cofe.v 29.1 KB
Newer Older
1
From iris.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 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;
55
  mixin_conv_compl n c : compl c {n} c 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

(** Bundeled version *)
61
Structure cofeT := CofeT' {
Robbert Krebbers's avatar
Robbert Krebbers committed
62 63 64 65
  cofe_car :> Type;
  cofe_equiv : Equiv cofe_car;
  cofe_dist : Dist cofe_car;
  cofe_compl : Compl cofe_car;
66 67
  cofe_mixin : CofeMixin cofe_car;
  cofe_car' : Type
Robbert Krebbers's avatar
Robbert Krebbers committed
68
}.
69 70
Arguments CofeT' _ {_ _ _} _ _.
Notation CofeT A m := (CofeT' A m A).
Robbert Krebbers's avatar
Robbert Krebbers committed
71
Add Printing Constructor cofeT.
72 73 74
Hint Extern 0 (Equiv _) => eapply (@cofe_equiv _) : typeclass_instances.
Hint Extern 0 (Dist _) => eapply (@cofe_dist _) : typeclass_instances.
Hint Extern 0 (Compl _) => eapply (@cofe_compl _) : typeclass_instances.
75 76 77 78 79 80 81 82 83 84
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.
85
  Lemma equiv_dist x y : x  y   n, x {n} y.
86 87 88
  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.
89
  Lemma dist_S n x y : x {S n} y  x {n} y.
90
  Proof. apply (mixin_dist_S _ (cofe_mixin A)). Qed.
91
  Lemma conv_compl n (c : chain A) : compl c {n} c n.
92 93 94
  Proof. apply (mixin_conv_compl _ (cofe_mixin A)). Qed.
End cofe_mixin.

Robbert Krebbers's avatar
Robbert Krebbers committed
95 96
Hint Extern 1 (_ {_} _) => apply equiv_dist; assumption.

97
(** Discrete COFEs and Timeless elements *)
Ralf Jung's avatar
Ralf Jung committed
98
(* TODO: On paper, We called these "discrete elements". I think that makes
Ralf Jung's avatar
Ralf Jung committed
99
   more sense. *)
100 101
Class Timeless `{Equiv A, Dist A} (x : A) := timeless y : x {0} y  x  y.
Arguments timeless {_ _ _} _ {_} _ _.
102 103
Class Discrete (A : cofeT) := discrete_timeless (x : A) :> Timeless x.

Robbert Krebbers's avatar
Robbert Krebbers committed
104 105
(** General properties *)
Section cofe.
106 107
  Context {A : cofeT}.
  Implicit Types x y : A.
Robbert Krebbers's avatar
Robbert Krebbers committed
108 109 110
  Global Instance cofe_equivalence : Equivalence (() : relation A).
  Proof.
    split.
111 112
    - by intros x; rewrite equiv_dist.
    - by intros x y; rewrite !equiv_dist.
113
    - by intros x y z; rewrite !equiv_dist; intros; trans y.
Robbert Krebbers's avatar
Robbert Krebbers committed
114
  Qed.
115
  Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
116 117
  Proof.
    intros x1 x2 ? y1 y2 ?; split; intros.
118 119
    - by trans x1; [|trans y1].
    - by trans x2; [|trans y2].
Robbert Krebbers's avatar
Robbert Krebbers committed
120
  Qed.
121
  Global Instance dist_proper n : Proper (() ==> () ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
122
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
123
    by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
124 125 126
  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
127
  Lemma dist_le n n' x y : x {n} y  n'  n  x {n'} y.
Robbert Krebbers's avatar
Robbert Krebbers committed
128
  Proof. induction 2; eauto using dist_S. Qed.
129 130
  Lemma dist_le' n n' x y : n'  n  x {n} y  x {n'} y.
  Proof. intros; eauto using dist_le. Qed.
131
  Instance ne_proper {B : cofeT} (f : A  B)
Robbert Krebbers's avatar
Robbert Krebbers committed
132 133
    `{! 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.
134
  Instance ne_proper_2 {B C : cofeT} (f : A  B  C)
Robbert Krebbers's avatar
Robbert Krebbers committed
135 136 137 138
    `{! 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
139
     by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
140
  Qed.
141
  Lemma contractive_S {B : cofeT} (f : A  B) `{!Contractive f} n x y :
142 143
    x {n} y  f x {S n} f y.
  Proof. eauto using contractive, dist_le with omega. Qed.
144 145 146
  Lemma contractive_0 {B : cofeT} (f : A  B) `{!Contractive f} x y :
    f x {0} f y.
  Proof. eauto using contractive with omega. Qed.
147
  Global Instance contractive_ne {B : cofeT} (f : A  B) `{!Contractive f} n :
148
    Proper (dist n ==> dist n) f | 100.
149
  Proof. by intros x y ?; apply dist_S, contractive_S. Qed.
150
  Global Instance contractive_proper {B : cofeT} (f : A  B) `{!Contractive f} :
151
    Proper (() ==> ()) f | 100 := _.
152

