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
145
146
End cofe.

147
(** Contractive functions *)
148
149
150
151
152
153
154
155
Definition dist_later {A : ofeT} (n : nat) (x y : A) : Prop :=
  match n with 0 => True | S n => x {n} y end.
Arguments dist_later _ !_ _ _ /.

Global Instance dist_later_equivalence A n : Equivalence (@dist_later A n).
Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed.

Notation Contractive f := ( n, Proper (dist_later n ==> dist n) f).
156

157
Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x).
158
159
Proof. by intros n y1 y2. Qed.

160
161
162
163
164
Section contractive.
  Context {A B : ofeT} (f : A  B) `{!Contractive f}.
  Implicit Types x y : A.

  Lemma contractive_0 x y : f x {0} f y.
165
  Proof. by apply (_ : Contractive f). Qed.
166
  Lemma contractive_S n x y : x {n} y  f x {S n} f y.
167
  Proof. intros. by apply (_ : Contractive f). Qed.
168
169
170
171
172
173
174

  Global Instance contractive_ne n : Proper (dist n ==> dist n) f | 100.
  Proof. by intros x y ?; apply dist_S, contractive_S. Qed.
  Global Instance contractive_proper : Proper (() ==> ()) f | 100.
  Proof. apply (ne_proper _). Qed.
End contractive.

175
176
177
178
179
180
181
Ltac f_contractive :=
  match goal with
  | |- ?f _ {_} ?f _ => apply (_ : Proper (dist_later _ ==> _) f)
  | |- ?f _ _ {_} ?f _ _ => apply (_ : Proper (dist_later _ ==> _ ==> _) f)
  | |- ?f _ _ {_} ?f _ _ => apply (_ : Proper (_ ==> dist_later _ ==> _) f)
  end;
  try match goal with
182
183
  | |- @dist_later ?A ?n ?x ?y =>
         destruct n as [|n]; [done|change (@dist A _ n x y)]
184
185
186
187
188
189
  end;
  try reflexivity.

Ltac solve_contractive :=
  preprocess_solve_proper;
  solve [repeat (first [f_contractive|f_equiv]; try eassumption)].
Robbert Krebbers's avatar
Robbert Krebbers committed
190

Robbert Krebbers's avatar
Robbert Krebbers committed
191
(** Fixpoint *)
192
Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A  A)
193
  `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
194
Next Obligation.
195
  intros A ? f ? n.
196
  induction n as [|n IH]=> -[|i] //= ?; try omega.
197
198
  - apply (contractive_0 f).
  - apply (contractive_S f), IH; auto with omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
199
Qed.
200

201
Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A  A)
202
  `{!Contractive f} : A := compl (fixpoint_chain f).
203
Definition fixpoint_aux : { x | x = @fixpoint_def }. by eexists. Qed.
204
Definition fixpoint {A AC AiH} f {Hf} := proj1_sig fixpoint_aux A AC AiH f Hf.
205
Definition fixpoint_eq : @fixpoint = @fixpoint_def := proj2_sig fixpoint_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
206
207

Section fixpoint.
208
  Context `{Cofe A, Inhabited A} (f : A  A) `{!Contractive f}.
209

210
  Lemma fixpoint_unfold : fixpoint f  f (fixpoint f).
Robbert Krebbers's avatar
Robbert Krebbers committed
211
  Proof.
212
213
    apply equiv_dist=>n.
    rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //.
214
    induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
215
  Qed.
216
217
218

  Lemma fixpoint_unique (x : A) : x  f x  x  fixpoint f.
  Proof.
219
220
221
    rewrite !equiv_dist=> Hx n. induction n as [|n IH]; simpl in *.
    - rewrite Hx fixpoint_unfold; eauto using contractive_0.
    - rewrite Hx fixpoint_unfold. apply (contractive_S _), IH.
222
223
  Qed.

224
  Lemma fixpoint_ne (g : A  A) `{!Contractive g} n :
225
    ( z, f z {n} g z)  fixpoint f {n} fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
226
  Proof.
227
    intros Hfg. rewrite fixpoint_eq /fixpoint_def
Robbert Krebbers's avatar
Robbert Krebbers committed
228
      (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=.
229
230
    induction n as [|n IH]; simpl in *; [by rewrite !Hfg|].
    rewrite Hfg; apply contractive_S, IH; auto using dist_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
231
  Qed.
232
233
  Lemma fixpoint_proper (g : A  A) `{!Contractive g} :
    ( x, f x  g x)  fixpoint f  fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
234
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.
235
236

