collections.v 47.4 KB
Newer Older
1
(* Copyright (c) 2012-2017, Coq-std++ developers. *)
2
3
4
5
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on collections. Most
importantly, it implements some tactics to automatically solve goals involving
collections. *)
6
From stdpp Require Export orders list.
7
8
(* FIXME: This file needs a 'Proof Using' hint, but the default we use
   everywhere makes for lots of extra ssumptions. *)
9

10
11
Instance collection_equiv `{ElemOf A C} : Equiv C := λ X Y,
   x, x  X  x  Y.
12
13
Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y,
   x, x  X  x  Y.
14
15
16
Instance collection_disjoint `{ElemOf A C} : Disjoint C := λ X Y,
   x, x  X  x  Y  False.
Typeclasses Opaque collection_equiv collection_subseteq collection_disjoint.
17

18
19
(** * Setoids *)
Section setoids_simple.
20
  Context `{SimpleCollection A C}.
Robbert Krebbers's avatar
Robbert Krebbers committed
21

22
  Global Instance collection_equivalence: @Equivalence C ().
23
  Proof.
24
25
26
27
    split.
    - done.
    - intros X Y ? x. by symmetry.
    - intros X Y Z ?? x; by trans (x  Y).
28
  Qed.
29
30
31
32
33
34
  Global Instance singleton_proper : Proper ((=) ==> ()) (singleton (B:=C)).
  Proof. apply _. Qed.
  Global Instance elem_of_proper :
    Proper ((=) ==> () ==> iff) (@elem_of A C _) | 5.
  Proof. by intros x ? <- X Y. Qed.
  Global Instance disjoint_proper: Proper (() ==> () ==> iff) (@disjoint C _).
35
  Proof.
36
    intros X1 X2 HX Y1 Y2 HY; apply forall_proper; intros x. by rewrite HX, HY.
37
  Qed.
38
39
40
41
42
43
44
45
46
47
48
49
  Global Instance union_proper : Proper (() ==> () ==> ()) (@union C _).
  Proof. intros X1 X2 HX Y1 Y2 HY x. rewrite !elem_of_union. f_equiv; auto. Qed.
  Global Instance union_list_proper: Proper (() ==> ()) (union_list (A:=C)).
  Proof. by induction 1; simpl; try apply union_proper. Qed.
  Global Instance subseteq_proper : Proper (() ==> () ==> iff) (() : relation C).
  Proof.
    intros X1 X2 HX Y1 Y2 HY. apply forall_proper; intros x. by rewrite HX, HY.
  Qed.
End setoids_simple.

Section setoids.
  Context `{Collection A C}.
50

51
52
53
  (** * Setoids *)
  Global Instance intersection_proper :
    Proper (() ==> () ==> ()) (@intersection C _).
54
  Proof.
55
    intros X1 X2 HX Y1 Y2 HY x. by rewrite !elem_of_intersection, HX, HY.
56
  Qed.
57
58
  Global Instance difference_proper :
     Proper (() ==> () ==> ()) (@difference C _).
59
  Proof.
60
    intros X1 X2 HX Y1 Y2 HY x. by rewrite !elem_of_difference, HX, HY.
61
  Qed.
62
End setoids.
Robbert Krebbers's avatar
Robbert Krebbers committed
63

64
65
66
67
68
Section setoids_monad.
  Context `{CollectionMonad M}.

  Global Instance collection_fmap_proper {A B} :
    Proper (pointwise_relation _ (=) ==> () ==> ()) (@fmap M _ A B).
69
  Proof.
70
71
    intros f1 f2 Hf X1 X2 HX x. rewrite !elem_of_fmap. f_equiv; intros z.
    by rewrite HX, Hf.
72
  Qed.
73
  Global Instance collection_bind_proper {A B} :
74
    Proper (pointwise_relation _ () ==> () ==> ()) (@mbind M _ A B).
75
76
  Proof.
    intros f1 f2 Hf X1 X2 HX x. rewrite !elem_of_bind. f_equiv; intros z.
77
    by rewrite HX, (Hf z).
78
79
80
81
82
83
84
  Qed.
  Global Instance collection_join_proper {A} :
    Proper (() ==> ()) (@mjoin M _ A).
  Proof.
    intros X1 X2 HX x. rewrite !elem_of_join. f_equiv; intros z. by rewrite HX.
  Qed.
End setoids_monad.
85

86
87
88
89
90
(** * Tactics *)
(** The tactic [set_unfold] transforms all occurrences of [(∪)], [(∩)], [(∖)],
[(<$>)], [∅], [{[_]}], [(≡)], and [(⊆)] into logically equivalent propositions
involving just [∈]. For example, [A → x ∈ X ∪ ∅] becomes [A → x ∈ X ∨ False].

91
92
93
This transformation is implemented using type classes instead of setoid
rewriting to ensure that we traverse each term at most once and to be able to
deal with occurences of the set operations under binders. *)
94
Class SetUnfold (P Q : Prop) := { set_unfold : P  Q }.
95
Arguments set_unfold _ _ {_} : assert.
96
97
98
99
100
Hint Mode SetUnfold + - : typeclass_instances.

Class SetUnfoldSimpl (P Q : Prop) := { set_unfold_simpl : SetUnfold P Q }.
Hint Extern 0 (SetUnfoldSimpl _ _) => csimpl; constructor : typeclass_instances.

101
Instance set_unfold_default P : SetUnfold P P | 1000. done. Qed.
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
Definition set_unfold_1 `{SetUnfold P Q} : P  Q := proj1 (set_unfold P Q).
Definition set_unfold_2 `{SetUnfold P Q} : Q  P := proj2 (set_unfold P Q).

