ofe.v 60.1 KB
Newer Older
1
From iris.algebra Require Export base.
2
Set Default Proof Using "Type".
3
Set Primitive Projections.
Robbert Krebbers's avatar
Robbert Krebbers committed
4

5
(** This files defines (a shallow embedding of) the category of OFEs:
6 7 8 9
    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.
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
10
    This makes writing such functions much easier. It turns out that it many
11 12 13
    cases, we do not even need non-expansiveness.
*)

Paolo G. Giarrusso's avatar
Paolo G. Giarrusso committed
14
(** Unbundled version *)
Robbert Krebbers's avatar
Robbert Krebbers committed
15
Class Dist A := dist : nat  relation A.
16
Instance: Params (@dist) 3 := {}.
17 18
Notation "x ≡{ n }≡ y" := (dist n x y)
  (at level 70, n at next level, format "x  ≡{ n }≡  y").
19 20 21
Notation "x ≡{ n }@{ A }≡ y" := (dist (A:=A) n x y)
  (at level 70, n at next level, only parsing).

Tej Chajed's avatar
Tej Chajed committed
22 23
Hint Extern 0 (_ {_} _) => reflexivity : core.
Hint Extern 0 (_ {_} _) => symmetry; assumption : core.
24 25
Notation NonExpansive f := ( n, Proper (dist n ==> dist n) f).
Notation NonExpansive2 f := ( n, Proper (dist n ==> dist n ==> dist n) f).
26

27
Tactic Notation "ofe_subst" ident(x) :=
28
  repeat match goal with
29
  | _ => progress simplify_eq/=
30 31 32
  | 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.
33
Tactic Notation "ofe_subst" :=
34
  repeat match goal with
35
  | _ => progress simplify_eq/=
36 37
  | 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
38
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
39

40 41 42 43 44
Record OfeMixin A `{Equiv A, Dist A} := {
  mixin_equiv_dist x y : x  y   n, x {n} y;
  mixin_dist_equivalence n : Equivalence (dist n);
  mixin_dist_S n x y : x {S n} y  x {n} y
}.
Robbert Krebbers's avatar
Robbert Krebbers committed
45

Paolo G. Giarrusso's avatar
Paolo G. Giarrusso committed
46
(** Bundled version *)
47
Structure ofeT := OfeT {
48 49 50
  ofe_car :> Type;
  ofe_equiv : Equiv ofe_car;
  ofe_dist : Dist ofe_car;
51
  ofe_mixin : OfeMixin ofe_car
Robbert Krebbers's avatar
Robbert Krebbers committed
52
}.
53
Arguments OfeT _ {_ _} _.
54 55 56 57 58 59 60
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.
61

62 63 64
(** 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
65
different places (see for example the constructors [CmraT] and [UcmraT] in the
66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
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).

87
(** Lifting properties from the mixin *)
88 89
Section ofe_mixin.
  Context {A : ofeT}.
90
  Implicit Types x y : A.
91
  Lemma equiv_dist x y : x  y   n, x {n} y.
92
  Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed.
93
  Global Instance dist_equivalence n : Equivalence (@dist A _ n).
94
  Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed.
95
  Lemma dist_S n x y : x {S n} y  x {n} y.
96 97
  Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed.
End ofe_mixin.
98

Tej Chajed's avatar
Tej Chajed committed
99
Hint Extern 1 (_ {_} _) => apply equiv_dist; assumption : core.
Robbert Krebbers's avatar
Robbert Krebbers committed
100

101 102 103 104
(** Discrete OFEs and discrete OFE elements *)
Class Discrete {A : ofeT} (x : A) := discrete y : x {0} y  x  y.
Arguments discrete {_} _ {_} _ _.
Hint Mode Discrete + ! : typeclass_instances.
105
Instance: Params (@Discrete) 1 := {}.
106

107
Class OfeDiscrete (A : ofeT) := ofe_discrete_discrete (x : A) :> Discrete x.
108 109 110 111 112 113 114 115 116

(** 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 {_} _ _ _ _.