  Lemma fixpoint_ind (P : A  Prop) :
237
    Proper (() ==> impl) P 
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
    ( x, P x)  ( x, P x  P (f x)) 
    ( (c : chain A), ( n, P (c n))  P (compl c)) 
    P (fixpoint f).
  Proof.
    intros ? [x Hx] Hincr Hlim. set (chcar i := Nat.iter (S i) f x).
    assert (Hcauch :  n i : nat, n  i  chcar i {n} chcar n).
    { intros n. induction n as [|n IH]=> -[|i] //= ?; try omega.
      - apply (contractive_0 f).
      - apply (contractive_S f), IH; auto with omega. }
    set (fp2 := compl {| chain_cauchy := Hcauch |}).
    rewrite -(fixpoint_unique fp2); first by apply Hlim; induction n; apply Hincr.
    apply equiv_dist=>n.
    rewrite /fp2 (conv_compl n) /= /chcar.
    induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
253
254
End fixpoint.

Robbert Krebbers's avatar
Robbert Krebbers committed
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
(** Mutual fixpoints *)
Section fixpoint2.
  Context `{Cofe A, Cofe B, !Inhabited A, !Inhabited B}.
  Context (fA : A  B  A).
  Context (fB : A  B  B).
  Context `{ n, Proper (dist_later n ==> dist n ==> dist n) fA}.
  Context `{ n, Proper (dist_later n ==> dist_later n ==> dist n) fB}.

  Local Definition fixpoint_AB (x : A) : B := fixpoint (fB x).
  Local Instance fixpoint_AB_contractive : Contractive fixpoint_AB.
  Proof.
    intros n x x' Hx; rewrite /fixpoint_AB.
    apply fixpoint_ne=> y. by f_contractive.
  Qed.

  Local Definition fixpoint_AA (x : A) : A := fA x (fixpoint_AB x).
  Local Instance fixpoint_AA_contractive : Contractive fixpoint_AA.
  Proof. solve_contractive. Qed.

  Definition fixpoint_A : A := fixpoint fixpoint_AA.
  Definition fixpoint_B : B := fixpoint_AB fixpoint_A.

  Lemma fixpoint_A_unfold : fA fixpoint_A fixpoint_B  fixpoint_A.
  Proof. by rewrite {2}/fixpoint_A (fixpoint_unfold _). Qed.
  Lemma fixpoint_B_unfold : fB fixpoint_A fixpoint_B  fixpoint_B.
  Proof. by rewrite {2}/fixpoint_B /fixpoint_AB (fixpoint_unfold _). Qed.

  Instance: Proper (() ==> () ==> ()) fA.
  Proof.
    apply ne_proper_2=> n x x' ? y y' ?. f_contractive; auto using dist_S.
  Qed.
  Instance: Proper (() ==> () ==> ()) fB.
  Proof.
    apply ne_proper_2=> n x x' ? y y' ?. f_contractive; auto using dist_S.
  Qed.

  Lemma fixpoint_A_unique p q : fA p q  p  fB p q  q  p  fixpoint_A.
  Proof.
    intros HfA HfB. rewrite -HfA. apply fixpoint_unique. rewrite /fixpoint_AA.
    f_equiv=> //. apply fixpoint_unique. by rewrite HfA HfB.
  Qed.
  Lemma fixpoint_B_unique p q : fA p q  p  fB p q  q  q  fixpoint_B.
  Proof. intros. apply fixpoint_unique. by rewrite -fixpoint_A_unique. Qed.
End fixpoint2.

Section fixpoint2_ne.
  Context `{Cofe A, Cofe B, !Inhabited A, !Inhabited B}.
  Context (fA fA' : A  B  A).
  Context (fB fB' : A  B  B).
  Context `{ n, Proper (dist_later n ==> dist n ==> dist n) fA}.
  Context `{ n, Proper (dist_later n ==> dist n ==> dist n) fA'}.
  Context `{ n, Proper (dist_later n ==> dist_later n ==> dist n) fB}.
  Context `{ n, Proper (dist_later n ==> dist_later n ==> dist n) fB'}.

  Lemma fixpoint_A_ne n :
    ( x y, fA x y {n} fA' x y)  ( x y, fB x y {n} fB' x y) 
    fixpoint_A fA fB {n} fixpoint_A fA' fB'.
  Proof.
    intros HfA HfB. apply fixpoint_ne=> z.
    rewrite /fixpoint_AA /fixpoint_AB HfA. f_equiv. by apply fixpoint_ne.
  Qed.
  Lemma fixpoint_B_ne n :
    ( x y, fA x y {n} fA' x y)  ( x y, fB x y {n} fB' x y) 
    fixpoint_B fA fB {n} fixpoint_B fA' fB'.
  Proof.
    intros HfA HfB. apply fixpoint_ne=> z. rewrite HfB. f_contractive.
    apply fixpoint_A_ne; auto using dist_S.
  Qed.

