ofe.v 44.2 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
Notation NonExpansive f := ( n, Proper (dist n ==> dist n) f).
Notation NonExpansive2 f := ( n, Proper (dist n ==> dist n ==> dist n) f).
22

23
Tactic Notation "ofe_subst" ident(x) :=
24
  repeat match goal with
25
  | _ => progress simplify_eq/=
26 27 28
  | 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.
29
Tactic Notation "ofe_subst" :=
30
  repeat match goal with
31
  | _ => progress simplify_eq/=
32 33
  | 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
34
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
35

36
Record OfeMixin A `{Equiv A, Dist A} := {
37
  mixin_equiv_dist x y : x  y   n, x {n} y;
38
  mixin_dist_equivalence n : Equivalence (dist n);
39
  mixin_dist_S n x y : x {S n} y  x {n} y
Robbert Krebbers's avatar
Robbert Krebbers committed
40 41 42
}.

(** Bundeled version *)
43 44 45 46 47
Structure ofeT := OfeT' {
  ofe_car :> Type;
  ofe_equiv : Equiv ofe_car;
  ofe_dist : Dist ofe_car;
  ofe_mixin : OfeMixin ofe_car;
48
  _ : Type
Robbert Krebbers's avatar
Robbert Krebbers committed
49
}.
50 51 52 53 54 55 56 57 58
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.
59

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
(** When declaring instances of subclasses of OFE (like CMRAs and unital CMRAs)
we need Coq to *infer* the canonical OFE instance of a given type and take the
mixin out of it. This makes sure we do not use two different OFE instances in
different places (see for example the constructors [CMRAT] and [UCMRAT] in the
file [cmra.v].)

In order to infer the OFE instance, we use the definition [ofe_mixin_of'] which
is inspired by the [clone] trick in ssreflect. It works as follows, when type
checking [@ofe_mixin_of' A ?Ac id] Coq faces a unification problem:

  ofe_car ?Ac  ~  A

which will resolve [?Ac] to the canonical OFE instance corresponding to [A]. The
definition [@ofe_mixin_of' A ?Ac id] will then provide the corresponding mixin.
Note that type checking of [ofe_mixin_of' A id] will fail when [A] does not have
a canonical OFE instance.

The notation [ofe_mixin_of A] that we define on top of [ofe_mixin_of' A id]
hides the [id] and normalizes the mixin to head normal form. The latter is to
ensure that we do not end up with redundant canonical projections to the mixin,
i.e. them all being of the shape [ofe_mixin_of' A id]. *)
Definition ofe_mixin_of' A {Ac : ofeT} (f : Ac  A) : OfeMixin Ac := ofe_mixin Ac.
Notation ofe_mixin_of A :=
  ltac:(let H := eval hnf in (ofe_mixin_of' A id) in exact H) (only parsing).

85
(** Lifting properties from the mixin *)
86 87
Section ofe_mixin.
  Context {A : ofeT}.
88
  Implicit Types x y : A.
89
  Lemma equiv_dist x y : x  y   n, x {n} y.
90
  Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed.
91
  Global Instance dist_equivalence n : Equivalence (@dist A _ n).
92
  Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed.
93
  Lemma dist_S n x y : x {S n} y  x {n} y.
94 95
  Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed.
End ofe_mixin.
96

Robbert Krebbers's avatar
Robbert Krebbers committed
97 98
Hint Extern 1 (_ {_} _) => apply equiv_dist; assumption.

99
(** Discrete OFEs and Timeless elements *)
Ralf Jung's avatar
Ralf Jung committed
100
(* TODO: On paper, We called these "discrete elements". I think that makes
Ralf Jung's avatar
Ralf Jung committed
101
   more sense. *)
102 103 104 105
Class Timeless {A : ofeT} (x : A) := timeless y : x {0} y  x  y.
Arguments timeless {_} _ {_} _ _.
Hint Mode Timeless + ! : typeclass_instances.

106 107 108 109 110 111 112 113 114 115
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 {_} _ _ _ _.