Lemma set_unfold_impl P Q P' Q' :
  SetUnfold P P'  SetUnfold Q Q'  SetUnfold (P  Q) (P'  Q').
Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed.
Lemma set_unfold_and P Q P' Q' :
  SetUnfold P P'  SetUnfold Q Q'  SetUnfold (P  Q) (P'  Q').
Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed.
Lemma set_unfold_or P Q P' Q' :
  SetUnfold P P'  SetUnfold Q Q'  SetUnfold (P  Q) (P'  Q').
Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed.
Lemma set_unfold_iff P Q P' Q' :
  SetUnfold P P'  SetUnfold Q Q'  SetUnfold (P  Q) (P'  Q').
Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed.
Lemma set_unfold_not P P' : SetUnfold P P'  SetUnfold (¬P) (¬P').
Proof. constructor. by rewrite (set_unfold P P'). Qed.
Lemma set_unfold_forall {A} (P P' : A  Prop) :
  ( x, SetUnfold (P x) (P' x))  SetUnfold ( x, P x) ( x, P' x).
Proof. constructor. naive_solver. Qed.
Lemma set_unfold_exist {A} (P P' : A  Prop) :
  ( x, SetUnfold (P x) (P' x))  SetUnfold ( x, P x) ( x, P' x).
Proof. constructor. naive_solver. Qed.

(* Avoid too eager application of the above instances (and thus too eager
unfolding of type class transparent definitions). *)
Hint Extern 0 (SetUnfold (_  _) _) =>
  class_apply set_unfold_impl : typeclass_instances.
Hint Extern 0 (SetUnfold (_  _) _) =>
  class_apply set_unfold_and : typeclass_instances.
Hint Extern 0 (SetUnfold (_  _) _) =>
  class_apply set_unfold_or : typeclass_instances.
Hint Extern 0 (SetUnfold (_  _) _) =>
  class_apply set_unfold_iff : typeclass_instances.
Hint Extern 0 (SetUnfold (¬ _) _) =>
  class_apply set_unfold_not : typeclass_instances.
Hint Extern 1 (SetUnfold ( _, _) _) =>
  class_apply set_unfold_forall : typeclass_instances.
Hint Extern 0 (SetUnfold ( _, _) _) =>
  class_apply set_unfold_exist : typeclass_instances.

Section set_unfold_simple.
  Context `{SimpleCollection A C}.
  Implicit Types x y : A.
  Implicit Types X Y : C.

148
  Global Instance set_unfold_empty x : SetUnfold (x  ( : C)) False.
149
  Proof. constructor. split. apply not_elem_of_empty. done. Qed.
150
  Global Instance set_unfold_singleton x y : SetUnfold (x  ({[ y ]} : C)) (x = y).
151
152
153
154
155
156
157
158
159
160
161
162
  Proof. constructor; apply elem_of_singleton. Qed.
  Global Instance set_unfold_union x X Y P Q :
    SetUnfold (x  X) P  SetUnfold (x  Y) Q  SetUnfold (x  X  Y) (P  Q).
  Proof.
    intros ??; constructor.
    by rewrite elem_of_union, (set_unfold (x  X) P), (set_unfold (x  Y) Q).
  Qed.
  Global Instance set_unfold_equiv_same X : SetUnfold (X  X) True | 1.
  Proof. done. Qed.
  Global Instance set_unfold_equiv_empty_l X (P : A  Prop) :
    ( x, SetUnfold (x  X) (P x))  SetUnfold (  X) ( x, ¬P x) | 5.
  Proof.
163
    intros ?; constructor. unfold equiv, collection_equiv.
164
    pose proof (not_elem_of_empty (C:=C)); naive_solver.
165
  Qed.
166
  Global Instance set_unfold_equiv_empty_r (P : A  Prop) X :
167
    ( x, SetUnfold (x  X) (P x))  SetUnfold (X  ) ( x, ¬P x) | 5.
168
169
  Proof.
    intros ?; constructor. unfold equiv, collection_equiv.
170
    pose proof (not_elem_of_empty (C:=C)); naive_solver.
171
  Qed.
172
  Global Instance set_unfold_equiv (P Q : A  Prop) X :
173
174
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
    SetUnfold (X  Y) ( x, P x  Q x) | 10.
175
  Proof. constructor. apply forall_proper; naive_solver. Qed.
176
  Global Instance set_unfold_subseteq (P Q : A  Prop) X Y :
177
178
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
    SetUnfold (X  Y) ( x, P x  Q x).
179
  Proof. constructor. apply forall_proper; naive_solver. Qed.
180
  Global Instance set_unfold_subset (P Q : A  Prop) X :
181
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
182
    SetUnfold (X  Y) (( x, P x  Q x)  ¬∀ x, Q x  P x).
183
  Proof.
184
185
    constructor. unfold strict.
    repeat f_equiv; apply forall_proper; naive_solver.
186
  Qed.
187
  Global Instance set_unfold_disjoint (P Q : A  Prop) X Y :
Robbert Krebbers's avatar
Robbert Krebbers committed
188
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
189
    SetUnfold (X ## Y) ( x, P x  Q x  False).
190
  Proof. constructor. unfold disjoint, collection_disjoint. naive_solver. Qed.
191
192
193
194
195
196

  Context `{!LeibnizEquiv C}.
  Global Instance set_unfold_equiv_same_L X : SetUnfold (X = X) True | 1.
  Proof. done. Qed.
  Global Instance set_unfold_equiv_empty_l_L X (P : A  Prop) :
    ( x, SetUnfold (x  X) (P x))  SetUnfold ( = X) ( x, ¬P x) | 5.
197
  Proof. constructor. unfold_leibniz. by apply set_unfold_equiv_empty_l. Qed.
198
  Global Instance set_unfold_equiv_empty_r_L (P : A  Prop) X :
199
    ( x, SetUnfold (x  X) (P x))  SetUnfold (X = ) ( x, ¬P x) | 5.
200
  Proof. constructor. unfold_leibniz. by apply set_unfold_equiv_empty_r. Qed.
201
  Global Instance set_unfold_equiv_L (P Q : A  Prop) X Y :
202
203
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
    SetUnfold (X = Y) ( x, P x  Q x) | 10.
204
  Proof. constructor. unfold_leibniz. by apply set_unfold_equiv. Qed.
205
206
207
208
209
210
211
212
213
214
End set_unfold_simple.