  Lemma fixpoint_A_proper :
    ( x y, fA x y  fA' x y)  ( x y, fB x y  fB' x y) 
    fixpoint_A fA fB  fixpoint_A fA' fB'.
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_A_ne. Qed.
  Lemma fixpoint_B_proper :
    ( x y, fA x y  fA' x y)  ( x y, fB x y  fB' x y) 
    fixpoint_B fA fB  fixpoint_B fA' fB'.
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_B_ne. Qed.
End fixpoint2_ne.

334
(** Function space *)
335
(* We make [ofe_fun] a definition so that we can register it as a canonical
336
structure. *)
337
Definition ofe_fun (A : Type) (B : ofeT) := A  B.
338

339
340
341
342
343
Section ofe_fun.
  Context {A : Type} {B : ofeT}.
  Instance ofe_fun_equiv : Equiv (ofe_fun A B) := λ f g,  x, f x  g x.
  Instance ofe_fun_dist : Dist (ofe_fun A B) := λ n f g,  x, f x {n} g x.
  Definition ofe_fun_ofe_mixin : OfeMixin (ofe_fun A B).
344
345
346
347
348
349
350
351
352
353
  Proof.
    split.
    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
      intros Hfg k; apply equiv_dist=> n; apply Hfg.
    - intros n; split.
      + by intros f x.
      + by intros f g ? x.
      + by intros f g h ?? x; trans (g x).
    - by intros n f g ? x; apply dist_S.
  Qed.
354
  Canonical Structure ofe_funC := OfeT (ofe_fun A B) ofe_fun_ofe_mixin.
355

356
357
358
359
360
361
362
363
364
  Program Definition ofe_fun_chain `(c : chain ofe_funC)
    (x : A) : chain B := {| chain_car n := c n x |}.
  Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed.
  Global Program Instance ofe_fun_cofe `{Cofe B} : Cofe ofe_funC :=
    { compl c x := compl (ofe_fun_chain c x) }.
  Next Obligation. intros ? n c x. apply (conv_compl n (ofe_fun_chain c x)). Qed.
End ofe_fun.

Arguments ofe_funC : clear implicits.
365
Notation "A -c> B" :=
366
367
  (ofe_funC A B) (at level 99, B at level 200, right associativity).
Instance ofe_fun_inhabited {A} {B : ofeT} `{Inhabited B} :
368
369
  Inhabited (A -c> B) := populate (λ _, inhabitant).

370
(** Non-expansive function space *)
371
372
373
Record ofe_mor (A B : ofeT) : Type := CofeMor {
  ofe_mor_car :> A  B;
  ofe_mor_ne n : Proper (dist n ==> dist n) ofe_mor_car
Robbert Krebbers's avatar
Robbert Krebbers committed
374
375
}.
Arguments CofeMor {_ _} _ {_}.
376
377
Add Printing Constructor ofe_mor.
Existing Instance ofe_mor_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
378

379
380
381
382
Notation "'λne' x .. y , t" :=
  (@CofeMor _ _ (λ x, .. (@CofeMor _ _ (λ y, t) _) ..) _)
  (at level 200, x binder, y binder, right associativity).

383
384
385
386
387
388
389
Section ofe_mor.
  Context {A B : ofeT}.
  Global Instance ofe_mor_proper (f : ofe_mor A B) : Proper (() ==> ()) f.
  Proof. apply ne_proper, ofe_mor_ne. Qed.
  Instance ofe_mor_equiv : Equiv (ofe_mor A B) := λ f g,  x, f x  g x.
  Instance ofe_mor_dist : Dist (ofe_mor A B) := λ n f g,  x, f x {n} g x.
  Definition ofe_mor_ofe_mixin : OfeMixin (ofe_mor A B).
390
391
  Proof.
    split.
392
    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
Robbert Krebbers's avatar
Robbert Krebbers committed
393
      intros Hfg k; apply equiv_dist=> n; apply Hfg.
394
    - intros n; split.
395
396
      + by intros f x.
      + by intros f g ? x.
397
      + by intros f g h ?? x; trans (g x).
398
    - by intros n f g ? x; apply dist_S.
399
  Qed.
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
  Canonical Structure ofe_morC := OfeT (ofe_mor A B) ofe_mor_ofe_mixin.