116
Program Definition chain_map {A B : ofeT} (f : A  B)
117
    `{!NonExpansive f} (c : chain A) : chain B :=
118 119 120
  {| chain_car n := f (c n) |}.
Next Obligation. by intros A B f Hf c n i ?; apply Hf, chain_cauchy. Qed.

121 122 123 124 125 126
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.
127

128
Lemma compl_chain_map `{Cofe A, Cofe B} (f : A  B) c `(NonExpansive f) :
129 130 131
  compl (chain_map f c)  f (compl c).
Proof. apply equiv_dist=>n. by rewrite !conv_compl. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
132
(** General properties *)
133
Section ofe.
134
  Context {A : ofeT}.
135
  Implicit Types x y : A.
136
  Global Instance ofe_equivalence : Equivalence (() : relation A).
Robbert Krebbers's avatar
Robbert Krebbers committed
137 138
  Proof.
    split.
139 140
    - by intros x; rewrite equiv_dist.
    - by intros x y; rewrite !equiv_dist.
141
    - by intros x y z; rewrite !equiv_dist; intros; trans y.
Robbert Krebbers's avatar
Robbert Krebbers committed
142
  Qed.
143
  Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
144 145
  Proof.
    intros x1 x2 ? y1 y2 ?; split; intros.
146 147
    - by trans x1; [|trans y1].
    - by trans x2; [|trans y2].
Robbert Krebbers's avatar
Robbert Krebbers committed
148
  Qed.
149
  Global Instance dist_proper n : Proper (() ==> () ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
150
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
151
    by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
152 153 154
  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
155
  Lemma dist_le n n' x y : x {n} y  n'  n  x {n'} y.
Robbert Krebbers's avatar
Robbert Krebbers committed
156
  Proof. induction 2; eauto using dist_S. Qed.
157 158
  Lemma dist_le' n n' x y : n'  n  x {n} y  x {n'} y.
  Proof. intros; eauto using dist_le. Qed.
159
  Instance ne_proper {B : ofeT} (f : A  B)
160
    `{!NonExpansive f} : Proper (() ==> ()) f | 100.
Robbert Krebbers's avatar
Robbert Krebbers committed
161
  Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed.
162
  Instance ne_proper_2 {B C : ofeT} (f : A  B  C)
163
    `{!NonExpansive2 f} :
Robbert Krebbers's avatar
Robbert Krebbers committed
164 165 166
    Proper (() ==> () ==> ()) f | 100.
  Proof.
     unfold Proper, respectful; setoid_rewrite equiv_dist.
Robbert Krebbers's avatar
Robbert Krebbers committed
167
     by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
168
  Qed.
169

170
  Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c {n} c (S n).
171 172 173 174
  Proof.
    transitivity (c n); first by apply conv_compl. symmetry.
    apply chain_cauchy. omega.
  Qed.
175 176
  Lemma timeless_iff n (x : A) `{!Timeless x} y : x  y  x {n} y.
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
177
    split; intros; auto. apply (timeless _), dist_le with n; auto with lia.
178
  Qed.
179
End ofe.
Robbert Krebbers's avatar
Robbert Krebbers committed
180

181
(** Contractive functions *)
182 183 184 185 186 187 188 189
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).
190

191
Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x).
192 193
Proof. by intros n y1 y2. Qed.

194
Section contractive.
195
  Local Set Default Proof Using "Type*".
196 197 198 199
  Context {A B : ofeT} (f : A  B) `{!Contractive f}.
  Implicit Types x y : A.

  Lemma contractive_0 x y : f x {0} f y.
200
  Proof. by apply (_ : Contractive f). Qed.
201
  Lemma contractive_S n x y : x {n} y  f x {S n} f y.
202
  Proof. intros. by apply (_ : Contractive f). Qed.
203

204 205
  Global Instance contractive_ne : NonExpansive f | 100.
  Proof. by intros n x y ?; apply dist_S, contractive_S. Qed.
206 207 208 209
  Global Instance contractive_proper : Proper (() ==> ()) f | 100.
  Proof. apply (ne_proper _). Qed.
End contractive.