Section set_unfold.
  Context `{Collection A C}.
  Implicit Types x y : A.
  Implicit Types X Y : C.

  Global Instance set_unfold_intersection x X Y P Q :
    SetUnfold (x  X) P  SetUnfold (x  Y) Q  SetUnfold (x  X  Y) (P  Q).
  Proof.
215
216
    intros ??; constructor. rewrite elem_of_intersection.
    by rewrite (set_unfold (x  X) P), (set_unfold (x  Y) Q).
217
218
219
220
  Qed.
  Global Instance set_unfold_difference x X Y P Q :
    SetUnfold (x  X) P  SetUnfold (x  Y) Q  SetUnfold (x  X  Y) (P  ¬Q).
  Proof.
221
222
    intros ??; constructor. rewrite elem_of_difference.
    by rewrite (set_unfold (x  X) P), (set_unfold (x  Y) Q).
223
224
225
226
  Qed.
End set_unfold.

Section set_unfold_monad.
227
  Context `{CollectionMonad M}.
228

229
230
  Global Instance set_unfold_ret {A} (x y : A) :
    SetUnfold (x  mret (M:=M) y) (x = y).
231
  Proof. constructor; apply elem_of_ret. Qed.
232
  Global Instance set_unfold_bind {A B} (f : A  M B) X (P Q : A  Prop) :
233
234
235
    ( y, SetUnfold (y  X) (P y))  ( y, SetUnfold (x  f y) (Q y)) 
    SetUnfold (x  X = f) ( y, Q y  P y).
  Proof. constructor. rewrite elem_of_bind; naive_solver. Qed.
236
  Global Instance set_unfold_fmap {A B} (f : A  B) (X : M A) (P : A  Prop) :
237
238
239
    ( y, SetUnfold (y  X) (P y)) 
    SetUnfold (x  f <$> X) ( y, x = f y  P y).
  Proof. constructor. rewrite elem_of_fmap; naive_solver. Qed.
240
  Global Instance set_unfold_join {A} (X : M (M A)) (P : M A  Prop) :
241
242
243
244
    ( Y, SetUnfold (Y  X) (P Y))  SetUnfold (x  mjoin X) ( Y, x  Y  P Y).
  Proof. constructor. rewrite elem_of_join; naive_solver. Qed.
End set_unfold_monad.

Robbert Krebbers's avatar
Robbert Krebbers committed
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
Section set_unfold_list.
  Context {A : Type}.
  Implicit Types x : A.
  Implicit Types l : list A.

  Global Instance set_unfold_nil x : SetUnfold (x  []) False.
  Proof. constructor; apply elem_of_nil. Qed.
  Global Instance set_unfold_cons x y l P :
    SetUnfold (x  l) P  SetUnfold (x  y :: l) (x = y  P).
  Proof. constructor. by rewrite elem_of_cons, (set_unfold (x  l) P). Qed.
  Global Instance set_unfold_app x l k P Q :
    SetUnfold (x  l) P  SetUnfold (x  k) Q  SetUnfold (x  l ++ k) (P  Q).
  Proof.
    intros ??; constructor.
    by rewrite elem_of_app, (set_unfold (x  l) P), (set_unfold (x  k) Q).
  Qed.
  Global Instance set_unfold_included l k (P Q : A  Prop) :
    ( x, SetUnfold (x  l) (P x))  ( x, SetUnfold (x  k) (Q x)) 
    SetUnfold (l  k) ( x, P x  Q x).
  Proof.
    constructor; unfold subseteq, list_subseteq.
    apply forall_proper; naive_solver.
  Qed.
End set_unfold_list.

270
271
272
Ltac set_unfold :=
  let rec unfold_hyps :=
    try match goal with
273
274
275
276
277
278
279
    | H : ?P |- _ =>
       lazymatch type of P with
       | Prop =>
         apply set_unfold_1 in H; revert H;
         first [unfold_hyps; intros H | intros H; fail 1]
       | _ => fail
       end
280
281
282
    end in
  apply set_unfold_2; unfold_hyps; csimpl in *.

283
284
(** Since [firstorder] already fails or loops on very small goals generated by
[set_solver], we use the [naive_solver] tactic as a substitute. *)
285
Tactic Notation "set_solver" "by" tactic3(tac) :=
286
  try fast_done;
287
288
289
290
291
292
293
294
295
296
297
298
299
  intros; setoid_subst;
  set_unfold;
  intros; setoid_subst;
  try match goal with |- _  _ => apply dec_stable end;
  naive_solver tac.
Tactic Notation "set_solver" "-" hyp_list(Hs) "by" tactic3(tac) :=
  clear Hs; set_solver by tac.
Tactic Notation "set_solver" "+" hyp_list(Hs) "by" tactic3(tac) :=
  clear -Hs; set_solver by tac.
Tactic Notation "set_solver" := set_solver by idtac.
Tactic Notation "set_solver" "-" hyp_list(Hs) := clear Hs; set_solver.
Tactic Notation "set_solver" "+" hyp_list(Hs) := clear -Hs; set_solver.

300
301
302
303
Hint Extern 1000 (_  _) => set_solver : set_solver.
Hint Extern 1000 (_  _) => set_solver : set_solver.
Hint Extern 1000 (_  _) => set_solver : set_solver.

304

305
306
(** * Collections with [∪], [∅] and [{[_]}] *)
Section simple_collection.
307
  Context `{SimpleCollection A C}.
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
  Implicit Types x y : A.
  Implicit Types X Y : C.
  Implicit Types Xs Ys : list C.

  (** Equality *)
  Lemma elem_of_equiv X Y : X  Y   x, x  X  x  Y.
  Proof. set_solver. Qed.
  Lemma collection_equiv_spec X Y : X  Y  X  Y  Y  X.
  Proof. set_solver. Qed.

