ofe.v 39.8 KB
Newer Older
1
From iris.algebra Require Export base.
2
Set Default Proof Using "Type*".
Robbert Krebbers's avatar
Robbert Krebbers committed
3

4
(** This files defines (a shallow embedding of) the category of OFEs:
5
6
7
8
9
10
11
12
    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
13
14
(** Unbundeled version *)
Class Dist A := dist : nat  relation A.
15
Instance: Params (@dist) 3.
16
17
Notation "x ≡{ n }≡ y" := (dist n x y)
  (at level 70, n at next level, format "x  ≡{ n }≡  y").
18
Hint Extern 0 (_ {_} _) => reflexivity.
19
Hint Extern 0 (_ {_} _) => symmetry; assumption.
20
21
22

Tactic Notation "cofe_subst" ident(x) :=
  repeat match goal with
23
  | _ => progress simplify_eq/=
24
25
26
27
  | 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" :=
28
  repeat match goal with
29
  | _ => progress simplify_eq/=
30
31
  | 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
32
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
33

34
Record OfeMixin A `{Equiv A, Dist A} := {
35
  mixin_equiv_dist x y : x  y   n, x {n} y;
36
  mixin_dist_equivalence n : Equivalence (dist n);
37
  mixin_dist_S n x y : x {S n} y  x {n} y
Robbert Krebbers's avatar
Robbert Krebbers committed
38
39
40
}.

(** Bundeled version *)
41
42
43
44
45
Structure ofeT := OfeT' {
  ofe_car :> Type;
  ofe_equiv : Equiv ofe_car;
  ofe_dist : Dist ofe_car;
  ofe_mixin : OfeMixin ofe_car;
46
  _ : Type
Robbert Krebbers's avatar
Robbert Krebbers committed
47
}.
48
49
50
51
52
53
54
55
56
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.
57
58

(** Lifting properties from the mixin *)
59
60
Section ofe_mixin.
  Context {A : ofeT}.
61
  Implicit Types x y : A.
62
  Lemma equiv_dist x y : x  y   n, x {n} y.
63
  Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed.
64
  Global Instance dist_equivalence n : Equivalence (@dist A _ n).
65
  Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed.
66
  Lemma dist_S n x y : x {S n} y  x {n} y.
67
68
  Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed.
End ofe_mixin.
69

Robbert Krebbers's avatar
Robbert Krebbers committed
70
71
Hint Extern 1 (_ {_} _) => apply equiv_dist; assumption.

72
(** Discrete OFEs and Timeless elements *)
Ralf Jung's avatar
Ralf Jung committed
73
(* TODO: On paper, We called these "discrete elements". I think that makes
Ralf Jung's avatar
Ralf Jung committed
74
   more sense. *)
75
76
Class Timeless `{Equiv A, Dist A} (x : A) := timeless y : x {0} y  x  y.
Arguments timeless {_ _ _} _ {_} _ _.
77
78
79
80
81
82
83
84
85
86
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 {_} _ _ _ _.

87
88
89
90
91
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.

92
93
94
95
96
97
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.
98

Robbert Krebbers's avatar
Robbert Krebbers committed
99
100
(** General properties *)
Section cofe.
101
  Context {A : ofeT}.
102
  Implicit Types x y : A.
Robbert Krebbers's avatar
Robbert Krebbers committed
103
104
105
  Global Instance cofe_equivalence : Equivalence (() : relation A).
  Proof.
    split.
106
107
    - by intros x; rewrite equiv_dist.
    - by intros x y; rewrite !equiv_dist.
108
    - by intros x y z; rewrite !equiv_dist; intros; trans y.
Robbert Krebbers's avatar
Robbert Krebbers committed
109
  Qed.
110
  Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
111
112
  Proof.
    intros x1 x2 ? y1 y2 ?; split; intros.
113
114
    - by trans x1; [|trans y1].
    - by trans x2; [|trans y2].
Robbert Krebbers's avatar
Robbert Krebbers committed
115
  Qed.
116
  Global Instance dist_proper n : Proper (() ==> () ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
117
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
118
    by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
119
120
121
  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
122
  Lemma dist_le n n' x y : x {n} y  n'  n  x {n'} y.
Robbert Krebbers's avatar
Robbert Krebbers committed
123
  Proof. induction 2; eauto using dist_S. Qed.
124
125
  Lemma dist_le' n n' x y : n'  n  x {n} y  x {n'} y.
  Proof. intros; eauto using dist_le. Qed.
126
  Instance ne_proper {B : ofeT} (f : A  B)
Robbert Krebbers's avatar
Robbert Krebbers committed
127
128
    `{! 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.
129
  Instance ne_proper_2 {B C : ofeT} (f : A  B  C)
Robbert Krebbers's avatar
Robbert Krebbers committed
130
131
132
133
    `{! 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
134
     by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
135
  Qed.
136