210 211 212 213 214 215 216
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
217 218
  | |- @dist_later ?A ?n ?x ?y =>
         destruct n as [|n]; [done|change (@dist A _ n x y)]
219 220 221 222 223 224
  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
225

Robbert Krebbers's avatar
Robbert Krebbers committed
226
(** Fixpoint *)
227
Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A  A)
228
  `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
229
Next Obligation.
230
  intros A ? f ? n.
231
  induction n as [|n IH]=> -[|i] //= ?; try omega.
232 233
  - apply (contractive_0 f).
  - apply (contractive_S f), IH; auto with omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
234
Qed.
235

236
Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A  A)
237
  `{!Contractive f} : A := compl (fixpoint_chain f).
238 239 240
Definition fixpoint_aux : seal (@fixpoint_def). by eexists. Qed.
Definition fixpoint {A AC AiH} f {Hf} := unseal fixpoint_aux A AC AiH f Hf.
Definition fixpoint_eq : @fixpoint = @fixpoint_def := seal_eq fixpoint_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
241 242

Section fixpoint.
243
  Context `{Cofe A, Inhabited A} (f : A  A) `{!Contractive f}.
244

245
  Lemma fixpoint_unfold : fixpoint f  f (fixpoint f).
Robbert Krebbers's avatar
Robbert Krebbers committed
246
  Proof.
247 248
    apply equiv_dist=>n.
    rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //.
249
    induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
250
  Qed.
251 252 253

  Lemma fixpoint_unique (x : A) : x  f x  x  fixpoint f.
  Proof.
254 255 256
    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.
257 258
  Qed.

259
  Lemma fixpoint_ne (g : A  A) `{!Contractive g} n :
260
    ( z, f z {n} g z)  fixpoint f {n} fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
261
  Proof.
262
    intros Hfg. rewrite fixpoint_eq /fixpoint_def
Robbert Krebbers's avatar
Robbert Krebbers committed
263
      (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=.
264 265
    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
266
  Qed.
267 268
  Lemma fixpoint_proper (g : A  A) `{!Contractive g} :
    ( x, f x  g x)  fixpoint f  fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
269
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.
270 271

  Lemma fixpoint_ind (P : A  Prop) :
272
    Proper (() ==> impl) P 
273 274 275 276 277 278 279 280 281 282 283 284 285 286 287
    ( 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
288 289
End fixpoint.

290 291 292
(** Fixpoint of f when f^k is contractive. **)
Definition fixpointK `{Cofe A, Inhabited A} k (f : A  A)
  `{!Contractive (Nat.iter k f)} := fixpoint (Nat.iter k f).
293

294
Section fixpointK.
295
  Local Set Default Proof Using "Type*".
296 297
  Context `{Cofe A, Inhabited A} (f : A  A) (k : nat).
  Context `{f_contractive : !Contractive (Nat.iter k f)}.
298
  (* TODO: Can we get rid of this assumption, derive it from contractivity? *)
299
  Context {f_ne : NonExpansive f}.
300 301 302

  Let f_proper : Proper (() ==> ()) f := ne_proper f.
  Existing Instance f_proper.
303

304
  Lemma fixpointK_unfold : fixpointK k f  f (fixpointK k f).
305
  Proof.
306 307
    symmetry. rewrite /fixpointK. apply fixpoint_unique.
    by rewrite -Nat_iter_S_r Nat_iter_S -fixpoint_unfold.
308 309
  Qed.

310
  Lemma fixpointK_unique (x : A) : x  f x  x  fixpointK k f.
311
  Proof.
312 313
    intros Hf. apply fixpoint_unique. clear f_contractive.
    induction k as [|k' IH]=> //=. by rewrite -IH.
314 315
  Qed.

316
  Section fixpointK_ne.
317
    Context (g : A  A) `{g_contractive : !Contractive (Nat.iter k g)}.
318
    Context {g_ne : NonExpansive g}.
319

320
    Lemma fixpointK_ne n : ( z, f z {n} g z)  fixpointK k f {n} fixpointK k g.
321
    Proof.
322 323 324
      rewrite /fixpointK=> Hfg /=. apply fixpoint_ne=> z.
      clear f_contractive g_contractive.
      induction k as [|k' IH]=> //=. by rewrite IH Hfg.
325 326
    Qed.