153 154 155 156 157
  Lemma conv_compl' n (c : chain A) : compl c {n} c (S n).
  Proof.
    transitivity (c n); first by apply conv_compl. symmetry.
    apply chain_cauchy. omega.
  Qed.
158 159
  Lemma timeless_iff n (x : A) `{!Timeless x} y : x  y  x {n} y.
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
160
    split; intros; auto. apply (timeless _), dist_le with n; auto with lia.
161
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
162 163
End cofe.

164 165 166
Instance const_contractive {A B : cofeT} (x : A) : Contractive (@const A B x).
Proof. by intros n y1 y2. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
167 168 169 170
(** 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) |}.
171
Next Obligation. by intros ? A ? B f Hf c n i ?; apply Hf, chain_cauchy. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
172

Robbert Krebbers's avatar
Robbert Krebbers committed
173
(** Fixpoint *)
174
Program Definition fixpoint_chain {A : cofeT} `{Inhabited A} (f : A  A)
175
  `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
176
Next Obligation.
177 178
  intros A ? f ? n.
  induction n as [|n IH]; intros [|i] ?; simpl in *; try reflexivity || omega.
179 180
  - apply (contractive_0 f).
  - apply (contractive_S f), IH; auto with omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
181
Qed.
182 183

Program Definition fixpoint_def {A : cofeT} `{Inhabited A} (f : A  A)
184
  `{!Contractive f} : A := compl (fixpoint_chain f).
185 186 187
Definition fixpoint_aux : { x | x = @fixpoint_def }. by eexists. Qed.
Definition fixpoint {A AiH} f {Hf} := proj1_sig fixpoint_aux A AiH f Hf.
Definition fixpoint_eq : @fixpoint = @fixpoint_def := proj2_sig fixpoint_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
188 189

Section fixpoint.
190
  Context {A : cofeT} `{Inhabited A} (f : A  A) `{!Contractive f}.
191
  Lemma fixpoint_unfold : fixpoint f  f (fixpoint f).
Robbert Krebbers's avatar
Robbert Krebbers committed
192
  Proof.
193 194
    apply equiv_dist=>n.
    rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //.
195
    induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
196
  Qed.
197
  Lemma fixpoint_ne (g : A  A) `{!Contractive g} n :
198
    ( z, f z {n} g z)  fixpoint f {n} fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
199
  Proof.
200
    intros Hfg. rewrite fixpoint_eq /fixpoint_def
Robbert Krebbers's avatar
Robbert Krebbers committed
201
      (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=.
202 203
    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
204
  Qed.
205 206
  Lemma fixpoint_proper (g : A  A) `{!Contractive g} :
    ( x, f x  g x)  fixpoint f  fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
207 208 209 210
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.
End fixpoint.

(** Function space *)
Robbert Krebbers's avatar
Robbert Krebbers committed
211
Record cofeMor (A B : cofeT) : Type := CofeMor {
Robbert Krebbers's avatar
Robbert Krebbers committed
212 213 214 215 216 217 218
  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.

