fin_maps.v 38.2 KB
Newer Older
1
2
3
4
5
6
(* Copyright (c) 2012, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** Finite maps associate data to keys. This file defines an interface for
finite maps and collects some theory on it. Most importantly, it proves useful
induction principles for finite maps and implements the tactic [simplify_map]
to simplify goals involving finite maps. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
7
Require Export prelude.
8
9
(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
10
11
12
13
14
course limits the space of finite map implementations, but since we are mainly
interested in finite maps with numbers as indexes, we do not consider this to
be a serious limitation. The main application of finite maps is to implement
the memory, where extensionality of Leibniz equality is very important for a
convenient use in the assertions of our axiomatic semantics. *)
15

16
17
18
(** Finiteness is axiomatized by requiring each map to have a finite domain.
Since we may have multiple implementations of finite sets, the [dom] function is
parametrized by an implementation of finite sets over the map's key type. *)
19

20
21
22
(** Finite map implementations are required to implement the [merge] function
which enables us to give a generic implementation of [union_with],
[intersection_with], and [difference_with]. *)
23
24
25
26

Class FinMap K M `{!FMap M}
    `{ A, Lookup K (M A) A} `{ A, Empty (M A)}
    `{ A, PartialAlter K (M A) A} `{ A, Dom K (M A)} `{!Merge M}
27
    `{ i j : K, Decision (i = j)} := {
28
29
30
31
32
33
34
35
36
37
38
39
  finmap_eq {A} (m1 m2 : M A) :
    ( i, m1 !! i = m2 !! i)  m1 = m2;
  lookup_empty {A} i :
    ( : M A) !! i = None;
  lookup_partial_alter {A} f (m : M A) i :
    partial_alter f i m !! i = f (m !! i);
  lookup_partial_alter_ne {A} f (m : M A) i j :
    i  j  partial_alter f i m !! j = m !! j;
  lookup_fmap {A B} (f : A  B) (m : M A) i :
    (f <$> m) !! i = f <$> m !! i;
  elem_of_dom C {A} `{Collection K C} (m : M A) i :
    i  dom C m  is_Some (m !! i);
40
  merge_spec {A} f `{!PropHolds (f None None = None)}
Robbert Krebbers's avatar
Robbert Krebbers committed
41
42
43
    (m1 m2 : M A) i : merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
}.

44
45
46
(** * Derived operations *)
(** All of the following functions are defined in a generic way for arbitrary
finite map implementations. These generic implementations do not cause a
47
48
49
50
51
significant performance loss to make including them in the finite map interface
worthwhile. *)
Instance finmap_insert `{PartialAlter K M A} : Insert K M A := λ i x,
  partial_alter (λ _, Some x) i.
Instance finmap_alter `{PartialAlter K M A} : Alter K M A := λ f,
52
  partial_alter (fmap f).
53
Instance finmap_delete `{PartialAlter K M A} : Delete K M :=
54
  partial_alter (λ _, None).
Robbert Krebbers's avatar
Robbert Krebbers committed
55

56
57
58
59
Instance finmap_singleton `{PartialAlter K M A}
  `{Empty M} : Singleton (K * A) M := λ p, <[fst p:=snd p]>.
Definition list_to_map `{Insert K M A} `{Empty M}
  (l : list (K * A)) : M := insert_list l .
Robbert Krebbers's avatar
Robbert Krebbers committed
60

Robbert Krebbers's avatar
Robbert Krebbers committed
61
Instance finmap_union_with `{Merge M} : UnionWith M := λ A f,
62
  merge (union_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
63
Instance finmap_intersection_with `{Merge M} : IntersectionWith M := λ A f,
64
  merge (intersection_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
65
Instance finmap_difference_with `{Merge M} : DifferenceWith M := λ A f,
66
  merge (difference_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
67

68
69
70
71
72
73
(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
Definition intersection_forall `{Lookup K M A} (R : relation A) : relation M :=
  λ m1 m2,  i x1 x2, m1 !! i = Some x1  m2 !! i = Some x2  R x1 x2.
Instance finmap_disjoint `{ A, Lookup K (M A) A} : Disjoint (M A) := λ A,
  intersection_forall (λ _ _, False).
Robbert Krebbers's avatar
Robbert Krebbers committed
74
75
76
77
78

(** The union of two finite maps only has a meaningful definition for maps
that are disjoint. However, as working with partial functions is inconvenient
in Coq, we define the union as a total function. In case both finite maps
have a value at the same index, we take the value of the first map. *)
79
80
81
82
83
84
85
86
Instance finmap_union `{Merge M} {A} : Union (M A) :=
  union_with (λ x _, x).
Instance finmap_intersection `{Merge M} {A} : Intersection (M A) :=
  union_with (λ x _, x).
(** The difference operation removes all values from the first map whose
index contains a value in the second map as well. *)
Instance finmap_difference `{Merge M} {A} : Difference (M A) :=
  difference_with (λ _ _, None).
Robbert Krebbers's avatar
Robbert Krebbers committed
87

88
(** * General theorems *)
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
Section finmap_common.
  Context `{FinMap K M} {A : Type}.