  (** Subset relation *)
  Global Instance collection_subseteq_antisymm: AntiSymm () (() : relation C).
  Proof. intros ??. set_solver. Qed.

  Global Instance collection_subseteq_preorder: PreOrder (() : relation C).
  Proof. split. by intros ??. intros ???; set_solver. Qed.

  Lemma subseteq_union X Y : X  Y  X  Y  Y.
  Proof. set_solver. Qed.
  Lemma subseteq_union_1 X Y : X  Y  X  Y  Y.
  Proof. by rewrite subseteq_union. Qed.
  Lemma subseteq_union_2 X Y : X  Y  Y  X  Y.
  Proof. by rewrite subseteq_union. Qed.

  Lemma union_subseteq_l X Y : X  X  Y.
  Proof. set_solver. Qed.
  Lemma union_subseteq_r X Y : Y  X  Y.
  Proof. set_solver. Qed.
  Lemma union_least X Y Z : X  Z  Y  Z  X  Y  Z.
  Proof. set_solver. Qed.

  Lemma elem_of_subseteq X Y : X  Y   x, x  X  x  Y.
  Proof. done. Qed.
  Lemma elem_of_subset X Y : X  Y  ( x, x  X  x  Y)  ¬( x, x  Y  x  X).
  Proof. set_solver. Qed.

  (** Union *)
345
346
  Lemma union_subseteq X Y Z : X  Y  Z  X  Z  Y  Z.
  Proof. set_solver. Qed.
347
348
349
350
351
352
  Lemma not_elem_of_union x X Y : x  X  Y  x  X  x  Y.
  Proof. set_solver. Qed.
  Lemma elem_of_union_l x X Y : x  X  x  X  Y.
  Proof. set_solver. Qed.
  Lemma elem_of_union_r x X Y : x  Y  x  X  Y.
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
353
  Lemma union_mono_l X Y1 Y2 : Y1  Y2  X  Y1  X  Y2.
354
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
355
  Lemma union_mono_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
356
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
357
  Lemma union_mono X1 X2 Y1 Y2 : X1  X2  Y1  Y2  X1  Y1  X2  Y2.
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
  Proof. set_solver. Qed.

  Global Instance union_idemp : IdemP (() : relation C) ().
  Proof. intros X. set_solver. Qed.
  Global Instance union_empty_l : LeftId (() : relation C)  ().
  Proof. intros X. set_solver. Qed.
  Global Instance union_empty_r : RightId (() : relation C)  ().
  Proof. intros X. set_solver. Qed.
  Global Instance union_comm : Comm (() : relation C) ().
  Proof. intros X Y. set_solver. Qed.
  Global Instance union_assoc : Assoc (() : relation C) ().
  Proof. intros X Y Z. set_solver. Qed.

  Lemma empty_union X Y : X  Y    X    Y  .
  Proof. set_solver. Qed.

374
  Lemma union_cancel_l X Y Z : Z ## X  Z ## Y  Z  X  Z  Y  X  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
375
  Proof. set_solver. Qed.
376
  Lemma union_cancel_r X Y Z : X ## Z  Y ## Z  X  Z  Y  Z  X  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
377
378
  Proof. set_solver. Qed.

379
  (** Empty *)
Robbert Krebbers's avatar
Robbert Krebbers committed
380
381
  Lemma empty_subseteq X :   X.
  Proof. set_solver. Qed.
382
383
  Lemma elem_of_equiv_empty X : X     x, x  X.
  Proof. set_solver. Qed.
384
  Lemma elem_of_empty x : x  ( : C)  False.
385
386
387
388
389
390
391
392
393
394
395
  Proof. set_solver. Qed.
  Lemma equiv_empty X : X    X  .
  Proof. set_solver. Qed.
  Lemma union_positive_l X Y : X  Y    X  .
  Proof. set_solver. Qed.
  Lemma union_positive_l_alt X Y : X    X  Y  .
  Proof. set_solver. Qed.
  Lemma non_empty_inhabited x X : x  X  X  .
  Proof. set_solver. Qed.

  (** Singleton *)
396
  Lemma elem_of_singleton_1 x y : x  ({[y]} : C)  x = y.
397
  Proof. by rewrite elem_of_singleton. Qed.
398
  Lemma elem_of_singleton_2 x y : x = y  x  ({[y]} : C).
399
400
401
402
403
  Proof. by rewrite elem_of_singleton. Qed.
  Lemma elem_of_subseteq_singleton x X : x  X  {[ x ]}  X.
  Proof. set_solver. Qed.
  Lemma non_empty_singleton x : ({[ x ]} : C)  .
  Proof. set_solver. Qed.
404
  Lemma not_elem_of_singleton x y : x  ({[ y ]} : C)  x  y.
405
406
407
  Proof. by rewrite elem_of_singleton. Qed.

  (** Disjointness *)
408
  Lemma elem_of_disjoint X Y : X ## Y   x, x  X  x  Y  False.
409
410
411
412
  Proof. done. Qed.

  Global Instance disjoint_sym : Symmetric (@disjoint C _).
  Proof. intros X Y. set_solver. Qed.
413
  Lemma disjoint_empty_l Y :  ## Y.
414
  Proof. set_solver. Qed.
415
  Lemma disjoint_empty_r X : X ## .
416
  Proof. set_solver. Qed.
417
  Lemma disjoint_singleton_l x Y : {[ x ]} ## Y  x  Y.
418
  Proof. set_solver. Qed.
419
  Lemma disjoint_singleton_r y X : X ## {[ y ]}  y  X.
420
  Proof. set_solver. Qed.
421
  Lemma disjoint_union_l X1 X2 Y : X1  X2 ## Y  X1 ## Y  X2 ## Y.
422
  Proof. set_solver. Qed.
423
  Lemma disjoint_union_r X Y1 Y2 : X ## Y1  Y2  X ## Y1  X ## Y2.
424
425
426
427
  Proof. set_solver. Qed.