  Program Definition ofe_mor_chain (c : chain ofe_morC)
    (x : A) : chain B := {| chain_car n := c n x |}.
  Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed.
  Program Definition ofe_mor_compl `{Cofe B} : Compl ofe_morC := λ c,
    {| ofe_mor_car x := compl (ofe_mor_chain c x) |}.
  Next Obligation.
    intros ? c n x y Hx. by rewrite (conv_compl n (ofe_mor_chain c x))
      (conv_compl n (ofe_mor_chain c y)) /= Hx.
  Qed.
  Global Program Instance ofe_more_cofe `{Cofe B} : Cofe ofe_morC :=
    {| compl := ofe_mor_compl |}.
  Next Obligation.
    intros ? n c x; simpl.
    by rewrite (conv_compl n (ofe_mor_chain c x)) /=.
  Qed.
417

418
419
  Global Instance ofe_mor_car_ne n :
    Proper (dist n ==> dist n ==> dist n) (@ofe_mor_car A B).
420
  Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed.
421
422
423
  Global Instance ofe_mor_car_proper :
    Proper (() ==> () ==> ()) (@ofe_mor_car A B) := ne_proper_2 _.
  Lemma ofe_mor_ext (f g : ofe_mor A B) : f  g   x, f x  g x.
424
  Proof. done. Qed.
425
End ofe_mor.
426

427
Arguments ofe_morC : clear implicits.
428
Notation "A -n> B" :=
429
430
  (ofe_morC A B) (at level 99, B at level 200, right associativity).
Instance ofe_mor_inhabited {A B : ofeT} `{Inhabited B} :
431
  Inhabited (A -n> B) := populate (λne _, inhabitant).
Robbert Krebbers's avatar
Robbert Krebbers committed
432

433
(** Identity and composition and constant function *)
Robbert Krebbers's avatar
Robbert Krebbers committed
434
435
Definition cid {A} : A -n> A := CofeMor id.
Instance: Params (@cid) 1.
436
Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x).
437
Instance: Params (@cconst) 2.
438

Robbert Krebbers's avatar
Robbert Krebbers committed
439
440
441
442
443
Definition ccompose {A B C}
  (f : B -n> C) (g : A -n> B) : A -n> C := CofeMor (f  g).
Instance: Params (@ccompose) 3.
Infix "◎" := ccompose (at level 40, left associativity).
Lemma ccompose_ne {A B C} (f1 f2 : B -n> C) (g1 g2 : A -n> B) n :
444
  f1 {n} f2  g1 {n} g2  f1  g1 {n} f2  g2.
Robbert Krebbers's avatar
Robbert Krebbers committed
445
Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
446

Ralf Jung's avatar
Ralf Jung committed
447
(* Function space maps *)
448
Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B')
Ralf Jung's avatar
Ralf Jung committed
449
  (h : A -n> B) : A' -n> B' := g  h  f.
450
451
Instance ofe_mor_map_ne {A A' B B'} n :
  Proper (dist n ==> dist n ==> dist n ==> dist n) (@ofe_mor_map A A' B B').
452
Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed.
Ralf Jung's avatar
Ralf Jung committed
453

454
455
456
457
Definition ofe_morC_map {A A' B B'} (f : A' -n> A) (g : B -n> B') :
  (A -n> B) -n> (A' -n>  B') := CofeMor (ofe_mor_map f g).
Instance ofe_morC_map_ne {A A' B B'} n :
  Proper (dist n ==> dist n ==> dist n) (@ofe_morC_map A A' B B').
Ralf Jung's avatar
Ralf Jung committed
458
Proof.
459
  intros f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map.
460
  by repeat apply ccompose_ne.
Ralf Jung's avatar
Ralf Jung committed
461
462
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
463
(** unit *)
464
465
Section unit.
  Instance unit_dist : Dist unit := λ _ _ _, True.
466
  Definition unit_ofe_mixin : OfeMixin unit.
467
  Proof. by repeat split; try exists 0. Qed.
468
  Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin.
Robbert Krebbers's avatar
Robbert Krebbers committed
469

470
471
  Global Program Instance unit_cofe : Cofe unitC := { compl x := () }.
  Next Obligation. by repeat split; try exists 0. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
472
473

  Global Instance unit_discrete_cofe : Discrete unitC.
Robbert Krebbers's avatar
Robbert Krebbers committed
474
  Proof. done. Qed.
475
End unit.
Robbert Krebbers's avatar
Robbert Krebbers committed
476
477

(** Product *)
478
Section product.
479
  Context {A B : ofeT}.
480
481
482
483
484
485

  Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n).
  Global Instance pair_ne :
    Proper (dist n ==> dist n ==> dist n) (@pair A B) := _.
  Global Instance fst_ne : Proper (dist n ==> dist n) (@fst A B) := _.
  Global Instance snd_ne : Proper (dist n ==> dist n) (@snd A B) := _.
486
  Definition prod_ofe_mixin : OfeMixin (A * B).
487
488
  Proof.
    split.
489
    - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
490
      rewrite !equiv_dist; naive_solver.
491
492
    - apply _.
    - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
493
  Qed.
494
495
496
497
498
499
500
501
502
  Canonical Structure prodC : ofeT := OfeT (A * B) prod_ofe_mixin.

