cofe.v 15.6 KB
Newer Older
1
From algebra Require Export base.
Robbert Krebbers's avatar
Robbert Krebbers committed
2

3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
(** This files defines (a shallow embedding of) the category of COFEs:
    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.

    In principle, it would be possible to perform a large part of the
    development on OFEs, i.e., on bisected metric spaces that are not
    necessary complete. This is because the function space A → B has a
    completion if B has one - for A, the metric itself suffices.
    That would result in a simplification of some constructions, becuase
    no completion would have to be provided. However, on the other hand,
    we would have to introduce the notion of OFEs into our alebraic
    hierarchy, which we'd rather avoid. Furthermore, on paper, justifying
    this mix of OFEs and COFEs is a little fuzzy.
*)

Robbert Krebbers's avatar
Robbert Krebbers committed
22
23
(** Unbundeled version *)
Class Dist A := dist : nat  relation A.
24
Instance: Params (@dist) 3.
25
26
Notation "x ≡{ n }≡ y" := (dist n x y)
  (at level 70, n at next level, format "x  ≡{ n }≡  y").
27
Hint Extern 0 (_ {_} _) => reflexivity.
28
Hint Extern 0 (_ {_} _) => symmetry; assumption.
29
30
31

Tactic Notation "cofe_subst" ident(x) :=
  repeat match goal with
32
  | _ => progress simplify_eq/=
33
34
35
36
  | 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" :=
37
  repeat match goal with
38
  | _ => progress simplify_eq/=
39
40
  | 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
41
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
42
43
44

Record chain (A : Type) `{Dist A} := {
  chain_car :> nat  A;
45
  chain_cauchy n i : n < i  chain_car i {n} chain_car (S n)
Robbert Krebbers's avatar
Robbert Krebbers committed
46
47
48
49
50
}.
Arguments chain_car {_ _} _ _.
Arguments chain_cauchy {_ _} _ _ _ _.
Class Compl A `{Dist A} := compl : chain A  A.

51
Record CofeMixin A `{Equiv A, Compl A} := {
52
  mixin_equiv_dist x y : x  y   n, x {n} y;
53
  mixin_dist_equivalence n : Equivalence (dist n);
54
  mixin_dist_S n x y : x {S n} y  x {n} y;
Robbert Krebbers's avatar
Robbert Krebbers committed
55
  mixin_conv_compl n c : compl c {n} c (S n)
Robbert Krebbers's avatar
Robbert Krebbers committed
56
}.
57
Class Contractive `{Dist A, Dist B} (f : A  B) :=
58
  contractive n x y : ( i, i < n  x {i} y)  f x {n} f y.
Robbert Krebbers's avatar
Robbert Krebbers committed
59
60
61
62
63
64
65

(** Bundeled version *)
Structure cofeT := CofeT {
  cofe_car :> Type;
  cofe_equiv : Equiv cofe_car;
  cofe_dist : Dist cofe_car;
  cofe_compl : Compl cofe_car;
66
  cofe_mixin : CofeMixin cofe_car
Robbert Krebbers's avatar
Robbert Krebbers committed
67
}.
68
Arguments CofeT {_ _ _ _} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
69
Add Printing Constructor cofeT.
70
71
72
73
74
75
76
77
78
79
80
Existing Instances cofe_equiv cofe_dist cofe_compl.
Arguments cofe_car : simpl never.
Arguments cofe_equiv : simpl never.
Arguments cofe_dist : simpl never.
Arguments cofe_compl : simpl never.
Arguments cofe_mixin : simpl never.

(** Lifting properties from the mixin *)
Section cofe_mixin.
  Context {A : cofeT}.
  Implicit Types x y : A.
81
  Lemma equiv_dist x y : x  y   n, x {n} y.
82
83
84
  Proof. apply (mixin_equiv_dist _ (cofe_mixin A)). Qed.
  Global Instance dist_equivalence n : Equivalence (@dist A _ n).
  Proof. apply (mixin_dist_equivalence _ (cofe_mixin A)). Qed.
85
  Lemma dist_S n x y : x {S n} y  x {n} y.
86
  Proof. apply (mixin_dist_S _ (cofe_mixin A)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
87
  Lemma conv_compl n (c : chain A) : compl c {n} c (S n).
88
89
90
  Proof. apply (mixin_conv_compl _ (cofe_mixin A)). Qed.
End cofe_mixin.