  (** Big unions *)
  Lemma elem_of_union_list Xs x : x   Xs   X, X  Xs  x  X.
428
429
  Proof.
    split.
430
431
    - induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|].
      setoid_rewrite elem_of_cons. apply elem_of_union in HXs. naive_solver.
Ralf Jung's avatar
Ralf Jung committed
432
    - intros [X [Hx]]. induction Hx; simpl; [by apply elem_of_union_l |].
433
      intros. apply elem_of_union_r; auto.
434
  Qed.
435

436
437
438
439
440
441
442
  Lemma union_list_nil :  @nil C = .
  Proof. done. Qed.
  Lemma union_list_cons X Xs :  (X :: Xs) = X   Xs.
  Proof. done. Qed.
  Lemma union_list_singleton X :  [X]  X.
  Proof. simpl. by rewrite (right_id  _). Qed.
  Lemma union_list_app Xs1 Xs2 :  (Xs1 ++ Xs2)   Xs1   Xs2.
443
  Proof.
444
445
    induction Xs1 as [|X Xs1 IH]; simpl; [by rewrite (left_id  _)|].
    by rewrite IH, (assoc _).
446
  Qed.
447
  Lemma union_list_reverse Xs :  (reverse Xs)   Xs.
448
  Proof.
449
450
451
    induction Xs as [|X Xs IH]; simpl; [done |].
    by rewrite reverse_cons, union_list_app,
      union_list_singleton, (comm _), IH.
452
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
453
454
  Lemma union_list_mono Xs Ys : Xs * Ys   Xs   Ys.
  Proof. induction 1; simpl; auto using union_mono. Qed.
455
  Lemma empty_union_list Xs :  Xs    Forall ( ) Xs.
456
  Proof.
457
458
459
    split.
    - induction Xs; simpl; rewrite ?empty_union; intuition.
    - induction 1 as [|?? E1 ? E2]; simpl. done. by apply empty_union.
460
  Qed.
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
  Section leibniz.
    Context `{!LeibnizEquiv C}.

    Lemma elem_of_equiv_L X Y : X = Y   x, x  X  x  Y.
    Proof. unfold_leibniz. apply elem_of_equiv. Qed.
    Lemma collection_equiv_spec_L X Y : X = Y  X  Y  Y  X.
    Proof. unfold_leibniz. apply collection_equiv_spec. Qed.

    (** Subset relation *)
    Global Instance collection_subseteq_partialorder :
      PartialOrder (() : relation C).
    Proof. split. apply _. intros ??. unfold_leibniz. apply (anti_symm _). Qed.

    Lemma subseteq_union_L X Y : X  Y  X  Y = Y.
    Proof. unfold_leibniz. apply subseteq_union. Qed.
    Lemma subseteq_union_1_L X Y : X  Y  X  Y = Y.
    Proof. unfold_leibniz. apply subseteq_union_1. Qed.
    Lemma subseteq_union_2_L X Y : X  Y = Y  X  Y.
    Proof. unfold_leibniz. apply subseteq_union_2. Qed.

    (** Union *)
    Global Instance union_idemp_L : IdemP (@eq C) ().
    Proof. intros ?. unfold_leibniz. apply (idemp _). Qed.
    Global Instance union_empty_l_L : LeftId (@eq C)  ().
    Proof. intros ?. unfold_leibniz. apply (left_id _ _). Qed.
    Global Instance union_empty_r_L : RightId (@eq C)  ().
    Proof. intros ?. unfold_leibniz. apply (right_id _ _). Qed.
    Global Instance union_comm_L : Comm (@eq C) ().
    Proof. intros ??. unfold_leibniz. apply (comm _). Qed.
    Global Instance union_assoc_L : Assoc (@eq C) ().
    Proof. intros ???. unfold_leibniz. apply (assoc _). Qed.

    Lemma empty_union_L X Y : X  Y =   X =   Y = .
    Proof. unfold_leibniz. apply empty_union. Qed.

497
    Lemma union_cancel_l_L X Y Z : Z ## X  Z ## Y  Z  X = Z  Y  X = Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
498
    Proof. unfold_leibniz. apply union_cancel_l. Qed.
499
    Lemma union_cancel_r_L X Y Z : X ## Z  Y ## Z  X  Z = Y  Z  X = Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
500
501
    Proof. unfold_leibniz. apply union_cancel_r. Qed.

502
503
504
505
506
507
508
509
510
511
512
513
514
    (** Empty *)
    Lemma elem_of_equiv_empty_L X : X =    x, x  X.
    Proof. unfold_leibniz. apply elem_of_equiv_empty. Qed.
    Lemma equiv_empty_L X : X    X = .
    Proof. unfold_leibniz. apply equiv_empty. Qed.
    Lemma union_positive_l_L X Y : X  Y =   X = .
    Proof. unfold_leibniz. apply union_positive_l. Qed.
    Lemma union_positive_l_alt_L X Y : X    X  Y  .
    Proof. unfold_leibniz. apply union_positive_l_alt. Qed.
    Lemma non_empty_inhabited_L x X : x  X  X  .
    Proof. unfold_leibniz. apply non_empty_inhabited. Qed.

    (** Singleton *)
515
    Lemma non_empty_singleton_L x : {[ x ]}  ( : C).
516
517
518
519
520
521
522
523
524
525
526
527
528
529
    Proof. unfold_leibniz. apply non_empty_singleton. Qed.

    (** Big unions *)
    Lemma union_list_singleton_L X :  [X] = X.
    Proof. unfold_leibniz. apply union_list_singleton. Qed.
    Lemma union_list_app_L Xs1 Xs2 :  (Xs1 ++ Xs2) =  Xs1   Xs2.
    Proof. unfold_leibniz. apply union_list_app. Qed.
    Lemma union_list_reverse_L Xs :  (reverse Xs) =  Xs.
    Proof. unfold_leibniz. apply union_list_reverse. Qed.
    Lemma empty_union_list_L Xs :  Xs =   Forall (= ) Xs.
    Proof. unfold_leibniz. by rewrite empty_union_list. Qed. 
  End leibniz.