327 328 329 330
    Lemma fixpointK_proper : ( z, f z  g z)  fixpointK k f  fixpointK k g.
    Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpointK_ne. Qed.
  End fixpointK_ne.
End fixpointK.
331

Robbert Krebbers's avatar
Robbert Krebbers committed
332
(** Mutual fixpoints *)
Ralf Jung's avatar
Ralf Jung committed
333
Section fixpointAB.
334 335
  Local Unset Default Proof Using.

Robbert Krebbers's avatar
Robbert Krebbers committed
336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376
  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.
Ralf Jung's avatar
Ralf Jung committed
377
End fixpointAB.
Robbert Krebbers's avatar
Robbert Krebbers committed
378

Ralf Jung's avatar
Ralf Jung committed
379
Section fixpointAB_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410
  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.
Ralf Jung's avatar
Ralf Jung committed
411
End fixpointAB_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
412

413
(** Function space *)
414
(* We make [ofe_fun] a definition so that we can register it as a canonical
415
structure. *)
416
Definition ofe_fun (A : Type) (B : ofeT) := A  B.
417

418 419 420 421 422
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).
423 424 425 426 427 428 429 430 431 432
  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.
433
  Canonical Structure ofe_funC := OfeT (ofe_fun A B) ofe_fun_ofe_mixin.
434

435 436 437 438 439 440 441 442 443
  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.
444
Notation "A -c> B" :=
445 446
  (ofe_funC A B) (at level 99, B at level 200, right associativity).
Instance ofe_fun_inhabited {A} {B : ofeT} `{Inhabited B} :
447 448
  Inhabited (A -c> B) := populate (λ _, inhabitant).

449
(** Non-expansive function space *)
450 451
Record ofe_mor (A B : ofeT) : Type := CofeMor {
  ofe_mor_car :> A  B;
452
  ofe_mor_ne : NonExpansive ofe_mor_car
Robbert Krebbers's avatar
Robbert Krebbers committed
453 454
}.
Arguments CofeMor {_ _} _ {_}.
455 456
Add Printing Constructor ofe_mor.
Existing Instance ofe_mor_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
457

458 459 460 461
Notation "'λne' x .. y , t" :=
  (@CofeMor _ _ (λ x, .. (@CofeMor _ _ (λ y, t) _) ..) _)
  (at level 200, x binder, y binder, right associativity).

462 463 464 465 466 467 468
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).
469 470
  Proof.
    split.
471
    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
Robbert Krebbers's avatar
Robbert Krebbers committed
472
      intros Hfg k; apply equiv_dist=> n; apply Hfg.
473
    - intros n; split.
474 475
      + by intros f x.
      + by intros f g ? x.
476
      + by intros f g h ?? x; trans (g x).
477
    - by intros n f g ? x; apply dist_S.
478
  Qed.
479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495
  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.
496

497 498 499
  Global Instance ofe_mor_car_ne :
    NonExpansive2 (@ofe_mor_car A B).
  Proof. intros n f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed.
500 501 502
  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.
503
  Proof. done. Qed.
504
End ofe_mor.
505

506
Arguments ofe_morC : clear implicits.
507
Notation "A -n> B" :=
508 509
  (ofe_morC A B) (at level 99, B at level 200, right associativity).
Instance ofe_mor_inhabited {A B : ofeT} `{Inhabited B} :
510
  Inhabited (A -n> B) := populate (λne _, inhabitant).
Robbert Krebbers's avatar
Robbert Krebbers committed
511

512
(** Identity and composition and constant function *)
Robbert Krebbers's avatar
Robbert Krebbers committed
513 514
Definition cid {A} : A -n> A := CofeMor id.
Instance: Params (@cid) 1.
515
Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x).
516
Instance: Params (@cconst) 2.
517

Robbert Krebbers's avatar
Robbert Krebbers committed
518 519 520 521 522
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 :
523
  f1 {n} f2  g1 {n} g2  f1  g1 {n} f2  g2.
Robbert Krebbers's avatar
Robbert Krebbers committed
524
Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
525