  Global Instance finmap_subseteq: SubsetEq (M A) := λ m n,
     i x, m !! i = Some x  n !! i = Some x.
  Global Instance: BoundedPreOrder (M A).
  Proof. split; [firstorder |]. intros m i x. by rewrite lookup_empty. Qed.

  Lemma lookup_weaken (m1 m2 : M A) i x :
    m1 !! i = Some x  m1  m2  m2 !! i = Some x.
  Proof. auto. Qed.
  Lemma lookup_weaken_None (m1 m2 : M A) i :
    m2 !! i = None  m1  m2  m1 !! i = None.
  Proof.
    rewrite eq_None_not_Some. intros Hm2 Hm1m2.
    specialize (Hm1m2 i). destruct (m1 !! i); naive_solver.
  Qed.

  Lemma lookup_weaken_inv (m1 m2 : M A) i x y :
108
    m1 !! i = Some x 
109
    m1  m2 
110
    m2 !! i = Some y 
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
    x = y.
  Proof.
    intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto.
    congruence.
  Qed.

  Lemma lookup_ne (m : M A) i j : m !! i  m !! j  i  j.
  Proof. congruence. Qed.

  Lemma not_elem_of_dom C `{Collection K C} (m : M A) i :
    i  dom C m  m !! i = None.
  Proof. by rewrite (elem_of_dom C), eq_None_not_Some. Qed.

  Lemma finmap_empty (m : M A) : ( i, m !! i = None)  m = .
  Proof. intros Hm. apply finmap_eq. intros. by rewrite Hm, lookup_empty. Qed.
  Lemma dom_empty C `{Collection K C} : dom C ( : M A)  .
127
  Proof.
128
129
130
    split; intro.
    * rewrite (elem_of_dom C), lookup_empty. by inversion 1.
    * solve_elem_of.
131
  Qed.
132
  Lemma dom_empty_inv C `{Collection K C} (m : M A) : dom C m    m = .
133
  Proof.
134
135
    intros E. apply finmap_empty. intros. apply (not_elem_of_dom C).
    rewrite E. solve_elem_of.
136
  Qed.
137
138
139
140
141
142
143
144

  Lemma lookup_empty_not i : ¬is_Some (( : M A) !! i).
  Proof. rewrite lookup_empty. by inversion 1. Qed.
  Lemma lookup_empty_Some i (x : A) : ¬ !! i = Some x.
  Proof. by rewrite lookup_empty. Qed.

  Lemma partial_alter_compose (m : M A) i f g :
    partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Robbert Krebbers's avatar
Robbert Krebbers committed
145
  Proof.
146
147
148
    intros. apply finmap_eq. intros ii. case (decide (i = ii)).
    * intros. subst. by rewrite !lookup_partial_alter.
    * intros. by rewrite !lookup_partial_alter_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
149
  Qed.
150
151
152
153
  Lemma partial_alter_comm (m : M A) i j f g :
    i  j 
      partial_alter f i (partial_alter g j m)
    = partial_alter g j (partial_alter f i m).
Robbert Krebbers's avatar
Robbert Krebbers committed
154
  Proof.
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
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
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
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
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
    intros. apply finmap_eq. intros jj.
    destruct (decide (jj = j)).
    * subst. by rewrite lookup_partial_alter_ne,
       !lookup_partial_alter, lookup_partial_alter_ne.
    * destruct (decide (jj = i)).
      + subst. by rewrite lookup_partial_alter,
         !lookup_partial_alter_ne, lookup_partial_alter by congruence.
      + by rewrite !lookup_partial_alter_ne by congruence.
  Qed.
  Lemma partial_alter_self_alt (m : M A) i x :
    x = m !! i  partial_alter (λ _, x) i m = m.
  Proof.
    intros. apply finmap_eq. intros ii.
    destruct (decide (i = ii)).
    * subst. by rewrite lookup_partial_alter.
    * by rewrite lookup_partial_alter_ne.
  Qed.
  Lemma partial_alter_self (m : M A) i : partial_alter (λ _, m !! i) i m = m.
  Proof. by apply partial_alter_self_alt. Qed.

  Lemma lookup_insert (m : M A) i x : <[i:=x]>m !! i = Some x.
  Proof. unfold insert. apply lookup_partial_alter. Qed.
  Lemma lookup_insert_rev (m : M A) i x y : <[i:= x ]>m !! i = Some y  x = y.
  Proof. rewrite lookup_insert. congruence. Qed.
  Lemma lookup_insert_ne (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
  Proof. unfold insert. apply lookup_partial_alter_ne. Qed.
  Lemma insert_comm (m : M A) i j x y :
    i  j  <[i:=x]>(<[j:=y]>m) = <[j:=y]>(<[i:=x]>m).
  Proof. apply partial_alter_comm. Qed.