  Section dec.
530
    Context `{!RelDecision (@equiv C _)}.
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
    Lemma collection_subseteq_inv X Y : X  Y  X  Y  X  Y.
    Proof. destruct (decide (X  Y)); [by right|left;set_solver]. Qed.
    Lemma collection_not_subset_inv X Y : X  Y  X  Y  X  Y.
    Proof. destruct (decide (X  Y)); [by right|left;set_solver]. Qed.

    Lemma non_empty_union X Y : X  Y    X    Y  .
    Proof. rewrite empty_union. destruct (decide (X  )); intuition. Qed.
    Lemma non_empty_union_list Xs :  Xs    Exists ( ) Xs.
    Proof. rewrite empty_union_list. apply (not_Forall_Exists _). Qed.

    Context `{!LeibnizEquiv C}.
    Lemma collection_subseteq_inv_L X Y : X  Y  X  Y  X = Y.
    Proof. unfold_leibniz. apply collection_subseteq_inv. Qed.
    Lemma collection_not_subset_inv_L X Y : X  Y  X  Y  X = Y.
    Proof. unfold_leibniz. apply collection_not_subset_inv. Qed.
    Lemma non_empty_union_L X Y : X  Y    X    Y  .
    Proof. unfold_leibniz. apply non_empty_union. Qed.
    Lemma non_empty_union_list_L Xs :  Xs    Exists ( ) Xs.
    Proof. unfold_leibniz. apply non_empty_union_list. Qed.
  End dec.
End simple_collection.


(** * Collections with [∪], [∩], [∖], [∅] and [{[_]}] *)
Robbert Krebbers's avatar
Robbert Krebbers committed
555
556
Section collection.
  Context `{Collection A C}.
557
  Implicit Types x y : A.
558
  Implicit Types X Y : C.
Robbert Krebbers's avatar
Robbert Krebbers committed
559

560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
  (** Intersection *)
  Lemma subseteq_intersection X Y : X  Y  X  Y  X.
  Proof. set_solver. Qed. 
  Lemma subseteq_intersection_1 X Y : X  Y  X  Y  X.
  Proof. apply subseteq_intersection. Qed.
  Lemma subseteq_intersection_2 X Y : X  Y  X  X  Y.
  Proof. apply subseteq_intersection. Qed.

  Lemma intersection_subseteq_l X Y : X  Y  X.
  Proof. set_solver. Qed.
  Lemma intersection_subseteq_r X Y : X  Y  Y.
  Proof. set_solver. Qed.
  Lemma intersection_greatest X Y Z : Z  X  Z  Y  Z  X  Y.
  Proof. set_solver. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
575
  Lemma intersection_mono_l X Y1 Y2 : Y1  Y2  X  Y1  X  Y2.
576
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
577
  Lemma intersection_mono_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
578
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
579
  Lemma intersection_mono X1 X2 Y1 Y2 :
580
    X1  X2  Y1  Y2  X1  Y1  X2  Y2.
581
  Proof. set_solver. Qed.
582
583
584
585
586
587
588
589
590
591
592
593

  Global Instance intersection_idemp : IdemP (() : relation C) ().
  Proof. intros X; set_solver. Qed.
  Global Instance intersection_comm : Comm (() : relation C) ().
  Proof. intros X Y; set_solver. Qed.
  Global Instance intersection_assoc : Assoc (() : relation C) ().
  Proof. intros X Y Z; set_solver. Qed.
  Global Instance intersection_empty_l : LeftAbsorb (() : relation C)  ().
  Proof. intros X; set_solver. Qed.
  Global Instance intersection_empty_r: RightAbsorb (() : relation C)  ().
  Proof. intros X; set_solver. Qed.

594
  Lemma intersection_singletons x : ({[x]} : C)  {[x]}  {[x]}.
595
  Proof. set_solver. Qed.
596
597
598
599
600
601
602
603
604
605
606

  Lemma union_intersection_l X Y Z : X  (Y  Z)  (X  Y)  (X  Z).
  Proof. set_solver. Qed.
  Lemma union_intersection_r X Y Z : (X  Y)  Z  (X  Z)  (Y  Z).
  Proof. set_solver. Qed.
  Lemma intersection_union_l X Y Z : X  (Y  Z)  (X  Y)  (X  Z).
  Proof. set_solver. Qed.
  Lemma intersection_union_r X Y Z : (X  Y)  Z  (X  Z)  (Y  Z).
  Proof. set_solver. Qed.

  (** Difference *)
Robbert Krebbers's avatar
Robbert Krebbers committed
607
  Lemma difference_twice X Y : (X  Y)  Y  X  Y.
608
  Proof. set_solver. Qed.
609
  Lemma subseteq_empty_difference X Y : X  Y  X  Y  .
610
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
611
  Lemma difference_diag X : X  X  .
612
  Proof. set_solver. Qed.
613
614
  Lemma difference_empty X : X    X.
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
615
  Lemma difference_union_distr_l X Y Z : (X  Y)  Z  X  Z  Y  Z.
616
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
617
  Lemma difference_union_distr_r X Y Z : Z  (X  Y)  (Z  X)  (Z  Y).
618
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
619
  Lemma difference_intersection_distr_l X Y Z : (X  Y)  Z  X  Z  Y  Z.
620
  Proof. set_solver. Qed.
621
  Lemma difference_disjoint X Y : X ## Y  X  Y  X.
622
  Proof. set_solver. Qed.
623
624
625
  Lemma subset_difference_elem_of {x: A} {s: C} (inx: x  s): s  {[ x ]}  s.
  Proof. set_solver. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
626

Robbert Krebbers's avatar
Robbert Krebbers committed
627
  Lemma difference_mono X1 X2 Y1 Y2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
628
629
    X1  X2  Y2  Y1  X1  Y1  X2  Y2.
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
630
  Lemma difference_mono_l X Y1 Y2 : Y2  Y1  X  Y1  X  Y2.
Robbert Krebbers's avatar
Robbert Krebbers committed
631
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
632
  Lemma difference_mono_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
633
634
  Proof. set_solver. Qed.