137
  Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c {n} c (S n).
138
139
140
141
  Proof.
    transitivity (c n); first by apply conv_compl. symmetry.
    apply chain_cauchy. omega.
  Qed.
142
143
  Lemma timeless_iff n (x : A) `{!Timeless x} y : x  y  x {n} y.
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
144
    split; intros; auto. apply (timeless _), dist_le with n; auto with lia.
145
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
146
147
End cofe.

148
(** Contractive functions *)
149
150
151
152
153
154
155
156
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).
157

158
Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x).
159
160
Proof. by intros n y1 y2. Qed.

161
162
163
164
165
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.
166
  Proof. by apply (_ : Contractive f). Qed.
167
  Lemma contractive_S n x y : x {n} y  f x {S n} f y.
168
  Proof. intros. by apply (_ : Contractive f). Qed.
169
170
171
172
173
174
175

  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.

176
177
178
179
180
181
182
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
183
184
  | |- @dist_later ?A ?n ?x ?y =>
         destruct n as [|n]; [done|change (@dist A _ n x y)]
185
186
187
188
189
190
  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
191

Robbert Krebbers's avatar
Robbert Krebbers committed
192
(** Fixpoint *)
193
Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A  A)
194
  `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
195
Next Obligation.
196
  intros A ? f ? n.
197
  induction n as [|n IH]=> -[|i] //= ?; try omega.
198
199
  - apply (contractive_0 f).
  - apply (contractive_S f), IH; auto with omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
200
Qed.
201

202
Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A  A)
203
  `{!Contractive f} : A := compl (fixpoint_chain f).
204
Definition fixpoint_aux : { x | x = @fixpoint_def }. by eexists. Qed.
205
Definition fixpoint {A AC AiH} f {Hf} := proj1_sig fixpoint_aux A AC AiH f Hf.
206
Definition fixpoint_eq : @fixpoint = @fixpoint_def := proj2_sig fixpoint_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
207
208

Section fixpoint.
209
  Context `{Cofe A, Inhabited A} (f : A  A) `{!Contractive f}.
210

211
  Lemma fixpoint_unfold : fixpoint f  f (fixpoint f).
Robbert Krebbers's avatar
Robbert Krebbers committed
212
  Proof.
213
214
    apply equiv_dist=>n.
    rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //.
215
    induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
216
  Qed.
217
218
219

  Lemma fixpoint_unique (x : A) : x  f x  x  fixpoint f.
  Proof.
220
221
222
    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.
223
224
  Qed.

225
  Lemma fixpoint_ne (g : A  A) `{!Contractive g} n :
226
    ( z, f z {n} g z)  fixpoint f {n} fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
227
  Proof.
228
    intros Hfg. rewrite fixpoint_eq /fixpoint_def
Robbert Krebbers's avatar
Robbert Krebbers committed
229
      (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=.
230
231
    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
232
  Qed.
233
234
  Lemma fixpoint_proper (g : A  A) `{!Contractive g} :
    ( x, f x  g x)  fixpoint f  fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
235
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.
236
237

  Lemma fixpoint_ind (P : A  Prop) :
238
    Proper (() ==> impl) P 
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
    ( 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
254
255
End fixpoint.