Ralf Jung's avatar
Ralf Jung committed
526
(* Function space maps *)
527
Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B')
Ralf Jung's avatar
Ralf Jung committed
528
  (h : A -n> B) : A' -n> B' := g  h  f.
529 530
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').
531
Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed.
Ralf Jung's avatar
Ralf Jung committed
532

533 534
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).
535 536
Instance ofe_morC_map_ne {A A' B B'} :
  NonExpansive2 (@ofe_morC_map A A' B B').
Ralf Jung's avatar
Ralf Jung committed
537
Proof.
538
  intros n f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map.
539
  by repeat apply ccompose_ne.
Ralf Jung's avatar
Ralf Jung committed
540 541
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
542
(** unit *)
543 544
Section unit.
  Instance unit_dist : Dist unit := λ _ _ _, True.
545
  Definition unit_ofe_mixin : OfeMixin unit.
546
  Proof. by repeat split; try exists 0. Qed.
547
  Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin.
Robbert Krebbers's avatar
Robbert Krebbers committed
548

549 550
  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
551 552

  Global Instance unit_discrete_cofe : Discrete unitC.
Robbert Krebbers's avatar
Robbert Krebbers committed
553
  Proof. done. Qed.
554
End unit.
Robbert Krebbers's avatar
Robbert Krebbers committed
555 556

(** Product *)
557
Section product.
558
  Context {A B : ofeT}.
559 560 561

  Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n).
  Global Instance pair_ne :
562 563 564
    NonExpansive2 (@pair A B) := _.
  Global Instance fst_ne : NonExpansive (@fst A B) := _.
  Global Instance snd_ne : NonExpansive (@snd A B) := _.
565
  Definition prod_ofe_mixin : OfeMixin (A * B).
566 567
  Proof.
    split.
568
    - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
569
      rewrite !equiv_dist; naive_solver.
570 571
    - apply _.
    - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
572
  Qed.
573 574 575 576 577 578 579 580 581
  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.

582 583 584
  Global Instance prod_timeless (x : A * B) :
    Timeless (x.1)  Timeless (x.2)  Timeless x.
  Proof. by intros ???[??]; split; apply (timeless _). Qed.
585 586
  Global Instance prod_discrete_cofe : Discrete A  Discrete B  Discrete prodC.
  Proof. intros ?? [??]; apply _. Qed.
587 588 589 590 591
End product.

Arguments prodC : clear implicits.
Typeclasses Opaque prod_dist.

592
Instance prod_map_ne {A A' B B' : ofeT} n :
Robbert Krebbers's avatar
Robbert Krebbers committed
593 594 595 596 597
  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).
598 599 600
Instance prodC_map_ne {A A' B B'} :
  NonExpansive2 (@prodC_map A A' B B').
Proof. intros n f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
601

602 603
(** Functors *)
Structure cFunctor := CFunctor {
604
  cFunctor_car : ofeT  ofeT  ofeT;
605 606
  cFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  cFunctor_car A1 B1 -n> cFunctor_car A2 B2;
607 608
  cFunctor_ne {A1 A2 B1 B2} :
    NonExpansive (@cFunctor_map A1 A2 B1 B2);
609
  cFunctor_id {A B : ofeT} (x : cFunctor_car A B) :
610 611 612 613 614
    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)
}.
615
Existing Instance cFunctor_ne.
616 617
Instance: Params (@cFunctor_map) 5.

618 619 620
Delimit Scope cFunctor_scope with CF.
Bind Scope cFunctor_scope with cFunctor.

621 622 623
Class cFunctorContractive (F : cFunctor) :=
  cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2).

624
Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A.
625 626
Coercion cFunctor_diag : cFunctor >-> Funclass.