91
92
93
94
95
(** Discrete COFEs and Timeless elements *)
Class Timeless {A : cofeT} (x : A) := timeless y : x {0} y  x  y.
Arguments timeless {_} _ {_} _ _.
Class Discrete (A : cofeT) := discrete_timeless (x : A) :> Timeless x.

Robbert Krebbers's avatar
Robbert Krebbers committed
96
97
(** General properties *)
Section cofe.
98
99
  Context {A : cofeT}.
  Implicit Types x y : A.
Robbert Krebbers's avatar
Robbert Krebbers committed
100
101
102
  Global Instance cofe_equivalence : Equivalence (() : relation A).
  Proof.
    split.
103
104
    - by intros x; rewrite equiv_dist.
    - by intros x y; rewrite !equiv_dist.
105
    - by intros x y z; rewrite !equiv_dist; intros; trans y.
Robbert Krebbers's avatar
Robbert Krebbers committed
106
  Qed.
107
  Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
108
109
  Proof.
    intros x1 x2 ? y1 y2 ?; split; intros.
110
111
    - by trans x1; [|trans y1].
    - by trans x2; [|trans y2].
Robbert Krebbers's avatar
Robbert Krebbers committed
112
  Qed.
113
  Global Instance dist_proper n : Proper (() ==> () ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
114
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
115
    by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
116
117
118
  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
119
  Lemma dist_le n n' x y : x {n} y  n'  n  x {n'} y.
Robbert Krebbers's avatar
Robbert Krebbers committed
120
  Proof. induction 2; eauto using dist_S. Qed.
121
  Instance ne_proper {B : cofeT} (f : A  B)
Robbert Krebbers's avatar
Robbert Krebbers committed
122
123
    `{! 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.
124
  Instance ne_proper_2 {B C : cofeT} (f : A  B  C)
Robbert Krebbers's avatar
Robbert Krebbers committed
125
126
127
128
    `{! 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
129
     by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
130
  Qed.
131
  Lemma contractive_S {B : cofeT} (f : A  B) `{!Contractive f} n x y :
132
133
    x {n} y  f x {S n} f y.
  Proof. eauto using contractive, dist_le with omega. Qed.
134
135
136
  Lemma contractive_0 {B : cofeT} (f : A  B) `{!Contractive f} x y :
    f x {0} f y.
  Proof. eauto using contractive with omega. Qed.
137
  Global Instance contractive_ne {B : cofeT} (f : A  B) `{!Contractive f} n :
138
    Proper (dist n ==> dist n) f | 100.
139
  Proof. by intros x y ?; apply dist_S, contractive_S. Qed.
140
  Global Instance contractive_proper {B : cofeT} (f : A  B) `{!Contractive f} :
141
    Proper (() ==> ()) f | 100 := _.
142
143
144
145
146
147

  Lemma timeless_iff n (x : A) `{!Timeless x} y : x  y  x {n} y.
  Proof.
    split; intros; [by apply equiv_dist|].
    apply (timeless _), dist_le with n; auto with lia.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
148
149
End cofe.