  Lemma lookup_insert_Some (m : M A) i j x y :
    <[i:=x]>m !! j = Some y  (i = j  x = y)  (i  j  m !! j = Some y).
  Proof.
    split.
    * destruct (decide (i = j)); subst;
        rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
    * intros [[??]|[??]].
      + subst. apply lookup_insert.
      + by rewrite lookup_insert_ne.
  Qed.
  Lemma lookup_insert_None (m : M A) i j x :
    <[i:=x]>m !! j = None  m !! j = None  i  j.
  Proof.
    split.
    * destruct (decide (i = j)); subst;
        rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
    * intros [??]. by rewrite lookup_insert_ne.
  Qed.

  Lemma lookup_singleton_Some i j (x y : A) :
    {[(i, x)]} !! j = Some y  i = j  x = y.
  Proof.
    unfold singleton, finmap_singleton.
    rewrite lookup_insert_Some, lookup_empty. simpl.
    intuition congruence.
  Qed.
  Lemma lookup_singleton_None i j (x : A) :
    {[(i, x)]} !! j = None  i  j.
  Proof.
    unfold singleton, finmap_singleton.
    rewrite lookup_insert_None, lookup_empty. simpl. tauto.
  Qed.
  Lemma insert_singleton i (x y : A) : <[i:=y]>{[(i, x)]} = {[(i, y)]}.
  Proof.
    unfold singleton, finmap_singleton, insert, finmap_insert.
    by rewrite <-partial_alter_compose.
  Qed.

  Lemma lookup_singleton i (x : A) : {[(i, x)]} !! i = Some x.
  Proof. by rewrite lookup_singleton_Some. Qed.
  Lemma lookup_singleton_ne i j (x : A) : i  j  {[(i, x)]} !! j = None.
  Proof. by rewrite lookup_singleton_None. Qed.

  Lemma lookup_delete (m : M A) i : delete i m !! i = None.
  Proof. apply lookup_partial_alter. Qed.
  Lemma lookup_delete_ne (m : M A) i j : i  j  delete i m !! j = m !! j.
  Proof. apply lookup_partial_alter_ne. Qed.

  Lemma lookup_delete_Some (m : M A) i j y :
    delete i m !! j = Some y  i  j  m !! j = Some y.
  Proof.
    split.
    * destruct (decide (i = j)); subst;
        rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
    * intros [??]. by rewrite lookup_delete_ne.
  Qed.
  Lemma lookup_delete_None (m : M A) i j :
    delete i m !! j = None  i = j  m !! j = None.
  Proof.
    destruct (decide (i = j)).
    * subst. rewrite lookup_delete. tauto.
    * rewrite lookup_delete_ne; tauto.
  Qed.

  Lemma delete_empty i : delete i ( : M A) = .
  Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
  Lemma delete_singleton i (x : A) : delete i {[(i, x)]} = .
  Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
  Lemma delete_comm (m : M A) i j :
    delete i (delete j m) = delete j (delete i m).
  Proof. destruct (decide (i = j)). by subst. by apply partial_alter_comm. Qed.
  Lemma delete_insert_comm (m : M A) i j x :
    i  j  delete i (<[j:=x]>m) = <[j:=x]>(delete i m).
  Proof. intro. by apply partial_alter_comm. Qed.

  Lemma delete_notin (m : M A) i :
    m !! i = None  delete i m = m.
  Proof.
    intros. apply finmap_eq. intros j.
    destruct (decide (i = j)).
    * subst. by rewrite lookup_delete.
    * by apply lookup_delete_ne.
  Qed.

  Lemma delete_partial_alter (m : M A) i f :
    m !! i = None  delete i (partial_alter f i m) = m.
  Proof.
    intros. unfold delete, finmap_delete. rewrite <-partial_alter_compose.
    rapply partial_alter_self_alt. congruence.
  Qed.
  Lemma delete_partial_alter_dom C `{Collection K C} (m : M A) i f :
    i  dom C m  delete i (partial_alter f i m) = m.
  Proof. rewrite (not_elem_of_dom C). apply delete_partial_alter. Qed.
  Lemma delete_insert (m : M A) i x :
    m !! i = None  delete i (<[i:=x]>m) = m.
  Proof. apply delete_partial_alter. Qed.
  Lemma delete_insert_dom C `{Collection K C} (m : M A) i x :
    i  dom C m  delete i (<[i:=x]>m) = m.
  Proof. rewrite (not_elem_of_dom C). apply delete_partial_alter. Qed.
  Lemma insert_delete (m : M A) i x :
    m !! i = Some x  <[i:=x]>(delete i m) = m.
  Proof.
    intros Hmi. unfold delete, finmap_delete, insert, finmap_insert.
    rewrite <-partial_alter_compose. unfold compose. rewrite <-Hmi.
    by apply partial_alter_self_alt.
  Qed.

  Lemma elem_of_dom_delete C `{Collection K C} (m : M A) i j :
    i  dom C (delete j m)  i  j  i  dom C m.
  Proof.
    rewrite !(elem_of_dom C), <-!not_eq_None_Some.
    rewrite lookup_delete_None. intuition.
  Qed.
  Lemma not_elem_of_dom_delete C `{Collection K C} (m : M A) i :
    i  dom C (delete i m).
  Proof. apply (not_elem_of_dom C), lookup_delete. Qed.