627
Program Definition constCF (B : ofeT) : cFunctor :=
628 629
  {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}.
Solve Obligations with done.
630
Coercion constCF : ofeT >-> cFunctor.
631

632
Instance constCF_contractive B : cFunctorContractive (constCF B).
633
Proof. rewrite /cFunctorContractive; apply _. Qed.
634 635 636 637

Program Definition idCF : cFunctor :=
  {| cFunctor_car A1 A2 := A2; cFunctor_map A1 A2 B1 B2 f := f.2 |}.
Solve Obligations with done.
638
Notation "∙" := idCF : cFunctor_scope.
639

640 641 642 643 644
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)
|}.
645 646 647
Next Obligation.
  intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne.
Qed.
648 649 650 651 652
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.
653
Notation "F1 * F2" := (prodCF F1%CF F2%CF) : cFunctor_scope.
654

655 656 657 658 659 660 661 662
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.

663 664 665
Instance compose_ne {A} {B B' : ofeT} (f : B -n> B') :
  NonExpansive (compose f : (A -c> B)  A -c> B').
Proof. intros n g g' Hf x; simpl. by rewrite (Hf x). Qed.
666

667
Definition ofe_funC_map {A B B'} (f : B -n> B') : (A -c> B) -n> (A -c> B') :=
668
  @CofeMor (_ -c> _) (_ -c> _) (compose f) _.
669 670 671
Instance ofe_funC_map_ne {A B B'} :
  NonExpansive (@ofe_funC_map A B B').
Proof. intros n f f' Hf g x. apply Hf. Qed.
672

673 674 675
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)
676 677
|}.
Next Obligation.
678
  intros ?? A1 A2 B1 B2 n ???; by apply ofe_funC_map_ne; apply cFunctor_ne.
679 680 681 682 683 684
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.
685
Notation "T -c> F" := (ofe_funCF T%type F%CF) : cFunctor_scope.
686

687 688
Instance ofe_funCF_contractive (T : Type) (F : cFunctor) :
  cFunctorContractive F  cFunctorContractive (ofe_funCF T F).
689 690
Proof.
  intros ?? A1 A2 B1 B2 n ???;
691
    by apply ofe_funC_map_ne; apply cFunctor_contractive.
692 693
Qed.

694
Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {|
695
  cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B;
Ralf Jung's avatar
Ralf Jung committed
696
  cFunctor_map A1 A2 B1 B2 fg :=
697
    ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg)
Ralf Jung's avatar
Ralf Jung committed
698
|}.
699 700
Next Obligation.
  intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *.
701
  apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg.
702
Qed.
Ralf Jung's avatar
Ralf Jung committed
703
Next Obligation.
704 705
  intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id.
  apply (ne_proper f). apply cFunctor_id.
Ralf Jung's avatar
Ralf Jung committed
706 707
Qed.
Next Obligation.
708 709
  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
710
Qed.
711
Notation "F1 -n> F2" := (ofe_morCF F1%CF F2%CF) : cFunctor_scope.
Ralf Jung's avatar
Ralf Jung committed
712

713
Instance ofe_morCF_contractive F1 F2 :
714
  cFunctorContractive F1  cFunctorContractive F2 
715
  cFunctorContractive (ofe_morCF F1 F2).
716 717
Proof.
  intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *.
718
  apply ofe_morC_map_ne; apply cFunctor_contractive; destruct n, Hfg; by split.
719 720
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
721 722
(** Sum *)
Section sum.
723
  Context {A B : ofeT}.
Robbert Krebbers's avatar
Robbert Krebbers committed
724 725

  Instance sum_dist : Dist (A + B) := λ n, sum_relation (dist n) (dist n).
726 727
  Global Instance inl_ne : NonExpansive (@inl A B) := _.
  Global Instance inr_ne : NonExpansive (@inr A B) := _.
Robbert Krebbers's avatar
Robbert Krebbers committed
728 729 730
  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) := _.

731 732 733 734 735 736 737 738 739 740 741 742
  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
743 744
    {| 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.
745
  Program Definition inr_chain (c : chain sumC) (b : B) : chain B :=
Robbert Krebbers's avatar
Robbert Krebbers committed
746 747 748
    {| 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.

749
  Definition sum_compl `{Cofe A, Cofe B} : Compl sumC := λ c,
Robbert Krebbers's avatar
Robbert Krebbers committed
750 751 752 753
    match c 0 with
    | inl a => inl (compl (inl_chain c a))
    | inr b => inr (compl (inr_chain c b))
    end.
754 755 756 757 758 759 760