635
  (** Disjointness *)
636
  Lemma disjoint_intersection X Y : X ## Y  X  Y  .
637
638
  Proof. set_solver. Qed.

639
640
  Section leibniz.
    Context `{!LeibnizEquiv C}.
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660

    (** Intersection *)
    Lemma subseteq_intersection_L X Y : X  Y  X  Y = X.
    Proof. unfold_leibniz. apply subseteq_intersection. Qed.
    Lemma subseteq_intersection_1_L X Y : X  Y  X  Y = X.
    Proof. unfold_leibniz. apply subseteq_intersection_1. Qed.
    Lemma subseteq_intersection_2_L X Y : X  Y = X  X  Y.
    Proof. unfold_leibniz. apply subseteq_intersection_2. Qed.

    Global Instance intersection_idemp_L : IdemP ((=) : relation C) ().
    Proof. intros ?. unfold_leibniz. apply (idemp _). Qed.
    Global Instance intersection_comm_L : Comm ((=) : relation C) ().
    Proof. intros ??. unfold_leibniz. apply (comm _). Qed.
    Global Instance intersection_assoc_L : Assoc ((=) : relation C) ().
    Proof. intros ???. unfold_leibniz. apply (assoc _). Qed.
    Global Instance intersection_empty_l_L: LeftAbsorb ((=) : relation C)  ().
    Proof. intros ?. unfold_leibniz. apply (left_absorb _ _). Qed.
    Global Instance intersection_empty_r_L: RightAbsorb ((=) : relation C)  ().
    Proof. intros ?. unfold_leibniz. apply (right_absorb _ _). Qed.

661
    Lemma intersection_singletons_L x : {[x]}  {[x]} = ({[x]} : C).
662
    Proof. unfold_leibniz. apply intersection_singletons. Qed.
663
664
665
666
667

    Lemma union_intersection_l_L X Y Z : X  (Y  Z) = (X  Y)  (X  Z).
    Proof. unfold_leibniz; apply union_intersection_l. Qed.
    Lemma union_intersection_r_L X Y Z : (X  Y)  Z = (X  Z)  (Y  Z).
    Proof. unfold_leibniz; apply union_intersection_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
668
    Lemma intersection_union_l_L X Y Z : X  (Y  Z) = (X  Y)  (X  Z).
669
    Proof. unfold_leibniz; apply intersection_union_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
670
    Lemma intersection_union_r_L X Y Z : (X  Y)  Z = (X  Z)  (Y  Z).
671
672
673
    Proof. unfold_leibniz; apply intersection_union_r. Qed.

    (** Difference *)
674
675
    Lemma difference_twice_L X Y : (X  Y)  Y = X  Y.
    Proof. unfold_leibniz. apply difference_twice. Qed.
676
677
    Lemma subseteq_empty_difference_L X Y : X  Y  X  Y = .
    Proof. unfold_leibniz. apply subseteq_empty_difference. Qed.
678
679
    Lemma difference_diag_L X : X  X = .
    Proof. unfold_leibniz. apply difference_diag. Qed.
680
681
    Lemma difference_empty_L X : X   = X.
    Proof. unfold_leibniz. apply difference_empty. Qed.
682
683
    Lemma difference_union_distr_l_L X Y Z : (X  Y)  Z = X  Z  Y  Z.
    Proof. unfold_leibniz. apply difference_union_distr_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
684
685
    Lemma difference_union_distr_r_L X Y Z : Z  (X  Y) = (Z  X)  (Z  Y).
    Proof. unfold_leibniz. apply difference_union_distr_r. Qed.
686
687
688
    Lemma difference_intersection_distr_l_L X Y Z :
      (X  Y)  Z = X  Z  Y  Z.
    Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed.
689
    Lemma difference_disjoint_L X Y : X ## Y  X  Y = X.
690
    Proof. unfold_leibniz. apply difference_disjoint. Qed.
691
692

    (** Disjointness *)
693
    Lemma disjoint_intersection_L X Y : X ## Y  X  Y = .
694
    Proof. unfold_leibniz. apply disjoint_intersection. Qed.
695
696
697
  End leibniz.

  Section dec.
698
    Context `{!RelDecision (@elem_of A C _)}.
699
    Lemma not_elem_of_intersection x X Y : x  X  Y  x  X  x  Y.
700
    Proof. rewrite elem_of_intersection. destruct (decide (x  X)); tauto. Qed.
701
    Lemma not_elem_of_difference x X Y : x  X  Y  x  X  x  Y.
702
    Proof. rewrite elem_of_difference. destruct (decide (x  Y)); tauto. Qed.
703
704
    Lemma union_difference X Y : X  Y  Y  X  Y  X.
    Proof.
705
      intros ? x; split; rewrite !elem_of_union, elem_of_difference; [|intuition].
706
      destruct (decide (x  X)); intuition.
707
    Qed.
708
709
710
711
712
    Lemma difference_union X Y : X  Y  Y  X  Y.
    Proof.
      intros x. rewrite !elem_of_union; rewrite elem_of_difference.
      split; [ | destruct (decide (x  Y)) ]; intuition.
    Qed.
713
    Lemma subseteq_disjoint_union X Y : X  Y   Z, Y  X  Z  X ## Z.
714
715
716
717
    Proof.
      split; [|set_solver].
      exists (Y  X); split; [auto using union_difference|set_solver].
    Qed.
718
    Lemma non_empty_difference X Y : X  Y  Y  X  .
719
    Proof. intros [HXY1 HXY2] Hdiff. destruct HXY2. set_solver. Qed.
720
    Lemma empty_difference_subseteq X Y : X  Y    X  Y.
721
    Proof. set_solver. Qed.
722
723
724
725
    Lemma singleton_union_difference X Y x :
      {[x]}  (X  Y)  ({[x]}  X)  (Y  {[x]}).
    Proof.
      intro y; split; intros Hy; [ set_solver | ].
726
      destruct (decide (y  ({[x]} : C))); set_solver.
727
    Qed.
728