  Lemma lookup_delete_list (m : M A) is j :
    j  is  delete_list is m !! j = None.
  Proof.
    induction 1 as [|i j is]; simpl.
    * by rewrite lookup_delete.
    * destruct (decide (i = j)).
      + subst. by rewrite lookup_delete.
      + rewrite lookup_delete_ne; auto.
  Qed.
  Lemma lookup_delete_list_not_elem_of (m : M A) is j :
    j  is  delete_list is m !! j = m !! j.
  Proof.
    induction is; simpl; [done |].
    rewrite elem_of_cons. intros.
    intros. rewrite lookup_delete_ne; intuition.
  Qed.
  Lemma delete_list_notin (m : M A) is :
    Forall (λ i, m !! i = None) is  delete_list is m = m.
  Proof.
    induction 1; simpl; [done |].
    rewrite delete_notin; congruence.
  Qed.

  Lemma delete_list_insert_comm (m : M A) is j x :
    j  is  delete_list is (<[j:=x]>m) = <[j:=x]>(delete_list is m).
  Proof.
    induction is; simpl; [done |].
    rewrite elem_of_cons. intros.
    rewrite IHis, delete_insert_comm; intuition.
  Qed.

  (** * Induction principles *)
  (** We use the induction principle on finite collections to prove the
  following induction principle on finite maps. *)
  Lemma finmap_ind_alt C (P : M A  Prop) `{FinCollection K C} :
    P  
    ( i x m, i  dom C m  P m  P (<[i:=x]>m)) 
     m, P m.
  Proof.
    intros Hemp Hinsert m.
    apply (collection_ind (λ X,  m, dom C m  X  P m)) with (dom C m).
    * solve_proper.
    * clear m. intros m Hm. rewrite finmap_empty.
      + done.
      + intros. rewrite <-(not_elem_of_dom C), Hm.
        by solve_elem_of.
    * clear m. intros i X Hi IH m Hdom.
      assert ( x, m !! i = Some x) as [x ?].
      { apply is_Some_alt, (elem_of_dom C).
        rewrite Hdom. clear Hdom.
        by solve_elem_of. }
      rewrite <-(insert_delete m i x) by done.
      apply Hinsert.
      { by apply (not_elem_of_dom_delete C). }
      apply IH. apply elem_of_equiv. intros.
      rewrite (elem_of_dom_delete C).
      esolve_elem_of.
    * done.
  Qed.

  (** We use the [listset] implementation to prove an induction principle that
  does not use the map's domain. *)
  Lemma finmap_ind (P : M A  Prop) :
    P  
    ( i x m, m !! i = None  P m  P (<[i:=x]>m)) 
     m, P m.
  Proof.
    setoid_rewrite <-(not_elem_of_dom (listset _)).
    apply (finmap_ind_alt (listset _) P).
Robbert Krebbers's avatar
Robbert Krebbers committed
371
372
  Qed.

373
374
375
376
377
378
379
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
411
412
413
414
415
  (** * Properties of the merge operation *)
  Section merge_with.
    Context (f : option A  option A  option A).

    Global Instance: LeftId (=) None f  LeftId (=)  (merge f).
    Proof.
      intros ??. apply finmap_eq. intros.
      by rewrite !(merge_spec f), lookup_empty, (left_id None f).
    Qed.
    Global Instance: RightId (=) None f  RightId (=)  (merge f).
    Proof.
      intros ??. apply finmap_eq. intros.
      by rewrite !(merge_spec f), lookup_empty, (right_id None f).
    Qed.
    Global Instance: Idempotent (=) f  Idempotent (=) (merge f).
    Proof. intros ??. apply finmap_eq. intros. by rewrite !(merge_spec f). Qed.

    Context `{!PropHolds (f None None = None)}.

    Lemma merge_spec_alt m1 m2 m :
      ( i, m !! i = f (m1 !! i) (m2 !! i))  merge f m1 m2 = m.
    Proof.
      split; [| intro; subst; apply (merge_spec _) ].
      intros Hlookup. apply finmap_eq. intros. rewrite Hlookup.
      apply (merge_spec _).
    Qed.

    Lemma merge_comm m1 m2 :
      ( i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) 
      merge f m1 m2 = merge f m2 m1.
    Proof. intros. apply finmap_eq. intros. by rewrite !(merge_spec f). Qed.
    Global Instance: Commutative (=) f  Commutative (=) (merge f).
    Proof. intros ???. apply merge_comm. intros. by apply (commutative f). Qed.