219 220 221 222 223
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.
224
  Instance cofe_mor_dist : Dist (cofeMor A B) := λ n f g,  x, f x {n} g x.
225 226 227 228 229 230
  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
231 232
    intros c n x y Hx. by rewrite (conv_compl n (fun_chain c x))
      (conv_compl n (fun_chain c y)) /= Hx.
233 234 235 236
  Qed.
  Definition cofe_mor_cofe_mixin : CofeMixin (cofeMor A B).
  Proof.
    split.
237
    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
Robbert Krebbers's avatar
Robbert Krebbers committed
238
      intros Hfg k; apply equiv_dist=> n; apply Hfg.
239
    - intros n; split.
240 241
      + by intros f x.
      + by intros f g ? x.
242
      + by intros f g h ?? x; trans (g x).
243
    - by intros n f g ? x; apply dist_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
244 245
    - intros n c x; simpl.
      by rewrite (conv_compl n (fun_chain c x)) /=.
246
  Qed.
247
  Canonical Structure cofe_mor : cofeT := CofeT (cofeMor A B) cofe_mor_cofe_mixin.
248 249 250 251 252 253 254 255 256 257 258

  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
259
Infix "-n>" := cofe_mor (at level 45, right associativity).
260 261
Instance cofe_more_inhabited {A B : cofeT} `{Inhabited B} :
  Inhabited (A -n> B) := populate (CofeMor (λ _, inhabitant)).
Robbert Krebbers's avatar
Robbert Krebbers committed
262

263
(** Identity and composition and constant function *)
Robbert Krebbers's avatar
Robbert Krebbers committed
264 265
Definition cid {A} : A -n> A := CofeMor id.
Instance: Params (@cid) 1.
266 267
Definition cconst {A B : cofeT} (x : B) : A -n> B := CofeMor (const x).
Instance: Params (@cconst) 2.
268

Robbert Krebbers's avatar
Robbert Krebbers committed
269 270 271 272 273
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 :
274
  f1 {n} f2  g1 {n} g2  f1  g1 {n} f2  g2.
Robbert Krebbers's avatar
Robbert Krebbers committed
275
Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
276

Ralf Jung's avatar
Ralf Jung committed
277
(* Function space maps *)
278
Definition cofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B')
Ralf Jung's avatar
Ralf Jung committed
279
  (h : A -n> B) : A' -n> B' := g  h  f.
280
Instance cofe_mor_map_ne {A A' B B'} n :
Ralf Jung's avatar
Ralf Jung committed
281
  Proper (dist n ==> dist n ==> dist n ==> dist n) (@cofe_mor_map A A' B B').
282
Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed.
Ralf Jung's avatar
Ralf Jung committed
283

284
Definition cofe_morC_map {A A' B B'} (f : A' -n> A) (g : B -n> B') :
Ralf Jung's avatar
Ralf Jung committed
285
  (A -n> B) -n> (A' -n>  B') := CofeMor (cofe_mor_map f g).
286
Instance cofe_morC_map_ne {A A' B B'} n :
Ralf Jung's avatar
Ralf Jung committed
287 288 289
  Proper (dist n ==> dist n ==> dist n) (@cofe_morC_map A A' B B').
Proof.
  intros f f' Hf g g' Hg ?. rewrite /= /cofe_mor_map.
290
  by repeat apply ccompose_ne.
Ralf Jung's avatar
Ralf Jung committed
291 292
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
293
(** unit *)
294 295 296 297 298
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.
299
  Canonical Structure unitC : cofeT := CofeT unit unit_cofe_mixin.
300
  Global Instance unit_discrete_cofe : Discrete unitC.
Robbert Krebbers's avatar
Robbert Krebbers committed
301
  Proof. done. Qed.
302
End unit.
Robbert Krebbers's avatar
Robbert Krebbers committed
303 304