  Global Program Instance prod_cofe `{Cofe A, Cofe B} : Cofe prodC :=
    { compl c := (compl (chain_map fst c), compl (chain_map snd c)) }.
  Next Obligation.
    intros ?? n c; split. apply (conv_compl n (chain_map fst c)).
    apply (conv_compl n (chain_map snd c)).
  Qed.

503
504
505
  Global Instance prod_timeless (x : A * B) :
    Timeless (x.1)  Timeless (x.2)  Timeless x.
  Proof. by intros ???[??]; split; apply (timeless _). Qed.
506
507
  Global Instance prod_discrete_cofe : Discrete A  Discrete B  Discrete prodC.
  Proof. intros ?? [??]; apply _. Qed.
508
509
510
511
512
End product.

Arguments prodC : clear implicits.
Typeclasses Opaque prod_dist.

513
Instance prod_map_ne {A A' B B' : ofeT} n :
Robbert Krebbers's avatar
Robbert Krebbers committed
514
515
516
517
518
519
520
521
522
  Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==>
           dist n ==> dist n) (@prod_map A A' B B').
Proof. by intros f f' Hf g g' Hg ?? [??]; split; [apply Hf|apply Hg]. Qed.
Definition prodC_map {A A' B B'} (f : A -n> A') (g : B -n> B') :
  prodC A B -n> prodC A' B' := CofeMor (prod_map f g).
Instance prodC_map_ne {A A' B B'} n :
  Proper (dist n ==> dist n ==> dist n) (@prodC_map A A' B B').
Proof. intros f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed.

523
524
(** Functors *)
Structure cFunctor := CFunctor {
525
  cFunctor_car : ofeT  ofeT  ofeT;
526
527
  cFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  cFunctor_car A1 B1 -n> cFunctor_car A2 B2;
528
529
  cFunctor_ne {A1 A2 B1 B2} n :
    Proper (dist n ==> dist n) (@cFunctor_map A1 A2 B1 B2);
530
  cFunctor_id {A B : ofeT} (x : cFunctor_car A B) :
531
532
533
534
535
    cFunctor_map (cid,cid) x  x;
  cFunctor_compose {A1 A2 A3 B1 B2 B3}
      (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x :
    cFunctor_map (fg, g'f') x  cFunctor_map (g,g') (cFunctor_map (f,f') x)
}.
536
Existing Instance cFunctor_ne.
537
538
Instance: Params (@cFunctor_map) 5.

539
540
541
Delimit Scope cFunctor_scope with CF.
Bind Scope cFunctor_scope with cFunctor.

542
543
544
Class cFunctorContractive (F : cFunctor) :=
  cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2).

545
Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A.
546
547
Coercion cFunctor_diag : cFunctor >-> Funclass.

548
Program Definition constCF (B : ofeT) : cFunctor :=
549
550
551
  {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}.
Solve Obligations with done.

552
Instance constCF_contractive B : cFunctorContractive (constCF B).
553
Proof. rewrite /cFunctorContractive; apply _. Qed.
554
555
556
557
558

Program Definition idCF : cFunctor :=
  {| cFunctor_car A1 A2 := A2; cFunctor_map A1 A2 B1 B2 f := f.2 |}.
Solve Obligations with done.

559
560
561
562
563
Program Definition prodCF (F1 F2 : cFunctor) : cFunctor := {|
  cFunctor_car A B := prodC (cFunctor_car F1 A B) (cFunctor_car F2 A B);
  cFunctor_map A1 A2 B1 B2 fg :=
    prodC_map (cFunctor_map F1 fg) (cFunctor_map F2 fg)
|}.
564
565
566
Next Obligation.
  intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne.
Qed.
567
568
569
570
571
572
Next Obligation. by intros F1 F2 A B [??]; rewrite /= !cFunctor_id. Qed.
Next Obligation.
  intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [??]; simpl.
  by rewrite !cFunctor_compose.
Qed.

573
574
575
576
577
578
579
580
Instance prodCF_contractive F1 F2 :
  cFunctorContractive F1  cFunctorContractive F2 
  cFunctorContractive (prodCF F1 F2).
Proof.
  intros ?? A1 A2 B1 B2 n ???;
    by apply prodC_map_ne; apply cFunctor_contractive.
Qed.