729
730
731
    Context `{!LeibnizEquiv C}.
    Lemma union_difference_L X Y : X  Y  Y = X  Y  X.
    Proof. unfold_leibniz. apply union_difference. Qed.
732
733
    Lemma difference_union_L X Y : X  Y  Y = X  Y.
    Proof. unfold_leibniz. apply difference_union. Qed.
734
735
    Lemma non_empty_difference_L X Y : X  Y  Y  X  .
    Proof. unfold_leibniz. apply non_empty_difference. Qed.
736
737
    Lemma empty_difference_subseteq_L X Y : X  Y =   X  Y.
    Proof. unfold_leibniz. apply empty_difference_subseteq. Qed.
738
    Lemma subseteq_disjoint_union_L X Y : X  Y   Z, Y = X  Z  X ## Z.
739
    Proof. unfold_leibniz. apply subseteq_disjoint_union. Qed.
740
741
742
    Lemma singleton_union_difference_L X Y x :
      {[x]}  (X  Y) = ({[x]}  X)  (Y  {[x]}).
    Proof. unfold_leibniz. apply singleton_union_difference. Qed.
743
744
745
  End dec.
End collection.

746
747
748
749
750
751
752
753
754

(** * Conversion of option and list *)
Definition of_option `{Singleton A C, Empty C} (mx : option A) : C :=
  match mx with None =>  | Some x => {[ x ]} end.
Fixpoint of_list `{Singleton A C, Empty C, Union C} (l : list A) : C :=
  match l with [] =>  | x :: l => {[ x ]}  of_list l end.

Section of_option_list.
  Context `{SimpleCollection A C}.
Robbert Krebbers's avatar
Robbert Krebbers committed
755
756
  Implicit Types l : list A.

757
  Lemma elem_of_of_option (x : A) mx: x  of_option (C:=C) mx  mx = Some x.
758
  Proof. destruct mx; set_solver. Qed.
759
  Lemma not_elem_of_of_option (x : A) mx: x  of_option (C:=C) mx  mx  Some x.
Robbert Krebbers's avatar
Robbert Krebbers committed
760
761
  Proof. by rewrite elem_of_of_option. Qed.

762
  Lemma elem_of_of_list (x : A) l : x  of_list (C:=C) l  x  l.
763
764
765
766
767
768
  Proof.
    split.
    - induction l; simpl; [by rewrite elem_of_empty|].
      rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto.
    - induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto.
  Qed.
769
  Lemma not_elem_of_of_list (x : A) l : x  of_list (C:=C) l  x  l.
Robbert Krebbers's avatar
Robbert Krebbers committed
770
771
  Proof. by rewrite elem_of_of_list. Qed.

772
  Global Instance set_unfold_of_option (mx : option A) x :
773
    SetUnfold (x  of_option (C:=C) mx) (mx = Some x).
774
775
  Proof. constructor; apply elem_of_of_option. Qed.
  Global Instance set_unfold_of_list (l : list A) x P :
776
    SetUnfold (x  l) P  SetUnfold (x  of_list (C:=C) l) P.
777
778
  Proof. constructor. by rewrite elem_of_of_list, (set_unfold (x  l) P). Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
779
780
781
782
783
784
785
786
  Lemma of_list_nil : of_list (C:=C) [] = .
  Proof. done. Qed.
  Lemma of_list_cons x l : of_list (C:=C) (x :: l) = {[ x ]}  of_list l.
  Proof. done. Qed.
  Lemma of_list_app l1 l2 : of_list (C:=C) (l1 ++ l2)  of_list l1  of_list l2.
  Proof. set_solver. Qed.
  Global Instance of_list_perm : Proper (() ==> ()) (of_list (C:=C)).
  Proof. induction 1; set_solver. Qed.
787

Robbert Krebbers's avatar
Robbert Krebbers committed
788
789
790
791
792
793
  Context `{!LeibnizEquiv C}.
  Lemma of_list_app_L l1 l2 : of_list (C:=C) (l1 ++ l2) = of_list l1  of_list l2.
  Proof. set_solver. Qed.
  Global Instance of_list_perm_L : Proper (() ==> (=)) (of_list (C:=C)).
  Proof. induction 1; set_solver. Qed.
End of_option_list.
794
795
796
797
798
799
800
801
802


(** * Guard *)
Global Instance collection_guard `{CollectionMonad M} : MGuard M :=
  λ P dec A x, match dec with left H => x H | _ =>  end.

Section collection_monad_base.
  Context `{CollectionMonad M}.
  Lemma elem_of_guard `{Decision P} {A} (x : A) (X : M A) :
803
    (x  guard P; X)  P  x  X.
804
805
806
807
808
809
810
  Proof.
    unfold mguard, collection_guard; simpl; case_match;
      rewrite ?elem_of_empty; naive_solver.
  Qed.
  Lemma elem_of_guard_2 `{Decision P} {A} (x : A) (X : M A) :
    P  x  X  x  guard P; X.
  Proof. by rewrite elem_of_guard. Qed.
811
  Lemma guard_empty `{Decision P} {A} (X : M A) : (guard P; X)    ¬P  X  .
812
813
814
815
  Proof.
    rewrite !elem_of_equiv_empty; setoid_rewrite elem_of_guard.
    destruct (decide P); naive_solver.
  Qed.
816
  Global Instance set_unfold_guard `{Decision P} {A} (x : A) (X : M A) Q :
817
818
819
820
821
822
823
824
    SetUnfold (x  X) Q  SetUnfold (x  guard P; X) (P  Q).
  Proof. constructor. by rewrite elem_of_guard, (set_unfold (x  X) Q). Qed.
  Lemma bind_empty {A B} (f : A  M B) X :
    X = f    X     x, x  X  f x  .
  Proof. set_solver.