Robbert Krebbers's avatar
Robbert Krebbers committed
256
257
(** Mutual fixpoints *)
Section fixpoint2.
258
259
  Local Unset Default Proof Using.

Robbert Krebbers's avatar
Robbert Krebbers committed
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
334
335
336
  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.

337
(** Function space *)
338
(* We make [ofe_fun] a definition so that we can register it as a canonical
339
structure. *)
340
Definition ofe_fun (A : Type) (B : ofeT) := A  B.
341

342
343
344
345
346
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).
347
348
349
350
351
352
353
354
355
356
  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.
357
  Canonical Structure ofe_funC := OfeT (ofe_fun A B) ofe_fun_ofe_mixin.
358

359
360
361
362
363
364
365
366
367
  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.
368
Notation "A -c> B" :=
369
370
  (ofe_funC A B) (at level 99, B at level 200, right associativity).
Instance ofe_fun_inhabited {A} {B : ofeT} `{Inhabited B} :
371
372
  Inhabited (A -c> B) := populate (λ _, inhabitant).

373
(** Non-expansive function space *)
374
375
376
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
377
378
}.
Arguments CofeMor {_ _} _ {_}.
379
380
Add Printing Constructor ofe_mor.
Existing Instance ofe_mor_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
381

382
383
384
385
Notation "'λne' x .. y , t" :=
  (@CofeMor _ _ (λ x, .. (@CofeMor _ _ (λ y, t) _) ..) _)
  (at level 200, x binder, y binder, right associativity).

386
387
388
389
390
391
392
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).
393
394
  Proof.
    split.
395
    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
Robbert Krebbers's avatar
Robbert Krebbers committed
396
      intros Hfg k; apply equiv_dist=> n; apply Hfg.
397
    - intros n; split.
398
399
      + by intros f x.
      + by intros f g ? x.
400
      + by intros f g h ?? x; trans (g x).
401
    - by intros n f g ? x; apply dist_S.
402
  Qed.
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
  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.
420

421
422
  Global Instance ofe_mor_car_ne n :
    Proper (dist n ==> dist n ==> dist n) (@ofe_mor_car A B).
423
  Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed.
424
425
426
  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.
427
  Proof. done. Qed.
428
End ofe_mor.
429

430
Arguments ofe_morC : clear implicits.
431
Notation "A -n> B" :=
432
433
  (ofe_morC A B) (at level 99, B at level 200, right associativity).
Instance ofe_mor_inhabited {A B : ofeT} `{Inhabited B} :
434
  Inhabited (A -n> B) := populate (λne _, inhabitant).
Robbert Krebbers's avatar
Robbert Krebbers committed
435

436
(** Identity and composition and constant function *)
Robbert Krebbers's avatar
Robbert Krebbers committed
437
438
Definition cid {A} : A -n> A := CofeMor id.
Instance: Params (@cid) 1.
439
Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x).
440
Instance: Params (@cconst) 2.
441

Robbert Krebbers's avatar
Robbert Krebbers committed
442
443
444
445
446
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 :
447
  f1 {n} f2  g1 {n} g2  f1  g1 {n} f2  g2.
Robbert Krebbers's avatar
Robbert Krebbers committed
448
Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
449

Ralf Jung's avatar
Ralf Jung committed
450
(* Function space maps *)
451
Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B')
Ralf Jung's avatar
Ralf Jung committed
452
  (h : A -n> B) : A' -n> B' := g  h  f.
453
454
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').
455
Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed.
Ralf Jung's avatar
Ralf Jung committed
456

457
458
459
460
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
461
Proof.
462
  intros f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map.
463
  by repeat apply ccompose_ne.
Ralf Jung's avatar
Ralf Jung committed
464
465
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
466
(** unit *)
467
468
Section unit.
  Instance unit_dist : Dist unit := λ _ _ _, True.
469
  Definition unit_ofe_mixin : OfeMixin unit.
470
  Proof. by repeat split; try exists 0. Qed.
471
  Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin.
Robbert Krebbers's avatar
Robbert Krebbers committed
472