Robbert Krebbers's avatar
Robbert Krebbers committed
150
151
152
153
(** Mapping a chain *)
Program Definition chain_map `{Dist A, Dist B} (f : A  B)
    `{! n, Proper (dist n ==> dist n) f} (c : chain A) : chain B :=
  {| chain_car n := f (c n) |}.
154
Next Obligation. by intros ? A ? B f Hf c n i ?; apply Hf, chain_cauchy. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
155

Robbert Krebbers's avatar
Robbert Krebbers committed
156
(** Fixpoint *)
157
Program Definition fixpoint_chain {A : cofeT} `{Inhabited A} (f : A  A)
158
  `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
159
Next Obligation.
160
  intros A ? f ? n. induction n as [|n IH]; intros [|i] ?; simpl; try omega.
161
162
  - apply (contractive_0 f).
  - apply (contractive_S f), IH; auto with omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
163
Qed.
164
Program Definition fixpoint {A : cofeT} `{Inhabited A} (f : A  A)
165
  `{!Contractive f} : A := compl (fixpoint_chain f).
Robbert Krebbers's avatar
Robbert Krebbers committed
166
167

Section fixpoint.
168
  Context {A : cofeT} `{Inhabited A} (f : A  A) `{!Contractive f}.
169
  Lemma fixpoint_unfold : fixpoint f  f (fixpoint f).
Robbert Krebbers's avatar
Robbert Krebbers committed
170
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
171
    apply equiv_dist=>n; rewrite /fixpoint (conv_compl n (fixpoint_chain f)) //.
172
    induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
173
  Qed.
174
  Lemma fixpoint_ne (g : A  A) `{!Contractive g} n :
175
    ( z, f z {n} g z)  fixpoint f {n} fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
176
  Proof.
177
    intros Hfg. rewrite /fixpoint
Robbert Krebbers's avatar
Robbert Krebbers committed
178
      (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=.
179
180
    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
181
  Qed.
182
183
  Lemma fixpoint_proper (g : A  A) `{!Contractive g} :
    ( x, f x  g x)  fixpoint f  fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
184
185
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.
End fixpoint.
186
Global Opaque fixpoint.
Robbert Krebbers's avatar
Robbert Krebbers committed
187
188

(** Function space *)
Robbert Krebbers's avatar
Robbert Krebbers committed
189
Record cofeMor (A B : cofeT) : Type := CofeMor {
Robbert Krebbers's avatar
Robbert Krebbers committed
190
191
192
193
194
195
196
  cofe_mor_car :> A  B;
  cofe_mor_ne n : Proper (dist n ==> dist n) cofe_mor_car
}.
Arguments CofeMor {_ _} _ {_}.
Add Printing Constructor cofeMor.
Existing Instance cofe_mor_ne.

197
198
199
200
201
Section cofe_mor.
  Context {A B : cofeT}.
  Global Instance cofe_mor_proper (f : cofeMor A B) : Proper (() ==> ()) f.
  Proof. apply ne_proper, cofe_mor_ne. Qed.
  Instance cofe_mor_equiv : Equiv (cofeMor A B) := λ f g,  x, f x  g x.
202
  Instance cofe_mor_dist : Dist (cofeMor A B) := λ n f g,  x, f x {n} g x.
203
204
205
206
207
208
  Program Definition fun_chain `(c : chain (cofeMor A B)) (x : A) : chain B :=
    {| chain_car n := c n x |}.
  Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed.
  Program Instance cofe_mor_compl : Compl (cofeMor A B) := λ c,
    {| cofe_mor_car x := compl (fun_chain c x) |}.
  Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
209
210
    intros c n x y Hx. by rewrite (conv_compl n (fun_chain c x))
      (conv_compl n (fun_chain c y)) /= Hx.
211
212
213
214
  Qed.
  Definition cofe_mor_cofe_mixin : CofeMixin (cofeMor A B).
  Proof.
    split.
215
    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
Robbert Krebbers's avatar
Robbert Krebbers committed
216
      intros Hfg k; apply equiv_dist=> n; apply Hfg.
217
    - intros n; split.
218
219
      + by intros f x.
      + by intros f g ? x.
220
      + by intros f g h ?? x; trans (g x).
221
    - by intros n f g ? x; apply dist_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
222
223
    - intros n c x; simpl.
      by rewrite (conv_compl n (fun_chain c x)) /=.
224
225
226
227
228
229
230
231
232
233
234
235
236
  Qed.
  Canonical Structure cofe_mor : cofeT := CofeT cofe_mor_cofe_mixin.