(** Product *)
305 306 307 308 309 310 311 312 313 314 315 316 317
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.
318
    - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
319
      rewrite !equiv_dist; naive_solver.
320 321
    - apply _.
    - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
322 323
    - intros n c; split. apply (conv_compl n (chain_map fst c)).
      apply (conv_compl n (chain_map snd c)).
324
  Qed.
325
  Canonical Structure prodC : cofeT := CofeT (A * B) prod_cofe_mixin.
326 327 328
  Global Instance prod_timeless (x : A * B) :
    Timeless (x.1)  Timeless (x.2)  Timeless x.
  Proof. by intros ???[??]; split; apply (timeless _). Qed.
329 330
  Global Instance prod_discrete_cofe : Discrete A  Discrete B  Discrete prodC.
  Proof. intros ?? [??]; apply _. Qed.
331 332 333 334 335 336
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
337 338 339 340 341 342 343 344 345
  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.

346 347
(** Functors *)
Structure cFunctor := CFunctor {
348
  cFunctor_car : cofeT  cofeT  cofeT;
349 350
  cFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  cFunctor_car A1 B1 -n> cFunctor_car A2 B2;
351 352
  cFunctor_ne {A1 A2 B1 B2} n :
    Proper (dist n ==> dist n) (@cFunctor_map A1 A2 B1 B2);
353 354 355 356 357 358
  cFunctor_id {A B : cofeT} (x : cFunctor_car A B) :
    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)
}.
359
Existing Instance cFunctor_ne.
360 361
Instance: Params (@cFunctor_map) 5.

362 363 364
Delimit Scope cFunctor_scope with CF.
Bind Scope cFunctor_scope with cFunctor.

365 366 367
Class cFunctorContractive (F : cFunctor) :=
  cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2).

368 369 370 371 372 373 374
Definition cFunctor_diag (F: cFunctor) (A: cofeT) : cofeT := cFunctor_car F A A.
Coercion cFunctor_diag : cFunctor >-> Funclass.

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

375
Instance constCF_contractive B : cFunctorContractive (constCF B).
376
Proof. rewrite /cFunctorContractive; apply _. Qed.
377 378 379 380 381

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

382 383 384 385 386
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)
|}.
387 388 389
Next Obligation.
  intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne.
Qed.
390 391 392 393 394 395
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.

396 397 398 399 400 401 402 403
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.

Ralf Jung's avatar
Ralf Jung committed
404 405 406 407 408
Program Definition cofe_morCF (F1 F2 : cFunctor) : cFunctor := {|
  cFunctor_car A B := cofe_mor (cFunctor_car F1 B A) (cFunctor_car F2 A B);
  cFunctor_map A1 A2 B1 B2 fg :=
    cofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg)
|}.
409 410 411 412
Next Obligation.
  intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *.
  apply cofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg.
Qed.
Ralf Jung's avatar
Ralf Jung committed
413
Next Obligation.
414 415
  intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id.
  apply (ne_proper f). apply cFunctor_id.
Ralf Jung's avatar
Ralf Jung committed
416 417
Qed.
Next Obligation.
418 419
  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
420 421
Qed.

422 423 424 425 426 427 428 429
Instance cofe_morCF_contractive F1 F2 :
  cFunctorContractive F1  cFunctorContractive F2 
  cFunctorContractive (cofe_morCF F1 F2).
Proof.
  intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *.
  apply cofe_morC_map_ne; apply cFunctor_contractive=>i ?; split; by apply Hfg.
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512
(** Sum *)
Section sum.
  Context {A B : cofeT}.

  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) := _.

  Program Definition inl_chain (c : chain (A + B)) (a : A) : chain A :=
    {| 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.
  Program Definition inr_chain (c : chain (A + B)) (b : B) : chain B :=
    {| 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.

  Instance sum_compl : Compl (A + B) := λ c,
    match c 0 with
    | inl a => inl (compl (inl_chain c a))
    | inr b => inr (compl (inr_chain c b))
    end.

  Definition sum_cofe_mixin : CofeMixin (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.
    - intros n c; rewrite /compl /sum_compl.
      feed inversion (chain_cauchy c 0 n); first 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.
  Qed.
  Canonical Structure sumC : cofeT := CofeT (A + B) sum_cofe_mixin.

  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.