581
Instance compose_ne {A} {B B' : ofeT} (f : B -n> B') n :
582
583
584
  Proper (dist n ==> dist n) (compose f : (A -c> B)  A -c> B').
Proof. intros g g' Hf x; simpl. by rewrite (Hf x). Qed.

585
Definition ofe_funC_map {A B B'} (f : B -n> B') : (A -c> B) -n> (A -c> B') :=
586
  @CofeMor (_ -c> _) (_ -c> _) (compose f) _.
587
588
Instance ofe_funC_map_ne {A B B'} n :
  Proper (dist n ==> dist n) (@ofe_funC_map A B B').
589
590
Proof. intros f f' Hf g x. apply Hf. Qed.

591
592
593
Program Definition ofe_funCF (T : Type) (F : cFunctor) : cFunctor := {|
  cFunctor_car A B := ofe_funC T (cFunctor_car F A B);
  cFunctor_map A1 A2 B1 B2 fg := ofe_funC_map (cFunctor_map F fg)
594
595
|}.
Next Obligation.
596
  intros ?? A1 A2 B1 B2 n ???; by apply ofe_funC_map_ne; apply cFunctor_ne.
597
598
599
600
601
602
603
Qed.
Next Obligation. intros F1 F2 A B ??. by rewrite /= /compose /= !cFunctor_id. Qed.
Next Obligation.
  intros T F A1 A2 A3 B1 B2 B3 f g f' g' ??; simpl.
  by rewrite !cFunctor_compose.
Qed.

604
605
Instance ofe_funCF_contractive (T : Type) (F : cFunctor) :
  cFunctorContractive F  cFunctorContractive (ofe_funCF T F).
606
607
Proof.
  intros ?? A1 A2 B1 B2 n ???;
608
    by apply ofe_funC_map_ne; apply cFunctor_contractive.
609
610
Qed.

611
Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {|
612
  cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B;
Ralf Jung's avatar
Ralf Jung committed
613
  cFunctor_map A1 A2 B1 B2 fg :=
614
    ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg)
Ralf Jung's avatar
Ralf Jung committed
615
|}.
616
617
Next Obligation.
  intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *.
618
  apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg.
619
Qed.
Ralf Jung's avatar
Ralf Jung committed
620
Next Obligation.
621
622
  intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id.
  apply (ne_proper f). apply cFunctor_id.
Ralf Jung's avatar
Ralf Jung committed
623
624
Qed.
Next Obligation.
625
626
  intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [h ?] ?; simpl in *.
  rewrite -!cFunctor_compose. do 2 apply (ne_proper _). apply cFunctor_compose.
Ralf Jung's avatar
Ralf Jung committed
627
628
Qed.

629
Instance ofe_morCF_contractive F1 F2 :
630
  cFunctorContractive F1  cFunctorContractive F2 
631
  cFunctorContractive (ofe_morCF F1 F2).
632
633
Proof.
  intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *.
634
  apply ofe_morC_map_ne; apply cFunctor_contractive; destruct n, Hfg; by split.
635
636
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
637
638
(** Sum *)
Section sum.
639
  Context {A B : ofeT}.
Robbert Krebbers's avatar
Robbert Krebbers committed
640
641
642
643
644
645
646

  Instance sum_dist : Dist (A + B) := λ n, sum_relation (dist n) (dist n).
  Global Instance inl_ne : Proper (dist n ==> dist n) (@inl A B) := _.
  Global Instance inr_ne : Proper (dist n ==> dist n) (@inr A B) := _.
  Global Instance inl_ne_inj : Inj (dist n) (dist n) (@inl A B) := _.
  Global Instance inr_ne_inj : Inj (dist n) (dist n) (@inr A B) := _.

647
648
649
650
651
652
653
654
655
656
657
658
  Definition sum_ofe_mixin : OfeMixin (A + B).
  Proof.
    split.
    - intros x y; split=> Hx.
      + destruct Hx=> n; constructor; by apply equiv_dist.
      + destruct (Hx 0); constructor; apply equiv_dist=> n; by apply (inj _).
    - apply _.
    - destruct 1; constructor; by apply dist_S.
  Qed.
  Canonical Structure sumC : ofeT := OfeT (A + B) sum_ofe_mixin.

  Program Definition inl_chain (c : chain sumC) (a : A) : chain A :=
Robbert Krebbers's avatar
Robbert Krebbers committed
659
660
    {| chain_car n := match c n return _ with inl a' => a' | _ => a end |}.
  Next Obligation. intros c a n i ?; simpl. by destruct (chain_cauchy c n i). Qed.
661
  Program Definition inr_chain (c : chain sumC) (b : B) : chain B :=
Robbert Krebbers's avatar
Robbert Krebbers committed
662
663
664
    {| chain_car n := match c n return _ with inr b' => b' | _ => b end |}.
  Next Obligation. intros c b n i ?; simpl. by destruct (chain_cauchy c n i). Qed.