  Global Instance cofe_mor_car_ne n :
    Proper (dist n ==> dist n ==> dist n) (@cofe_mor_car A B).
  Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed.
  Global Instance cofe_mor_car_proper :
    Proper (() ==> () ==> ()) (@cofe_mor_car A B) := ne_proper_2 _.
  Lemma cofe_mor_ext (f g : cofeMor A B) : f  g   x, f x  g x.
  Proof. done. Qed.
End cofe_mor.

Arguments cofe_mor : clear implicits.
Robbert Krebbers's avatar
Robbert Krebbers committed
237
Infix "-n>" := cofe_mor (at level 45, right associativity).
238
239
Instance cofe_more_inhabited {A B : cofeT} `{Inhabited B} :
  Inhabited (A -n> B) := populate (CofeMor (λ _, inhabitant)).
Robbert Krebbers's avatar
Robbert Krebbers committed
240
241
242
243
244
245
246
247
248

(** Identity and composition *)
Definition cid {A} : A -n> A := CofeMor id.
Instance: Params (@cid) 1.
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 :
249
  f1 {n} f2  g1 {n} g2  f1  g1 {n} f2  g2.
Robbert Krebbers's avatar
Robbert Krebbers committed
250
Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
251
252

(** unit *)
253
254
255
256
257
258
Section unit.
  Instance unit_dist : Dist unit := λ _ _ _, True.
  Instance unit_compl : Compl unit := λ _, ().
  Definition unit_cofe_mixin : CofeMixin unit.
  Proof. by repeat split; try exists 0. Qed.
  Canonical Structure unitC : cofeT := CofeT unit_cofe_mixin.
259
  Global Instance unit_discrete_cofe : Discrete unitC.
Robbert Krebbers's avatar
Robbert Krebbers committed
260
  Proof. done. Qed.
261
End unit.
Robbert Krebbers's avatar
Robbert Krebbers committed
262
263

(** Product *)
264
265
266
267
268
269
270
271
272
273
274
275
276
Section product.
  Context {A B : cofeT}.

  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) := _.
  Instance prod_compl : Compl (A * B) := λ c,
    (compl (chain_map fst c), compl (chain_map snd c)).
  Definition prod_cofe_mixin : CofeMixin (A * B).
  Proof.
    split.
277
    - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
278
      rewrite !equiv_dist; naive_solver.
279
280
    - apply _.
    - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
281
282
    - intros n c; split. apply (conv_compl n (chain_map fst c)).
      apply (conv_compl n (chain_map snd c)).
283
284
285
286
287
  Qed.
  Canonical Structure prodC : cofeT := CofeT prod_cofe_mixin.
  Global Instance pair_timeless (x : A) (y : B) :
    Timeless x  Timeless y  Timeless (x,y).
  Proof. by intros ?? [x' y'] [??]; split; apply (timeless _). Qed.
288
289
  Global Instance prod_discrete_cofe : Discrete A  Discrete B  Discrete prodC.
  Proof. intros ?? [??]; apply _. Qed.
290
291
292
293
294
295
End product.

Arguments prodC : clear implicits.
Typeclasses Opaque prod_dist.

Instance prod_map_ne {A A' B B' : cofeT} n :
Robbert Krebbers's avatar
Robbert Krebbers committed
296
297
298
299
300
301
302
303
304
  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.