    Lemma merge_assoc m1 m2 m3 :
      ( i, f (m1 !! i) (f (m2 !! i) (m3 !! i)) =
            f (f (m1 !! i) (m2 !! i)) (m3 !! i)) 
      merge f m1 (merge f m2 m3) = merge f (merge f m1 m2) m3.
    Proof. intros. apply finmap_eq. intros. by rewrite !(merge_spec f). Qed.
    Global Instance: Associative (=) f  Associative (=) (merge f).
    Proof. intros ????. apply merge_assoc. intros. by apply (associative f). Qed.
  End merge_with.
End finmap_common.
416

417
418
(** * The finite map tactic *)
(** The tactic [simplify_map by tac] simplifies finite map expressions
419
occuring in the conclusion and hypotheses. It uses [tac] to discharge generated
420
inequalities. *)
421
Tactic Notation "simpl_map" "by" tactic3(tac) := repeat
422
423
424
  match goal with
  | H : context[  !! _ ] |- _ => rewrite lookup_empty in H
  | H : context[ (<[_:=_]>_) !! _ ] |- _ => rewrite lookup_insert in H
425
  | H : context[ (<[_:=_]>_) !! _ ] |- _ => rewrite lookup_insert_ne in H by tac
426
427
  | H : context[ (delete _ _) !! _] |- _ => rewrite lookup_delete in H
  | H : context[ (delete _ _) !! _] |- _ => rewrite lookup_delete_ne in H by tac
428
  | H : context[ {[ _ ]} !! _ ] |- _ => rewrite lookup_singleton in H
429
  | H : context[ {[ _ ]} !! _ ] |- _ => rewrite lookup_singleton_ne in H by tac
430
431
  | |- context[  !! _ ] => rewrite lookup_empty
  | |- context[ (<[_:=_]>_) !! _ ] => rewrite lookup_insert
432
  | |- context[ (<[_:=_]>_) !! _ ] => rewrite lookup_insert_ne by tac
433
  | |- context[ (delete _ _) !! _ ] => rewrite lookup_delete
434
  | |- context[ (delete _ _) !! _ ] => rewrite lookup_delete_ne by tac
435
  | |- context[ {[ _ ]} !! _ ] => rewrite lookup_singleton
436
  | |- context[ {[ _ ]} !! _ ] => rewrite lookup_singleton_ne by tac
437
  end.
438
Tactic Notation "simpl_map" := simpl_map by auto.
439

440
441
442
443
444
445
446
447
448
449
Tactic Notation "simplify_map_equality" "by" tactic3(tac) := repeat
  match goal with
  | _ => progress simpl_map by tac
  | _ => progress simplify_equality
  | H1 : ?m1 !! ?i = Some ?x, H2 : ?m2 !! ?i = Some ?y |- _ =>
    let H3 := fresh in
    feed pose proof (lookup_weaken_inv m1 m2 i x y) as H3;
      [done | by tac | done | ];
    clear H2; symmetry in H3
  end.
450
Tactic Notation "simplify_map_equality" := simplify_map_equality by auto.
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994

Section finmap_more.
  Context `{FinMap K M} {A : Type}.

  (** * Properties on the relation [intersection_forall] *)
  Section intersection_forall.
    Context (R : relation A).

    Global Instance intersection_forall_sym:
      Symmetric R  Symmetric (intersection_forall R).
    Proof. firstorder auto. Qed.
    Lemma intersection_forall_empty_l (m : M A) : intersection_forall R  m.
    Proof. intros ???. by simpl_map. Qed.
    Lemma intersection_forall_empty_r (m : M A) : intersection_forall R m .
    Proof. intros ???. by simpl_map. Qed.

    (** Due to the finiteness of finite maps, we can extract a witness are
    property does not hold for the intersection. *)
    Lemma not_intersection_forall `{ x y, Decision (R x y)} (m1 m2 : M A) :
      ¬intersection_forall R m1 m2
          i x1 x2, m1 !! i = Some x1  m2 !! i = Some x2  ¬R x1 x2.
    Proof.
      split.
      * intros Hdisjoint.
        set (Pi i :=  x1 x2, m1 !! i = Some x1  m2 !! i = Some x2  ¬R x1 x2).
        assert ( i, Decision (Pi i)).
        { intros i. unfold Decision, Pi.
          destruct (m1 !! i) as [x1|], (m2 !! i) as [x2|]; try (by left).
          destruct (decide (R x1 x2)).
          * naive_solver.
          * intuition congruence. }
        destruct (decide (cexists Pi (dom (listset _) m1  dom (listset _) m2)))
          as [[i [Hdom Hi]] | Hi].
        + rewrite elem_of_intersection in Hdom.
          rewrite !(elem_of_dom (listset _)), !is_Some_alt in Hdom.
          destruct Hdom as [[x1 ?] [x2 ?]]. exists i x1 x2; auto.
        + destruct Hdisjoint. intros i x1 x2 Hx1 Hx2.
          apply dec_stable. intros HP.
          destruct Hi. exists i.
          rewrite elem_of_intersection, !(elem_of_dom (listset _)).
          intuition eauto; congruence.
      * intros (i & x1 & x2 & Hx1 & Hx2 & Hx1x2) Hdisjoint.
        by apply Hx1x2, (Hdisjoint i x1 x2).
    Qed.
  End intersection_forall.

  (** * Properties on the disjoint maps *)
  Lemma finmap_disjoint_alt (m1 m2 : M A) :
    m1  m2   i, m1 !! i = None  m2 !! i = None.
  Proof.
    split; intros Hm1m2 i; specialize (Hm1m2 i);
      destruct (m1 !! i), (m2 !! i); naive_solver.
  Qed.    
  Lemma finmap_not_disjoint (m1 m2 : M A) :
    ¬m1  m2   i x1 x2, m1 !! i = Some x1  m2 !! i = Some x2.
  Proof.
    unfold disjoint, finmap_disjoint.
    rewrite not_intersection_forall.
    * firstorder auto.
    * right. auto.
  Qed.