473
474
  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
475
476

  Global Instance unit_discrete_cofe : Discrete unitC.
Robbert Krebbers's avatar
Robbert Krebbers committed
477
  Proof. done. Qed.
478
End unit.
Robbert Krebbers's avatar
Robbert Krebbers committed
479
480

(** Product *)
481
Section product.
482
  Context {A B : ofeT}.
483
484
485
486
487
488

  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) := _.
489
  Definition prod_ofe_mixin : OfeMixin (A * B).
490
491
  Proof.
    split.
492
    - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
493
      rewrite !equiv_dist; naive_solver.
494
495
    - apply _.
    - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
496
  Qed.
497
498
499
500
501
502
503
504
505
  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.

506
507
508
  Global Instance prod_timeless (x : A * B) :
    Timeless (x.1)  Timeless (x.2)  Timeless x.
  Proof. by intros ???[??]; split; apply (timeless _). Qed.
509
510
  Global Instance prod_discrete_cofe : Discrete A  Discrete B  Discrete prodC.
  Proof. intros ?? [??]; apply _. Qed.
511
512
513
514
515
End product.

Arguments prodC : clear implicits.
Typeclasses Opaque prod_dist.

516
Instance prod_map_ne {A A' B B' : ofeT} n :
Robbert Krebbers's avatar
Robbert Krebbers committed
517
518
519
520
521
522
523
524
525
  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.

526
527
(** Functors *)
Structure cFunctor := CFunctor {
528
  cFunctor_car : ofeT  ofeT  ofeT;
529
530
  cFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  cFunctor_car A1 B1 -n> cFunctor_car A2 B2;
531
532
  cFunctor_ne {A1 A2 B1 B2} n :
    Proper (dist n ==> dist n) (@cFunctor_map A1 A2 B1 B2);
533
  cFunctor_id {A B : ofeT} (x : cFunctor_car A B) :
534
535
536
537
538
    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)
}.
539
Existing Instance cFunctor_ne.
540
541
Instance: Params (@cFunctor_map) 5.

542
543
544
Delimit Scope cFunctor_scope with CF.
Bind Scope cFunctor_scope with cFunctor.

545
546
547
Class cFunctorContractive (F : cFunctor) :=
  cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2).

548
Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A.
549
550
Coercion cFunctor_diag : cFunctor >-> Funclass.

551
Program Definition constCF (B : ofeT) : cFunctor :=
552
553
554
  {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}.
Solve Obligations with done.

555
Instance constCF_contractive B : cFunctorContractive (constCF B).
556
Proof. rewrite /cFunctorContractive; apply _. Qed.
557
558
559
560
561

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

562
563
564
565
566
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)
|}.
567
568
569
Next Obligation.
  intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne.
Qed.
570
571
572
573
574
575
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.

576
577
578
579
580
581
582
583
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.

584
Instance compose_ne {A} {B B' : ofeT} (f : B -n> B') n :
585
586
587
  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.

588
Definition ofe_funC_map {A B B'} (f : B -n> B') : (A -c> B) -n> (A -c> B') :=
589
  @CofeMor (_ -c> _) (_ -c> _) (compose f) _.
590
591
Instance ofe_funC_map_ne {A B B'} n :
  Proper (dist n ==> dist n) (@ofe_funC_map A B B').
592
593
Proof. intros f f' Hf g x. apply Hf. Qed.

594
595
596
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)
597
598
|}.
Next Obligation.
599
  intros ?? A1 A2 B1 B2 n ???; by apply ofe_funC_map_ne; apply cFunctor_ne.
600
601
602
603
604
605
606
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.

607
608
Instance ofe_funCF_contractive (T : Type) (F : cFunctor) :
  cFunctorContractive F  cFunctorContractive (ofe_funCF T F).
609
610
Proof.
  intros ?? A1 A2 B1 B2 n ???;
611
    by apply ofe_funC_map_ne; apply cFunctor_contractive.
612
613
Qed.