305
306
307
(** Discrete cofe *)
Section discrete_cofe.
  Context `{Equiv A, @Equivalence A ()}.
308
  Instance discrete_dist : Dist A := λ n x y, x  y.
309
  Instance discrete_compl : Compl A := λ c, c 1.
310
  Definition discrete_cofe_mixin : CofeMixin A.
311
312
  Proof.
    split.
313
314
315
    - intros x y; split; [done|intros Hn; apply (Hn 0)].
    - done.
    - done.
Robbert Krebbers's avatar
Robbert Krebbers committed
316
    - intros n c. rewrite /compl /discrete_compl /=.
317
      symmetry; apply (chain_cauchy c 0 (S n)); omega.
318
  Qed.
319
  Definition discreteC : cofeT := CofeT discrete_cofe_mixin.
320
321
  Global Instance discrete_discrete_cofe : Discrete discreteC.
  Proof. by intros x y. Qed.
322
End discrete_cofe.
Robbert Krebbers's avatar
Robbert Krebbers committed
323
Arguments discreteC _ {_ _}.
324

Robbert Krebbers's avatar
Robbert Krebbers committed
325
Definition leibnizC (A : Type) : cofeT := @discreteC A equivL _.
326
327
328
Instance leibnizC_leibniz : LeibnizEquiv (leibnizC A).
Proof. by intros A x y. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
329
330
Canonical Structure natC := leibnizC nat.
Canonical Structure boolC := leibnizC bool.
331

332
(** Later *)
333
Inductive later (A : Type) : Type := Next { later_car : A }.
334
Add Printing Constructor later.
335
Arguments Next {_} _.
336
Arguments later_car {_} _.
337
Lemma later_eta {A} (x : later A) : Next (later_car x) = x.
Robbert Krebbers's avatar
Robbert Krebbers committed
338
Proof. by destruct x. Qed.
339

340
Section later.
341
342
343
  Context {A : cofeT}.
  Instance later_equiv : Equiv (later A) := λ x y, later_car x  later_car y.
  Instance later_dist : Dist (later A) := λ n x y,
344
    match n with 0 => True | S n => later_car x {n} later_car y end.
345
  Program Definition later_chain (c : chain (later A)) : chain A :=
346
    {| chain_car n := later_car (c (S n)) |}.
347
  Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed.
348
  Instance later_compl : Compl (later A) := λ c, Next (compl (later_chain c)).
349
  Definition later_cofe_mixin : CofeMixin (later A).
350
351
  Proof.
    split.
352
    - intros x y; unfold equiv, later_equiv; rewrite !equiv_dist.
353
      split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)).
354
    - intros [|n]; [by split|split]; unfold dist, later_dist.
355
356
      + by intros [x].
      + by intros [x] [y].
357
      + by intros [x] [y] [z] ??; trans y.
358
    - intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
359
    - intros [|n] c; [done|by apply (conv_compl n (later_chain c))].
360
  Qed.
361
  Canonical Structure laterC : cofeT := CofeT later_cofe_mixin.
362
363
  Global Instance Next_contractive : Contractive (@Next A).
  Proof. intros [|n] x y Hxy; [done|]; apply Hxy; lia. Qed.
364
  Global Instance Later_inj n : Inj (dist n) (dist (S n)) (@Next A).
Robbert Krebbers's avatar
Robbert Krebbers committed
365
  Proof. by intros x y. Qed.
366
End later.
367
368
369
370

Arguments laterC : clear implicits.

Definition later_map {A B} (f : A  B) (x : later A) : later B :=
371
  Next (f (later_car x)).
372
373
374
375
376
377
378
379
380
381
382
383
Instance later_map_ne {A B : cofeT} (f : A  B) n :
  Proper (dist (pred n) ==> dist (pred n)) f 
  Proper (dist n ==> dist n) (later_map f) | 0.
Proof. destruct n as [|n]; intros Hf [x] [y] ?; do 2 red; simpl; auto. Qed.
Lemma later_map_id {A} (x : later A) : later_map id x = x.
Proof. by destruct x. Qed.
Lemma later_map_compose {A B C} (f : A  B) (g : B  C) (x : later A) :
  later_map (g  f) x = later_map g (later_map f x).
Proof. by destruct x. Qed.
Definition laterC_map {A B} (f : A -n> B) : laterC A -n> laterC B :=
  CofeMor (later_map f).
Instance laterC_map_contractive (A B : cofeT) : Contractive (@laterC_map A B).
384
Proof. intros [|n] f g Hf n'; [done|]; apply Hf; lia. Qed.