  Global Instance: Symmetric (@disjoint (M A) _).
  Proof. apply intersection_forall_sym. auto. Qed.
  Lemma finmap_disjoint_empty_l (m : M A) :   m.
  Proof. apply intersection_forall_empty_l. Qed.
  Lemma finmap_disjoint_empty_r (m : M A) : m  .
  Proof. apply intersection_forall_empty_r. Qed.

  Lemma finmap_disjoint_weaken (m1 m1' m2 m2' : M A) :
    m1'  m2' 
    m1  m1'  m2  m2' 
    m1  m2.
  Proof.
    intros Hdisjoint Hm1 Hm2 i x1 x2 Hx1 Hx2.
    destruct (Hdisjoint i x1 x2); auto.
  Qed.
  Lemma finmap_disjoint_weaken_l (m1 m1' m2  : M A) :
    m1'  m2  m1  m1'  m1  m2.
  Proof. eauto using finmap_disjoint_weaken. Qed.
  Lemma finmap_disjoint_weaken_r (m1 m2 m2' : M A) :
    m1  m2'  m2  m2'  m1  m2.
  Proof. eauto using finmap_disjoint_weaken. Qed.

  Lemma finmap_disjoint_Some_l (m1 m2 : M A) i x:
    m1  m2 
    m1 !! i = Some x 
    m2 !! i = None.
  Proof.
    intros Hdisjoint ?. rewrite eq_None_not_Some, is_Some_alt.
    intros [x2 ?]. by apply (Hdisjoint i x x2).
  Qed.
  Lemma finmap_disjoint_Some_r (m1 m2 : M A) i x:
    m1  m2 
    m2 !! i = Some x 
    m1 !! i = None.
  Proof. rewrite (symmetry_iff ()). apply finmap_disjoint_Some_l. Qed.

  Lemma finmap_disjoint_singleton_l (m : M A) i x :
    {[(i, x)]}  m  m !! i = None.
  Proof.
    split.
    * intro. apply (finmap_disjoint_Some_l {[(i, x)]} _ _ x); by simpl_map.
    * intros ? j y1 y2 ??.
      destruct (decide (i = j)); simplify_map_equality; congruence.
  Qed.
  Lemma finmap_disjoint_singleton_r (m : M A) i x :
    m  {[(i, x)]}  m !! i = None.
  Proof. by rewrite (symmetry_iff ()), finmap_disjoint_singleton_l. Qed.

  Lemma finmap_disjoint_singleton_l_2 (m : M A) i x :
    m !! i = None  {[(i, x)]}  m.
  Proof. by rewrite finmap_disjoint_singleton_l. Qed.
  Lemma finmap_disjoint_singleton_r_2 (m : M A) i x :
    m !! i = None  m  {[(i, x)]}.
  Proof. by rewrite finmap_disjoint_singleton_r. Qed.

  (** * Properties of the union and intersection operation *)
  Section union_intersection_with.
    Context (f : A  A  A).

    Lemma finmap_union_with_Some m1 m2 i x y :
      m1 !! i = Some x 
      m2 !! i = Some y 
      union_with f m1 m2 !! i = Some (f x y).
    Proof.
      intros Hx Hy. unfold union_with, finmap_union_with.
      by rewrite (merge_spec _), Hx, Hy.
    Qed.
    Lemma finmap_union_with_Some_l m1 m2 i x :
      m1 !! i = Some x 
      m2 !! i = None 
      union_with f m1 m2 !! i = Some x.
    Proof.
      intros Hx Hy. unfold union_with, finmap_union_with.
      by rewrite (merge_spec _), Hx, Hy.
    Qed.
    Lemma finmap_union_with_Some_r m1 m2 i y :
      m1 !! i = None 
      m2 !! i = Some y 
      union_with f m1 m2 !! i = Some y.
    Proof.
      intros Hx Hy. unfold union_with, finmap_union_with.
      by rewrite (merge_spec _), Hx, Hy.
    Qed.
    Lemma finmap_union_with_None m1 m2 i :
      union_with f m1 m2 !! i = None  m1 !! i = None  m2 !! i = None.
    Proof.
      unfold union_with, finmap_union_with. rewrite (merge_spec _).
      destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
    Qed.

    Global Instance: LeftId (=)  (@union_with M _ _ f).
    Proof. unfold union_with, finmap_union_with. apply _. Qed.
    Global Instance: RightId (=)  (@union_with M _ _ f).
    Proof. unfold union_with, finmap_union_with. apply _. Qed.
    Global Instance: Commutative (=) f  Commutative (=) (@union_with M _ _ f).
    Proof. unfold union_with, finmap_union_with. apply _. Qed.
    Global Instance: Associative (=) f  Associative (=) (@union_with M _ _ f).
    Proof. unfold union_with, finmap_union_with. apply _. Qed.
    Global Instance: Idempotent (=) f  Idempotent (=) (@union_with M _ _ f).
    Proof. unfold union_with, finmap_union_with. apply _. Qed.
  End union_intersection_with.