614
Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {|
615
  cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B;
Ralf Jung's avatar
Ralf Jung committed
616
  cFunctor_map A1 A2 B1 B2 fg :=
617
    ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg)
Ralf Jung's avatar
Ralf Jung committed
618
|}.
619
620
Next Obligation.
  intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *.
621
  apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg.
622
Qed.
Ralf Jung's avatar
Ralf Jung committed
623
Next Obligation.
624
625
  intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id.
  apply (ne_proper f). apply cFunctor_id.
Ralf Jung's avatar
Ralf Jung committed
626
627
Qed.
Next Obligation.
628
629
  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
630
631
Qed.

632
Instance ofe_morCF_contractive F1 F2 :
633
  cFunctorContractive F1  cFunctorContractive F2 
634
  cFunctorContractive (ofe_morCF F1 F2).
635
636
Proof.
  intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *.
637
  apply ofe_morC_map_ne; apply cFunctor_contractive; destruct n, Hfg; by split.
638
639
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
640
641
(** Sum *)
Section sum.
642
  Context {A B : ofeT}.
Robbert Krebbers's avatar
Robbert Krebbers committed
643
644
645
646
647
648
649

  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) := _.

650
651
652
653
654
655
656
657
658
659
660
661
  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
662
663
    {| 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.
664
  Program Definition inr_chain (c : chain sumC) (b : B) : chain B :=
Robbert Krebbers's avatar
Robbert Krebbers committed
665
666
667
    {| 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.

668
  Definition sum_compl `{Cofe A, Cofe B} : Compl sumC := λ c,
Robbert Krebbers's avatar
Robbert Krebbers committed
669
670
671
672
    match c 0 with
    | inl a => inl (compl (inl_chain c a))
    | inr b => inr (compl (inr_chain c b))
    end.
673
674
675
676
677
678
679
  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
680
681
682
683
684
685
686
687
688
689
690
691
692
  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.

693
Instance sum_map_ne {A A' B B' : ofeT} n :
Robbert Krebbers's avatar
Robbert Krebbers committed
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
724
725
726
  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.

727
728
729
(** Discrete cofe *)
Section discrete_cofe.
  Context `{Equiv A, @Equivalence A ()}.
730

731
  Instance discrete_dist : Dist A := λ n x y, x  y.
732
  Definition discrete_ofe_mixin : OfeMixin A.
733
734
  Proof.
    split.
735
736
737
    - intros x y; split; [done|intros Hn; apply (Hn 0)].
    - done.
    - done.
738
  Qed.
739

740
741
742
743
744
  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.
745
746
747
  Qed.
End discrete_cofe.

748
749
Notation discreteC A := (OfeT A discrete_ofe_mixin).
Notation leibnizC A := (OfeT A (@discrete_ofe_mixin _ equivL _)).
750
751
752
753
754
755

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.
756

Robbert Krebbers's avatar
Robbert Krebbers committed
757
Canonical Structure boolC := leibnizC bool.
758
759
760
761
Canonical Structure natC := leibnizC nat.
Canonical Structure positiveC := leibnizC positive.
Canonical Structure NC := leibnizC N.
Canonical Structure ZC := leibnizC Z.
762

763
764
(* Option *)
Section option.
765
  Context {A : ofeT}.
766

767
  Instance option_dist : Dist (option A) := λ n, option_Forall2 (dist n).
768
  Lemma dist_option_Forall2 n mx my : mx {n} my  option_Forall2 (dist n) mx my.
769
  Proof. done. Qed.
770

771
  Definition option_ofe_mixin : OfeMixin (option A).
772
773
774
775
776
  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).
777
    - apply _.
778
779
    - destruct 1; constructor; by apply dist_S.
  Qed.
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
  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.

796
797
798
799
800
801
802
803
804
  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
805
806
807
  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.
808
809
810
811
812

  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.
813
814
815
816
817
818
819
820
821
822
823
824
825

  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.
826
827
End option.

828
Typeclasses Opaque option_dist.
829
830
Arguments optionC : clear implicits