117
Program Definition chain_map {A B : ofeT} (f : A  B)
118
    `{!NonExpansive f} (c : chain A) : chain B :=
119 120 121
  {| chain_car n := f (c n) |}.
Next Obligation. by intros A B f Hf c n i ?; apply Hf, chain_cauchy. Qed.

122 123 124 125 126 127
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
128
Hint Mode Cofe ! : typeclass_instances.
129

130
Lemma compl_chain_map `{Cofe A, Cofe B} (f : A  B) c `(NonExpansive f) :
131 132 133
  compl (chain_map f c)  f (compl c).
Proof. apply equiv_dist=>n. by rewrite !conv_compl. Qed.

134 135 136 137 138 139 140 141
Program Definition chain_const {A : ofeT} (a : A) : chain A :=
  {| chain_car n := a |}.
Next Obligation. by intros A a n i _. Qed.

Lemma compl_chain_const {A : ofeT} `{!Cofe A} (a : A) :
  compl (chain_const a)  a.
Proof. apply equiv_dist=>n. by rewrite conv_compl. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
142
(** General properties *)
143
Section ofe.
144
  Context {A : ofeT}.
145
  Implicit Types x y : A.
146
  Global Instance ofe_equivalence : Equivalence (() : relation A).
Robbert Krebbers's avatar
Robbert Krebbers committed
147 148
  Proof.
    split.
149 150
    - by intros x; rewrite equiv_dist.
    - by intros x y; rewrite !equiv_dist.
151
    - by intros x y z; rewrite !equiv_dist; intros; trans y.
Robbert Krebbers's avatar
Robbert Krebbers committed
152
  Qed.
153
  Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
154 155
  Proof.
    intros x1 x2 ? y1 y2 ?; split; intros.
156 157
    - by trans x1; [|trans y1].
    - by trans x2; [|trans y2].
Robbert Krebbers's avatar
Robbert Krebbers committed
158
  Qed.
159
  Global Instance dist_proper n : Proper (() ==> () ==> iff) (@dist A _ n).
Robbert Krebbers's avatar
Robbert Krebbers committed
160
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
161
    by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
162 163 164
  Qed.
  Global Instance dist_proper_2 n x : Proper (() ==> iff) (dist n x).
  Proof. by apply dist_proper. Qed.
165 166
  Global Instance Discrete_proper : Proper (() ==> iff) (@Discrete A).
  Proof. intros x y Hxy. rewrite /Discrete. by setoid_rewrite Hxy. Qed.
167

Robbert Krebbers's avatar
Robbert Krebbers committed
168
  Lemma dist_le n n' x y : x {n} y  n'  n  x {n'} y.
Robbert Krebbers's avatar
Robbert Krebbers committed
169
  Proof. induction 2; eauto using dist_S. Qed.
170 171
  Lemma dist_le' n n' x y : n'  n  x {n} y  x {n'} y.
  Proof. intros; eauto using dist_le. Qed.
172 173
  Instance ne_proper {B : ofeT} (f : A  B) `{!NonExpansive f} :
    Proper (() ==> ()) f | 100.
Robbert Krebbers's avatar
Robbert Krebbers committed
174
  Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed.
175
  Instance ne_proper_2 {B C : ofeT} (f : A  B  C) `{!NonExpansive2 f} :
Robbert Krebbers's avatar
Robbert Krebbers committed
176 177 178
    Proper (() ==> () ==> ()) f | 100.
  Proof.
     unfold Proper, respectful; setoid_rewrite equiv_dist.
Robbert Krebbers's avatar
Robbert Krebbers committed
179
     by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n).
Robbert Krebbers's avatar
Robbert Krebbers committed
180
  Qed.
181

182
  Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c {n} c (S n).
183 184
  Proof.
    transitivity (c n); first by apply conv_compl. symmetry.
Ralf Jung's avatar
Ralf Jung committed
185
    apply chain_cauchy. lia.
186
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
187