  Global Instance: LeftId (=)  (@union (M A) _) := _.
  Global Instance: RightId (=)  (@union (M A) _) := _.
  Global Instance: Associative (=) (@union (M A) _) := _.
  Global Instance: Idempotent (=) (@union (M A) _) := _.

  Lemma finmap_union_Some_raw (m1 m2 : M A) i x :
    (m1  m2) !! i = Some x 
      m1 !! i = Some x  (m1 !! i = None  m2 !! i = Some x).
  Proof.
    unfold union, finmap_union, union_with, finmap_union_with.
    rewrite (merge_spec _).
    destruct (m1 !! i), (m2 !! i); compute; try intuition congruence.
  Qed.
  Lemma finmap_union_None (m1 m2 : M A) i :
    (m1  m2) !! i = None  m1 !! i = None  m2 !! i = None.
  Proof. apply finmap_union_with_None. Qed.

  Lemma finmap_union_Some (m1 m2 : M A) i x :
    m1  m2 
    (m1  m2) !! i = Some x  m1 !! i = Some x  m2 !! i = Some x.
  Proof.
    intros Hdisjoint. rewrite finmap_union_Some_raw.
    intuition eauto using finmap_disjoint_Some_r.
  Qed.

  Lemma finmap_union_Some_l (m1 m2 : M A) i x :
    m1  m2 
    m1 !! i = Some x 
    (m1  m2) !! i = Some x.
  Proof. intro. rewrite finmap_union_Some; intuition. Qed.
  Lemma finmap_union_Some_r (m1 m2 : M A) i x :
    m1  m2 
    m2 !! i = Some x 
    (m1  m2) !! i = Some x.
  Proof. intro. rewrite finmap_union_Some; intuition. Qed.

  Lemma finmap_union_comm (m1 m2 : M A) :
    m1  m2 
    m1  m2 = m2  m1.
  Proof.
    intros Hdisjoint. apply (merge_comm (union_with (λ x _, x))).
    intros i. specialize (Hdisjoint i).
    destruct (m1 !! i), (m2 !! i); compute; naive_solver.
  Qed.

  Lemma finmap_subseteq_union (m1 m2 : M A) :
    m1  m2 
    m1  m2 = m2.
  Proof.
    intros Hm1m2.
    apply finmap_eq. intros i. apply option_eq. intros x.
    rewrite finmap_union_Some_raw. split; [by intuition |].
    intros Hm2. specialize (Hm1m2 i).
    destruct (m1 !! i) as [y|]; [| by auto].
    rewrite (Hm1m2 y eq_refl) in Hm2. intuition congruence.
  Qed.
  Lemma finmap_subseteq_union_l (m1 m2 : M A) :
    m1  m2 
    m1  m1  m2.
  Proof. intros ? i x. rewrite finmap_union_Some_raw. intuition. Qed.
  Lemma finmap_subseteq_union_r (m1 m2 : M A) :
    m1  m2 
    m2  m1  m2.
  Proof.
    intros. rewrite finmap_union_comm by done.
    by apply finmap_subseteq_union_l.
  Qed.

  Lemma finmap_disjoint_union_l (m1 m2 m3 : M A) :
    m1  m2  m3  m1  m3  m2  m3.
  Proof.
    rewrite !finmap_disjoint_alt.
    setoid_rewrite finmap_union_None. naive_solver.
  Qed.
  Lemma finmap_disjoint_union_r (m1 m2 m3 : M A) :
    m1  m2  m3  m1  m2  m1  m3.
  Proof.
    rewrite !finmap_disjoint_alt.
    setoid_rewrite finmap_union_None. naive_solver.
  Qed.
  Lemma finmap_disjoint_union_l_2 (m1 m2 m3 : M A) :
    m1  m3  m2  m3  m1  m2  m3.
  Proof. by rewrite finmap_disjoint_union_l. Qed.
  Lemma finmap_disjoint_union_r_2 (m1 m2 m3 : M A) :
    m1  m2  m1  m3  m1  m2  m3.
  Proof. by rewrite finmap_disjoint_union_r. Qed.
  Lemma finmap_union_cancel_l (m1 m2 m3 : M A) :
    m1  m3 
    m2  m3 
    m1  m3 = m2  m3 
    m1 = m2.
  Proof.
    revert m1 m2 m3.
    cut ( (m1 m2 m3 : M A) i x,
      m1  m3 
      m2  m3 
      m1  m3 = m2  m3 
      m1 !! i = Some x  m2 !! i = Some x).
    { intros. apply finmap_eq. intros i.
      apply option_eq. naive_solver. }
    intros m1 m2 m3 b v Hm1m3 Hm2m3 E ?.
    destruct (proj1 (finmap_union_Some m2 m3 b v Hm2m3)) as [E2|E2].
    * rewrite <-E. by apply finmap_union_Some_l.
    * done.
    * contradict E2. by apply eq_None_ne_Some, finmap_disjoint_Some_l with m1 v.
  Qed.
  Lemma finmap_union_cancel_r (m1 m2 m3 : M A) :
    m1  m3 
    m2  m3 
    m3  m1 = m3  m2 
    m1 = m2.
  Proof.
    intros ??. rewrite !(finmap_union_comm m3) by done.
    by apply finmap_union_cancel_l.
  Qed.