665
  Definition sum_compl `{Cofe A, Cofe B} : Compl sumC := λ c,
Robbert Krebbers's avatar
Robbert Krebbers committed
666
667
668
669
    match c 0 with
    | inl a => inl (compl (inl_chain c a))
    | inr b => inr (compl (inr_chain c b))
    end.
670
671
672
673
674
675
676
  Global Program Instance sum_cofe `{Cofe A, Cofe B} : Cofe sumC :=
    { compl := sum_compl }.
  Next Obligation.
    intros ?? n c; rewrite /compl /sum_compl.
    feed inversion (chain_cauchy c 0 n); first by auto with lia; constructor.
    - rewrite (conv_compl n (inl_chain c _)) /=. destruct (c n); naive_solver.
    - rewrite (conv_compl n (inr_chain c _)) /=. destruct (c n); naive_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
677
678
679
680
681
682
683
684
685
686
687
688
689
  Qed.

  Global Instance inl_timeless (x : A) : Timeless x  Timeless (inl x).
  Proof. inversion_clear 2; constructor; by apply (timeless _). Qed.
  Global Instance inr_timeless (y : B) : Timeless y  Timeless (inr y).
  Proof. inversion_clear 2; constructor; by apply (timeless _). Qed.
  Global Instance sum_discrete_cofe : Discrete A  Discrete B  Discrete sumC.
  Proof. intros ?? [?|?]; apply _. Qed.
End sum.

Arguments sumC : clear implicits.
Typeclasses Opaque sum_dist.

690
Instance sum_map_ne {A A' B B' : ofeT} n :
Robbert Krebbers's avatar
Robbert Krebbers committed
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
  Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==>
           dist n ==> dist n) (@sum_map A A' B B').
Proof.
  intros f f' Hf g g' Hg ??; destruct 1; constructor; [by apply Hf|by apply Hg].
Qed.
Definition sumC_map {A A' B B'} (f : A -n> A') (g : B -n> B') :
  sumC A B -n> sumC A' B' := CofeMor (sum_map f g).
Instance sumC_map_ne {A A' B B'} n :
  Proper (dist n ==> dist n ==> dist n) (@sumC_map A A' B B').
Proof. intros f f' Hf g g' Hg [?|?]; constructor; [apply Hf|apply Hg]. Qed.

Program Definition sumCF (F1 F2 : cFunctor) : cFunctor := {|
  cFunctor_car A B := sumC (cFunctor_car F1 A B) (cFunctor_car F2 A B);
  cFunctor_map A1 A2 B1 B2 fg :=
    sumC_map (cFunctor_map F1 fg) (cFunctor_map F2 fg)
|}.
Next Obligation.
  intros ?? A1 A2 B1 B2 n ???; by apply sumC_map_ne; apply cFunctor_ne.
Qed.
Next Obligation. by intros F1 F2 A B [?|?]; rewrite /= !cFunctor_id. Qed.
Next Obligation.
  intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [?|?]; simpl;
    by rewrite !cFunctor_compose.
Qed.

Instance sumCF_contractive F1 F2 :
  cFunctorContractive F1  cFunctorContractive F2 
  cFunctorContractive (sumCF F1 F2).
Proof.
  intros ?? A1 A2 B1 B2 n ???;
    by apply sumC_map_ne; apply cFunctor_contractive.
Qed.

724
725
726
(** Discrete cofe *)
Section discrete_cofe.
  Context `{Equiv A, @Equivalence A ()}.
727

728
  Instance discrete_dist : Dist A := λ n x y, x  y.
729
  Definition discrete_ofe_mixin : OfeMixin A.
730
731
  Proof.
    split.
732
733
734
    - intros x y; split; [done|intros Hn; apply (Hn 0)].
    - done.
    - done.
735
  Qed.
736

737
738
739
740
741
  Global Program Instance discrete_cofe : Cofe (OfeT A discrete_ofe_mixin) :=
    { compl c := c 0 }.
  Next Obligation.
    intros n c. rewrite /compl /=;
    symmetry; apply (chain_cauchy c 0 n). omega.
742
743
744
  Qed.
End discrete_cofe.