188
  Lemma discrete_iff n (x : A) `{!Discrete x} y : x  y  x {n} y.
189
  Proof.
190
    split; intros; auto. apply (discrete _), dist_le with n; auto with lia.
191
  Qed.
192
  Lemma discrete_iff_0 n (x : A) `{!Discrete x} y : x {0} y  x {n} y.
193
  Proof. by rewrite -!discrete_iff. Qed.
194
End ofe.
Robbert Krebbers's avatar
Robbert Krebbers committed
195

196
(** Contractive functions *)
197
Definition dist_later `{Dist A} (n : nat) (x y : A) : Prop :=
198
  match n with 0 => True | S n => x {n} y end.
199
Arguments dist_later _ _ !_ _ _ /.
200

201
Global Instance dist_later_equivalence (A : ofeT) n : Equivalence (@dist_later A _ n).
202 203
Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed.

204 205 206
Lemma dist_dist_later {A : ofeT} n (x y : A) : dist n x y  dist_later n x y.
Proof. intros Heq. destruct n; first done. exact: dist_S. Qed.

207 208 209 210 211 212 213 214 215 216 217
Lemma dist_later_dist {A : ofeT} n (x y : A) : dist_later (S n) x y  dist n x y.
Proof. done. Qed.

(* We don't actually need this lemma (as our tactics deal with this through
   other means), but technically speaking, this is the reason why
   pre-composing a non-expansive function to a contractive function
   preserves contractivity. *)
Lemma ne_dist_later {A B : ofeT} (f : A  B) :
  NonExpansive f   n, Proper (dist_later n ==> dist_later n) f.
Proof. intros Hf [|n]; last exact: Hf. hnf. by intros. Qed.

218
Notation Contractive f := ( n, Proper (dist_later n ==> dist n) f).
219

220
Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x).
221 222
Proof. by intros n y1 y2. Qed.

223
Section contractive.
224
  Local Set Default Proof Using "Type*".
225 226 227 228
  Context {A B : ofeT} (f : A  B) `{!Contractive f}.
  Implicit Types x y : A.

  Lemma contractive_0 x y : f x {0} f y.
229
  Proof. by apply (_ : Contractive f). Qed.
230
  Lemma contractive_S n x y : x {n} y  f x {S n} f y.
231
  Proof. intros. by apply (_ : Contractive f). Qed.
232

233 234
  Global Instance contractive_ne : NonExpansive f | 100.
  Proof. by intros n x y ?; apply dist_S, contractive_S. Qed.
235 236 237 238
  Global Instance contractive_proper : Proper (() ==> ()) f | 100.
  Proof. apply (ne_proper _). Qed.
End contractive.

239 240
Ltac f_contractive :=
  match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
241 242 243
  | |- ?f _ {_} ?f _ => simple apply (_ : Proper (dist_later _ ==> _) f)
  | |- ?f _ _ {_} ?f _ _ => simple apply (_ : Proper (dist_later _ ==> _ ==> _) f)
  | |- ?f _ _ {_} ?f _ _ => simple apply (_ : Proper (_ ==> dist_later _ ==> _) f)
244 245
  end;
  try match goal with
246
  | |- @dist_later ?A _ ?n ?x ?y =>
247
         destruct n as [|n]; [exact I|change (@dist A _ n x y)]
248
  end;
Robbert Krebbers's avatar
Robbert Krebbers committed
249
  try simple apply reflexivity.
250

Robbert Krebbers's avatar
Robbert Krebbers committed
251 252
Ltac solve_contractive :=
  solve_proper_core ltac:(fun _ => first [f_contractive | f_equiv]).
Robbert Krebbers's avatar
Robbert Krebbers committed
253

Robbert Krebbers's avatar
Robbert Krebbers committed
254 255 256 257 258 259 260 261 262 263 264 265 266 267
(** Limit preserving predicates *)
Class LimitPreserving `{!Cofe A} (P : A  Prop) : Prop :=
  limit_preserving (c : chain A) : ( n, P (c n))  P (compl c).
Hint Mode LimitPreserving + + ! : typeclass_instances.