Instance sum_map_ne {A A' B B' : cofeT} n :
  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.

513 514 515
(** Discrete cofe *)
Section discrete_cofe.
  Context `{Equiv A, @Equivalence A ()}.
516
  Instance discrete_dist : Dist A := λ n x y, x  y.
517
  Instance discrete_compl : Compl A := λ c, c 0.
518
  Definition discrete_cofe_mixin : CofeMixin A.
519 520
  Proof.
    split.
521 522 523
    - intros x y; split; [done|intros Hn; apply (Hn 0)].
    - done.
    - done.
524 525
    - intros n c. rewrite /compl /discrete_compl /=;
      symmetry; apply (chain_cauchy c 0 n). omega.
526 527 528
  Qed.
End discrete_cofe.

529 530 531 532 533 534 535 536
Notation discreteC A := (CofeT A discrete_cofe_mixin).
Notation leibnizC A := (CofeT A (@discrete_cofe_mixin _ equivL _)).

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.
537

Robbert Krebbers's avatar
Robbert Krebbers committed
538 539
Canonical Structure natC := leibnizC nat.
Canonical Structure boolC := leibnizC bool.
540

541 542 543 544
(* Option *)
Section option.
  Context {A : cofeT}.

545
  Instance option_dist : Dist (option A) := λ n, option_Forall2 (dist n).
546
  Lemma dist_option_Forall2 n mx my : mx {n} my  option_Forall2 (dist n) mx my.
547
  Proof. done. Qed.
548 549

  Program Definition option_chain (c : chain (option A)) (x : A) : chain A :=
550
    {| chain_car n := from_option id x (c n) |}.
551 552 553 554 555 556 557 558 559 560
  Next Obligation. intros c x n i ?; simpl. by destruct (chain_cauchy c n i). Qed.
  Instance option_compl : Compl (option A) := λ c,
    match c 0 with Some x => Some (compl (option_chain c x)) | None => None end.

  Definition option_cofe_mixin : CofeMixin (option A).
  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).
561
    - apply _.
562 563 564 565 566 567 568 569 570 571 572 573 574 575 576
    - destruct 1; constructor; by apply dist_S.
    - intros n c; rewrite /compl /option_compl.
      feed inversion (chain_cauchy c 0 n); first auto with lia; constructor.
      rewrite (conv_compl n (option_chain c _)) /=. destruct (c n); naive_solver.
  Qed.
  Canonical Structure optionC := CofeT (option A) option_cofe_mixin.
  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
577 578 579
  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.
580 581 582 583 584

  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.
585 586 587 588 589 590 591 592 593 594 595 596 597

  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.
598 599
End option.

600
Typeclasses Opaque option_dist.
601 602
Arguments optionC : clear implicits.

Robbert Krebbers's avatar
Robbert Krebbers committed
603 604 605
Instance option_fmap_ne {A B : cofeT} n:
  Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@fmap option _ A B).
Proof. intros f f' Hf ?? []; constructor; auto. Qed.
606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632
Definition optionC_map {A B} (f : A -n> B) : optionC A -n> optionC B :=
  CofeMor (fmap f : optionC A  optionC B).
Instance optionC_map_ne A B n : Proper (dist n ==> dist n) (@optionC_map A B).
Proof. by intros f f' Hf []; constructor; apply Hf. Qed.

Program Definition optionCF (F : cFunctor) : cFunctor := {|
  cFunctor_car A B := optionC (cFunctor_car F A B);
  cFunctor_map A1 A2 B1 B2 fg := optionC_map (cFunctor_map F fg)
|}.
Next Obligation.
  by intros F A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, cFunctor_ne.
Qed.
Next Obligation.
  intros F A B x. rewrite /= -{2}(option_fmap_id x).
  apply option_fmap_setoid_ext=>y; apply cFunctor_id.
Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -option_fmap_compose.
  apply option_fmap_setoid_ext=>y; apply cFunctor_compose.
Qed.

Instance optionCF_contractive F :
  cFunctorContractive F  cFunctorContractive (optionCF F).
Proof.
  by intros ? A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, cFunctor_contractive.
Qed.