745
746
Notation discreteC A := (OfeT A discrete_ofe_mixin).
Notation leibnizC A := (OfeT A (@discrete_ofe_mixin _ equivL _)).
747
748
749
750
751
752

Instance discrete_discrete_cofe `{Equiv A, @Equivalence A ()} :
  Discrete (discreteC A).
Proof. by intros x y. Qed.
Instance leibnizC_leibniz A : LeibnizEquiv (leibnizC A).
Proof. by intros x y. Qed.
753

Robbert Krebbers's avatar
Robbert Krebbers committed
754
Canonical Structure boolC := leibnizC bool.
755
756
757
758
Canonical Structure natC := leibnizC nat.
Canonical Structure positiveC := leibnizC positive.
Canonical Structure NC := leibnizC N.
Canonical Structure ZC := leibnizC Z.
759

760
761
(* Option *)
Section option.
762
  Context {A : ofeT}.
763

764
  Instance option_dist : Dist (option A) := λ n, option_Forall2 (dist n).
765
  Lemma dist_option_Forall2 n mx my : mx {n} my  option_Forall2 (dist n) mx my.
766
  Proof. done. Qed.
767

768
  Definition option_ofe_mixin : OfeMixin (option A).
769
770
771
772
773
  Proof.
    split.
    - intros mx my; split; [by destruct 1; constructor; apply equiv_dist|].
      intros Hxy; destruct (Hxy 0); constructor; apply equiv_dist.
      by intros n; feed inversion (Hxy n).
774
    - apply _.
775
776
    - destruct 1; constructor; by apply dist_S.
  Qed.
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
  Canonical Structure optionC := OfeT (option A) option_ofe_mixin.

  Program Definition option_chain (c : chain optionC) (x : A) : chain A :=
    {| chain_car n := from_option id x (c n) |}.
  Next Obligation. intros c x n i ?; simpl. by destruct (chain_cauchy c n i). Qed.
  Definition option_compl `{Cofe A} : Compl optionC := λ c,
    match c 0 with Some x => Some (compl (option_chain c x)) | None => None end.
  Global Program Instance option_cofe `{Cofe A} : Cofe optionC :=
    { compl := option_compl }.
  Next Obligation.
    intros ? n c; rewrite /compl /option_compl.
    feed inversion (chain_cauchy c 0 n); auto with lia; [].
    constructor. rewrite (conv_compl n (option_chain c _)) /=.
    destruct (c n); naive_solver.
  Qed.

793
794
795
796
797
798
799
800
801
  Global Instance option_discrete : Discrete A  Discrete optionC.
  Proof. destruct 2; constructor; by apply (timeless _). Qed.

  Global Instance Some_ne : Proper (dist n ==> dist n) (@Some A).
  Proof. by constructor. Qed.
  Global Instance is_Some_ne n : Proper (dist n ==> iff) (@is_Some A).
  Proof. destruct 1; split; eauto. Qed.
  Global Instance Some_dist_inj : Inj (dist n) (dist n) (@Some A).
  Proof. by inversion_clear 1. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
802
803
804
  Global Instance from_option_ne {B} (R : relation B) (f : A  B) n :
    Proper (dist n ==> R) f  Proper (R ==> dist n ==> R) (from_option f).
  Proof. destruct 3; simpl; auto. Qed.
805
806
807
808
809

  Global Instance None_timeless : Timeless (@None A).
  Proof. inversion_clear 1; constructor. Qed.
  Global Instance Some_timeless x : Timeless x  Timeless (Some x).
  Proof. by intros ?; inversion_clear 1; constructor; apply timeless. Qed.
810
811
812
813
814
815
816
817
818
819
820
821
822

  Lemma dist_None n mx : mx {n} None  mx = None.
  Proof. split; [by inversion_clear 1|by intros ->]. Qed.
  Lemma dist_Some_inv_l n mx my x :
    mx {n} my  mx = Some x   y, my = Some y  x {n} y.
  Proof. destruct 1; naive_solver. Qed.
  Lemma dist_Some_inv_r n mx my y :
    mx {n} my  my = Some y   x, mx = Some x  x {n} y.
  Proof. destruct 1; naive_solver. Qed.
  Lemma dist_Some_inv_l' n my x : Some x {n} my   x', Some x' = my  x {n} x'.
  Proof. intros ?%(dist_Some_inv_l _ _ _ x); naive_solver. Qed.
  Lemma dist_Some_inv_r' n mx y : mx {n} Some y   y', mx = Some y'  y {n} y'.
  Proof. intros ?%(dist_Some_inv_r _ _ _ y); naive_solver. Qed.
823
824
End option.

825
Typeclasses Opaque option_dist.
826
827
Arguments optionC : clear implicits.

828
Instance option_fmap_ne {A B : ofeT} n:
Robbert Krebbers's avatar
Robbert Krebbers committed
829
830
  Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@fmap option _ A B).
Proof. intros f f' Hf ?? []; constructor; auto. Qed.