Section limit_preserving.
  Context `{Cofe A}.
  (* These are not instances as they will never fire automatically...
     but they can still be helpful in proving things to be limit preserving. *)

  Lemma limit_preserving_ext (P Q : A  Prop) :
    ( x, P x  Q x)  LimitPreserving P  LimitPreserving Q.
  Proof. intros HP Hlimit c ?. apply HP, Hlimit=> n; by apply HP. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
268
  Global Instance limit_preserving_const (P : Prop) : LimitPreserving (λ _ : A, P).
Robbert Krebbers's avatar
Robbert Krebbers committed
269 270
  Proof. intros c HP. apply (HP 0). Qed.

271
  Lemma limit_preserving_discrete (P : A  Prop) :
Robbert Krebbers's avatar
Robbert Krebbers committed
272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293
    Proper (dist 0 ==> impl) P  LimitPreserving P.
  Proof. intros PH c Hc. by rewrite (conv_compl 0). Qed.

  Lemma limit_preserving_and (P1 P2 : A  Prop) :
    LimitPreserving P1  LimitPreserving P2 
    LimitPreserving (λ x, P1 x  P2 x).
  Proof. intros Hlim1 Hlim2 c Hc. split. apply Hlim1, Hc. apply Hlim2, Hc. Qed.

  Lemma limit_preserving_impl (P1 P2 : A  Prop) :
    Proper (dist 0 ==> impl) P1  LimitPreserving P2 
    LimitPreserving (λ x, P1 x  P2 x).
  Proof.
    intros Hlim1 Hlim2 c Hc HP1. apply Hlim2=> n; apply Hc.
    eapply Hlim1, HP1. apply dist_le with n; last lia. apply (conv_compl n).
  Qed.

  Lemma limit_preserving_forall {B} (P : B  A  Prop) :
    ( y, LimitPreserving (P y)) 
    LimitPreserving (λ x,  y, P y x).
  Proof. intros Hlim c Hc y. by apply Hlim. Qed.
End limit_preserving.

Robbert Krebbers's avatar
Robbert Krebbers committed
294
(** Fixpoint *)
295
Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A  A)
296
  `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
297
Next Obligation.
298
  intros A ? f ? n.
Ralf Jung's avatar
Ralf Jung committed
299
  induction n as [|n IH]=> -[|i] //= ?; try lia.
300
  - apply (contractive_0 f).
Ralf Jung's avatar
Ralf Jung committed
301
  - apply (contractive_S f), IH; auto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
302
Qed.
303

304
Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A  A)
305
  `{!Contractive f} : A := compl (fixpoint_chain f).
306
Definition fixpoint_aux : seal (@fixpoint_def). by eexists. Qed.
307 308
Definition fixpoint {A AC AiH} f {Hf} := fixpoint_aux.(unseal) A AC AiH f Hf.
Definition fixpoint_eq : @fixpoint = @fixpoint_def := fixpoint_aux.(seal_eq).
Robbert Krebbers's avatar
Robbert Krebbers committed
309 310

Section fixpoint.
311
  Context `{Cofe A, Inhabited A} (f : A  A) `{!Contractive f}.
312

313
  Lemma fixpoint_unfold : fixpoint f  f (fixpoint f).
Robbert Krebbers's avatar
Robbert Krebbers committed
314
  Proof.
315 316
    apply equiv_dist=>n.
    rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //.
317
    induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
318
  Qed.
319 320 321

  Lemma fixpoint_unique (x : A) : x  f x  x  fixpoint f.
  Proof.
322 323 324
    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.
325 326
  Qed.

327
  Lemma fixpoint_ne (g : A  A) `{!Contractive g} n :
328
    ( z, f z {n} g z)  fixpoint f {n} fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
329
  Proof.
330
    intros Hfg. rewrite fixpoint_eq /fixpoint_def
Robbert Krebbers's avatar
Robbert Krebbers committed
331
      (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=.
332 333
    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
334
  Qed.
335 336
  Lemma fixpoint_proper (g : A  A) `{!Contractive g} :
    ( x, f x  g x)  fixpoint f  fixpoint g.
Robbert Krebbers's avatar
Robbert Krebbers committed
337
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.
338 339