633
(** Later *)
634
Inductive later (A : Type) : Type := Next { later_car : A }.
635
Add Printing Constructor later.
636
Arguments Next {_} _.
637
Arguments later_car {_} _.
638
Lemma later_eta {A} (x : later A) : Next (later_car x) = x.
Robbert Krebbers's avatar
Robbert Krebbers committed
639
Proof. by destruct x. Qed.
640

641
Section later.
642 643 644
  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,
645
    match n with 0 => True | S n => later_car x {n} later_car y end.
646
  Program Definition later_chain (c : chain (later A)) : chain A :=
647
    {| chain_car n := later_car (c (S n)) |}.
648
  Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed.
649
  Instance later_compl : Compl (later A) := λ c, Next (compl (later_chain c)).
650
  Definition later_cofe_mixin : CofeMixin (later A).
651 652
  Proof.
    split.
653 654
    - intros x y; unfold equiv, later_equiv; rewrite !equiv_dist.
      split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)).
655
    - intros [|n]; [by split|split]; unfold dist, later_dist.
656 657
      + by intros [x].
      + by intros [x] [y].
658
      + by intros [x] [y] [z] ??; trans y.
659
    - intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
660
    - intros [|n] c; [done|by apply (conv_compl n (later_chain c))].
661
  Qed.
662
  Canonical Structure laterC : cofeT := CofeT (later A) later_cofe_mixin.
663 664
  Global Instance Next_contractive : Contractive (@Next A).
  Proof. intros [|n] x y Hxy; [done|]; apply Hxy; lia. Qed.
665
  Global Instance Later_inj n : Inj (dist n) (dist (S n)) (@Next A).
Robbert Krebbers's avatar
Robbert Krebbers committed
666
  Proof. by intros x y. Qed.
667
End later.
668 669 670 671

Arguments laterC : clear implicits.

Definition later_map {A B} (f : A  B) (x : later A) : later B :=
672
  Next (f (later_car x)).
673 674 675 676 677 678 679 680 681
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.
682 683 684
Lemma later_map_ext {A B : cofeT} (f g : A  B) x :
  ( x, f x  g x)  later_map f x  later_map g x.
Proof. destruct x; intros Hf; apply Hf. Qed.
685 686 687
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).
688
Proof. intros [|n] f g Hf n'; [done|]; apply Hf; lia. Qed.
689

690 691 692
Program Definition laterCF (F : cFunctor) : cFunctor := {|
  cFunctor_car A B := laterC (cFunctor_car F A B);
  cFunctor_map A1 A2 B1 B2 fg := laterC_map (cFunctor_map F fg)
693
|}.
694 695 696 697
Next Obligation.
  intros F A1 A2 B1 B2 n fg fg' ?.
  by apply (contractive_ne laterC_map), cFunctor_ne.
Qed.
698
Next Obligation.
699 700 701 702 703 704 705 706
  intros F A B x; simpl. rewrite -{2}(later_map_id x).
  apply later_map_ext=>y. by rewrite cFunctor_id.
Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x; simpl. rewrite -later_map_compose.
  apply later_map_ext=>y; apply cFunctor_compose.
Qed.

707
Instance laterCF_contractive F : cFunctorContractive (laterCF F).
708
Proof.
709
  intros A1 A2 B1 B2 n fg fg' Hfg.
710
  apply laterC_map_contractive => i ?. by apply cFunctor_ne, Hfg.
711
Qed.
712 713 714 715

(** Notation for writing functors *)
Notation "∙" := idCF : cFunctor_scope.
Notation "F1 -n> F2" := (cofe_morCF F1%CF F2%CF) : cFunctor_scope.
716
Notation "F1 * F2" := (prodCF F1%CF F2%CF) : cFunctor_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
717
Notation "F1 + F2" := (sumCF F1%CF F2%CF) : cFunctor_scope.
718 719
Notation "▶ F"  := (laterCF F%CF) (at level 20, right associativity) : cFunctor_scope.
Coercion constCF : cofeT >-> cFunctor.