  Lemma fixpoint_ind (P : A  Prop) :
340
    Proper (() ==> impl) P 
341
    ( x, P x)  ( x, P x  P (f x)) 
Robbert Krebbers's avatar
Robbert Krebbers committed
342
    LimitPreserving P 
343 344 345 346
    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).
Robbert Krebbers's avatar
Robbert Krebbers committed
347
    { intros n. rewrite /chcar. induction n as [|n IH]=> -[|i] //=;
Ralf Jung's avatar
Ralf Jung committed
348
        eauto using contractive_0, contractive_S with lia. }
349
    set (fp2 := compl {| chain_cauchy := Hcauch |}).
Robbert Krebbers's avatar
Robbert Krebbers committed
350 351 352 353
    assert (f fp2  fp2).
    { apply equiv_dist=>n. rewrite /fp2 (conv_compl n) /= /chcar.
      induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. }
    rewrite -(fixpoint_unique fp2) //.
Robbert Krebbers's avatar
Robbert Krebbers committed
354
    apply Hlim=> n /=. by apply Nat_iter_ind.
355
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
356 357
End fixpoint.

Robbert Krebbers's avatar
Robbert Krebbers committed
358

359 360 361
(** 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).
362

363
Section fixpointK.
364
  Local Set Default Proof Using "Type*".
365
  Context `{Cofe A, Inhabited A} (f : A  A) (k : nat).
366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388
  Context {f_contractive : Contractive (Nat.iter k f)} {f_ne : NonExpansive f}.
  (* Note than f_ne is crucial here:  there are functions f such that f^2 is contractive,
     but f is not non-expansive.
     Consider for example f: SPred → SPred (where SPred is "downclosed sets of natural numbers").
     Define f (using informative excluded middle) as follows:
     f(N) = N  (where N is the set of all natural numbers)
     f({0, ..., n}) = {0, ... n-1}  if n is even (so n-1 is at least -1, in which case we return the empty set)
     f({0, ..., n}) = {0, ..., n+2} if n is odd
     In other words, if we consider elements of SPred as ordinals, then we decreaste odd finite
     ordinals by 1 and increase even finite ordinals by 2.
     f is not non-expansive:  Consider f({0}) = ∅ and f({0,1}) = f({0,1,2,3}).
     The arguments are clearly 0-equal, but the results are not.

     Now consider g := f^2. We have
     g(N) = N
     g({0, ..., n}) = {0, ... n+1}  if n is even
     g({0, ..., n}) = {0, ..., n+4} if n is odd
     g is contractive.  All outputs contain 0, so they are all 0-equal.
     Now consider two n-equal inputs. We have to show that the outputs are n+1-equal.
     Either they both do not contain n in which case they have to be fully equal and
     hence so are the results.  Or else they both contain n, so the results will
     both contain n+1, so the results are n+1-equal.
   *)
389 390

  Let f_proper : Proper (() ==> ()) f := ne_proper f.
391
  Local Existing Instance f_proper.
392

393
  Lemma fixpointK_unfold : fixpointK k f  f (fixpointK k f).
394
  Proof.
395 396
    symmetry. rewrite /fixpointK. apply fixpoint_unique.
    by rewrite -Nat_iter_S_r Nat_iter_S -fixpoint_unfold.
397 398
  Qed.

399
  Lemma fixpointK_unique (x : A) : x  f x  x  fixpointK k f.
400
  Proof.
401 402
    intros Hf. apply fixpoint_unique. clear f_contractive.
    induction k as [|k' IH]=> //=. by rewrite -IH.
403 404
  Qed.

405
  Section fixpointK_ne.
406
    Context (g : A  A) `{g_contractive : !Contractive (Nat.iter k g)}.
407
    Context {g_ne : NonExpansive g}.
408

409
    Lemma fixpointK_ne n : ( z, f z {n} g z)  fixpointK k f {n} fixpointK k g.
410
    Proof.
411 412 413
      rewrite /fixpointK=> Hfg /=. apply fixpoint_ne=> z.
      clear f_contractive g_contractive.
      induction k as [|k' IH]=> //=. by rewrite IH Hfg.
414 415
    Qed.

416 417 418
    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.
Ralf Jung's avatar
Ralf Jung committed
419 420 421 422

  Lemma fixpointK_ind (P : A  Prop) :
    Proper (() ==> impl) P 
    ( x, P x)  ( x, P x  P (f x)) 
Robbert Krebbers's avatar
Robbert Krebbers committed
423
    LimitPreserving P 
Ralf Jung's avatar
Ralf Jung committed
424 425
    P (fixpointK k f).
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
426
    intros. rewrite /fixpointK. apply fixpoint_ind; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
427
    intros; apply Nat_iter_ind; auto.
Ralf Jung's avatar
Ralf Jung committed
428
  Qed.
429
End fixpointK.
430

Robbert Krebbers's avatar
Robbert Krebbers committed
431
(** Mutual fixpoints *)
Ralf Jung's avatar
Ralf Jung committed
432
Section fixpointAB.
433 434
  Local Unset Default Proof Using.

Robbert Krebbers's avatar
Robbert Krebbers committed
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
  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
476
End fixpointAB.
Robbert Krebbers's avatar
Robbert Krebbers committed
477

Ralf Jung's avatar
Ralf Jung committed
478
Section fixpointAB_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
  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
510
End fixpointAB_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
511

512
(** Non-expansive function space *)
513
Record ofe_mor (A B : ofeT) : Type := OfeMor {
514
  ofe_mor_car :> A  B;
515
  ofe_mor_ne : NonExpansive ofe_mor_car
Robbert Krebbers's avatar
Robbert Krebbers committed
516
}.
517
Arguments OfeMor {_ _} _ {_}.
518 519
Add Printing Constructor ofe_mor.
Existing Instance ofe_mor_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
520

521
Notation "'λne' x .. y , t" :=
522
  (@OfeMor _ _ (λ x, .. (@OfeMor _ _ (λ y, t) _) ..) _)
523 524
  (at level 200, x binder, y binder, right associativity).

525 526 527 528 529 530 531
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).
532 533
  Proof.
    split.
534
    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
Robbert Krebbers's avatar
Robbert Krebbers committed
535
      intros Hfg k; apply equiv_dist=> n; apply Hfg.
536
    - intros n; split.
537 538
      + by intros f x.
      + by intros f g ? x.
539
      + by intros f g h ?? x; trans (g x).
540
    - by intros n f g ? x; apply dist_S.
541
  Qed.
542
  Canonical Structure ofe_morO := OfeT (ofe_mor A B) ofe_mor_ofe_mixin.
543

544
  Program Definition ofe_mor_chain (c : chain ofe_morO)
545 546
    (x : A) : chain B := {| chain_car n := c n x |}.
  Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed.
547
  Program Definition ofe_mor_compl `{Cofe B} : Compl ofe_morO := λ c,
548 549 550 551 552
    {| 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.
553
  Global Program Instance ofe_mor_cofe `{Cofe B} : Cofe ofe_morO :=
554 555 556 557 558
    {| compl := ofe_mor_compl |}.
  Next Obligation.
    intros ? n c x; simpl.
    by rewrite (conv_compl n (ofe_mor_chain c x)) /=.
  Qed.
559

560 561 562
  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.
563 564 565
  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.
566
  Proof. done. Qed.
567
End ofe_mor.
568

569
Arguments ofe_morO : clear implicits.
570
Notation "A -n> B" :=
571
  (ofe_morO A B) (at level 99, B at level 200, right associativity).
572
Instance ofe_mor_inhabited {A B : ofeT} `{Inhabited B} :
573
  Inhabited (A -n> B) := populate (λne _, inhabitant).
Robbert Krebbers's avatar
Robbert Krebbers committed
574

575
(** Identity and composition and constant function *)
576
Definition cid {A} : A -n> A := OfeMor id.
577
Instance: Params (@cid) 1 := {}.
578
Definition cconst {A B : ofeT} (x : B) : A -n> B := OfeMor (const x).
579
Instance: Params (@cconst) 2 := {}.
580

Robbert Krebbers's avatar
Robbert Krebbers committed
581
Definition ccompose {A B C}
582
  (f : B -n> C) (g : A -n> B) : A -n> C := OfeMor (f  g).
583
Instance: Params (@ccompose) 3 := {}.
Robbert Krebbers's avatar
Robbert Krebbers committed
584
Infix "◎" := ccompose (at level 40, left associativity).
585 586 587
Global Instance ccompose_ne {A B C} :
  NonExpansive2 (@ccompose A B C).
Proof. intros n ?? Hf g1 g2 Hg x. rewrite /= (Hg x) (Hf (g2 x)) //. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
588

Ralf Jung's avatar
Ralf Jung committed
589
(* Function space maps *)
590
Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B')
Ralf Jung's avatar
Ralf Jung committed
591
  (h : A -n> B) : A' -n> B' := g  h  f.
592 593
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').
594
Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed.
Ralf Jung's avatar
Ralf Jung committed
595

596 597 598 599
Definition ofe_morO_map {A A' B B'} (f : A' -n> A) (g : B -n> B') :
  (A -n> B) -n> (A' -n>  B') := OfeMor (ofe_mor_map f g).
Instance ofe_morO_map_ne {A A' B B'} :
  NonExpansive2 (@ofe_morO_map A A' B B').
Ralf Jung's avatar
Ralf Jung committed
600
Proof.
601
  intros n f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map.
602
  by repeat apply ccompose_ne.
Ralf Jung's avatar
Ralf Jung committed
603 604
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
605
(** unit *)
606 607
Section unit.
  Instance unit_dist : Dist unit := λ _ _ _, True.
608
  Definition unit_ofe_mixin : OfeMixin unit.
609
  Proof. by repeat split; try exists 0. Qed.
610
  Canonical Structure unitO : ofeT := OfeT unit unit_ofe_mixin.
Robbert Krebbers's avatar
Robbert Krebbers committed
611

612
  Global Program Instance unit_cofe : Cofe unitO := { compl x := () }.
613
  Next Obligation. by repeat split; try exists 0. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
614

615
  Global Instance unit_ofe_discrete : OfeDiscrete unitO.
Robbert Krebbers's avatar
Robbert Krebbers committed
616
  Proof. done. Qed.
617
End unit.
Robbert Krebbers's avatar
Robbert Krebbers committed
618

619 620 621 622 623 624 625 626 627 628 629 630 631 632
(** empty *)
Section empty.
  Instance Empty_set_dist : Dist Empty_set := λ _ _ _, True.
  Definition Empty_set_ofe_mixin : OfeMixin Empty_set.
  Proof. by repeat split; try exists 0. Qed.
  Canonical Structure Empty_setO : ofeT := OfeT Empty_set Empty_set_ofe_mixin.

  Global Program Instance Empty_set_cofe : Cofe Empty_setO := { compl x := x 0 }.
  Next Obligation. by repeat split; try exists 0. Qed.

  Global Instance Empty_set_ofe_discrete : OfeDiscrete Empty_setO.
  Proof. done. Qed.
End empty.

Robbert Krebbers's avatar
Robbert Krebbers committed
633
(** Product *)
634
Section product.
635
  Context {A B : ofeT}.
636 637 638

  Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n).
  Global Instance pair_ne :
639 640 641
    NonExpansive2 (@pair A B) := _.
  Global Instance fst_ne : NonExpansive (@fst A B) := _.
  Global Instance snd_ne : NonExpansive (@snd A B) := _.
642
  Definition prod_ofe_mixin : OfeMixin (A * B).
643 644
  Proof.
    split.
645
    - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
646
      rewrite !equiv_dist; naive_solver.
647 648
    - apply _.
    - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
649
  Qed.
650
  Canonical Structure prodO : ofeT := OfeT (A * B) prod_ofe_mixin.
651

652
  Global Program Instance prod_cofe `{Cofe A, Cofe B} : Cofe prodO :=
653 654 655 656 657 658
    { 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.

659 660 661
  Global Instance prod_discrete (x : A * B) :
    Discrete (x.1)  Discrete (x.2)  Discrete x.
  Proof. by intros ???[??]; split; apply (discrete _). Qed.
662
  Global Instance prod_ofe_discrete :
663
    OfeDiscrete A  OfeDiscrete B  OfeDiscrete prodO.
664
  Proof. intros ?? [??]; apply _. Qed.
665 666
End product.

667
Arguments prodO : clear implicits.
668 669
Typeclasses Opaque prod_dist.

670
Instance prod_map_ne {A A' B B' : ofeT} n :
Robbert Krebbers's avatar
Robbert Krebbers committed
671 672 673
  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.
674 675 676 677
Definition prodO_map {A A' B B'} (f : A -n> A') (g : B -n> B') :
  prodO A B -n> prodO A' B' := OfeMor (prod_map f g).
Instance prodO_map_ne {A A' B B'} :
  NonExpansive2 (@prodO_map A A' B B').
678
Proof. intros n f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
679

680
(** Functors *)
681 682 683 684 685 686 687 688 689
Record oFunctor := OFunctor {
  oFunctor_car :  A `{!Cofe A} B `{!Cofe B}, ofeT;
  oFunctor_map `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :
    ((A2 -n> A1) * (B1 -n> B2))  oFunctor_car A1 B1 -n> oFunctor_car A2 B2;
  oFunctor_ne `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :
    NonExpansive (@oFunctor_map A1 _ A2 _ B1 _ B2 _);
  oFunctor_id `{!Cofe A, !Cofe B} (x : oFunctor_car A B) :
    oFunctor_map (cid,cid) x  x;
  oFunctor_compose `{!Cofe A1, !Cofe A2, !Cofe A3, !Cofe B1, !Cofe B2, !Cofe B3}
690
      (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x :
691
    oFunctor_map (fg, g'f') x  oFunctor_map (g,g') (oFunctor_map (f,f') x)
692
}.
693 694
Existing Instance oFunctor_ne.
Instance: Params (@oFunctor_map) 9 := {}.
695

696 697
Delimit Scope oFunctor_scope with OF.
Bind Scope oFunctor_scope with oFunctor.
698

699 700 701 702
Class oFunctorContractive (F : oFunctor) :=
  oFunctor_contractive `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :>
    Contractive (@oFunctor_map F A1 _ A2 _ B1 _ B2 _).
Hint Mode oFunctorContractive ! : typeclass_instances.
703

704 705
Definition oFunctor_diag (F: oFunctor) (A: ofeT) `{!Cofe A} : ofeT :=
  oFunctor_car F A A.
706
(** Note that the implicit argument [Cofe A] is not taken into account when
707
[oFunctor_diag] is used as a coercion. So, given [F : oFunctor] and [A : ofeT]
708
one has to write [F A _]. *)
709
Coercion oFunctor_diag : oFunctor >-> Funclass.
710

711 712
Program Definition constOF (B : ofeT) : oFunctor :=
  {| oFunctor_car A1 A2 _ _ := B; oFunctor_map A1 _ A2 _ B1 _ B2 _ f := cid |}.
713
Solve Obligations with done.
714
Coercion constOF : ofeT >-> oFunctor.
715

716 717
Instance constOF_contractive B : oFunctorContractive (constOF B).
Proof. rewrite /oFunctorContractive; apply _. Qed.
718

719 720
Program Definition idOF : oFunctor :=
  {| oFunctor_car A1 _ A2 _ := A2; oFunctor_map A1 _ A2 _ B1 _ B2 _ f := f.2 |}.
721
Solve Obligations with done.
722
Notation "∙" := idOF : oFunctor_scope.
723

724 725 726 727
Program Definition prodOF (F1 F2 : oFunctor) : oFunctor := {|
  oFunctor_car A _ B _ := prodO (oFunctor_car F1 A B) (oFunctor_car F2 A B);
  oFunctor_map A1 _ A2 _ B1 _ B2 _ fg :=
    prodO_map (oFunctor_map F1 fg) (oFunctor_map F2 fg)
728
|}.
Robbert Krebbers's avatar