  Lemma finmap_union_singleton_l (m : M A) i x :
    <[i:=x]>m = {[(i,x)]}  m.
  Proof.
    apply finmap_eq. intros j. apply option_eq. intros y.
    rewrite finmap_union_Some_raw.
    destruct (decide (i = j)); simplify_map_equality; intuition congruence.
  Qed.
  Lemma finmap_union_singleton_r (m : M A) i x :
    m !! i = None 
    <[i:=x]>m = m  {[(i,x)]}.
  Proof.
    intro. rewrite finmap_union_singleton_l, finmap_union_comm; [done |].
    by apply finmap_disjoint_singleton_l.
  Qed.

  Lemma finmap_disjoint_insert_l (m1 m2 : M A) i x :
    <[i:=x]>m1  m2  m2 !! i = None  m1  m2.
  Proof.
    rewrite finmap_union_singleton_l.
    by rewrite finmap_disjoint_union_l, finmap_disjoint_singleton_l.
  Qed.
  Lemma finmap_disjoint_insert_r (m1 m2 : M A) i x :
    m1  <[i:=x]>m2  m1 !! i = None  m1  m2.
  Proof.
    rewrite finmap_union_singleton_l.
    by rewrite finmap_disjoint_union_r, finmap_disjoint_singleton_r.
  Qed.

  Lemma finmap_disjoint_insert_l_2 (m1 m2 : M A) i x :
    m2 !! i = None  m1  m2  <[i:=x]>m1  m2.
  Proof. by rewrite finmap_disjoint_insert_l. Qed.
  Lemma finmap_disjoint_insert_r_2 (m1 m2 : M A) i x :
    m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
  Proof. by rewrite finmap_disjoint_insert_r. Qed.

  Lemma finmap_union_insert_l (m1 m2 : M A) i x :
    <[i:=x]>(m1  m2) = <[i:=x]>m1  m2.
  Proof. by rewrite !finmap_union_singleton_l, (associative ()). Qed.
  Lemma finmap_union_insert_r (m1 m2 : M A) i x :
    m1 !! i = None 
    <[i:=x]>(m1  m2) = m1  <[i:=x]>m2.
  Proof.
    intro. rewrite !finmap_union_singleton_l, !(associative ()).
    rewrite (finmap_union_comm m1); [done |].
    by apply finmap_disjoint_singleton_r.
  Qed.

  Lemma finmap_insert_list_union l (m : M A) :
    insert_list l m = list_to_map l  m.
  Proof.
    induction l; simpl.
    * by rewrite (left_id _ _).
    * by rewrite IHl, finmap_union_insert_l.
  Qed.

  Lemma finmap_subseteq_insert (m1 m2 : M A) i x :
    m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
  Proof.
    intros ?? j. destruct (decide (j = i)).
    * intros y ?. congruence.
    * intros. simpl_map. auto.
  Qed.

  (** * Properties of the delete operation *)
  Lemma finmap_disjoint_delete_l (m1 m2 : M A) i :
    m1  m2  delete i m1  m2.
  Proof.
    rewrite !finmap_disjoint_alt.
    intros Hdisjoint j. destruct (Hdisjoint j); auto.
    rewrite lookup_delete_None. tauto.
  Qed.
  Lemma finmap_disjoint_delete_r (m1 m2 : M A) i :
    m1  m2  m1  delete i m2.
  Proof. symmetry. by apply finmap_disjoint_delete_l. Qed.

  Lemma finmap_disjoint_delete_list_l (m1 m2 : M A) is :
    m1  m2  delete_list is m1  m2.
  Proof. induction is; simpl; auto using finmap_disjoint_delete_l. Qed.
  Lemma finmap_disjoint_delete_list_r (m1 m2 : M A) is :
    m1  m2  m1  delete_list is m2.
  Proof. induction is; simpl; auto using finmap_disjoint_delete_r. Qed.

  Lemma finmap_union_delete (m1 m2 : M A) i :
    delete i (m1  m2) = delete i m1  delete i m2.
  Proof.
    intros. apply finmap_eq. intros j. apply option_eq. intros y.
    destruct (decide (i = j)); simplify_map_equality;
     rewrite ?finmap_union_Some_raw; simpl_map; intuition congruence.
  Qed.
  Lemma finmap_union_delete_list (m1 m2 : M A) is :
    delete_list is (m1  m2) = delete_list is m1  delete_list is m2.
  Proof.
    induction is; simpl; [done |].
    by rewrite IHis, finmap_union_delete.
  Qed.

  Lemma finmap_disjoint_union_list_l (ms : list (M A)) (m : M A) :
     ms  m  Forall ( m) ms.
  Proof.
    split.
    * induction ms; simpl; rewrite ?finmap_disjoint_union_l; intuition.
    * induction 1; simpl.
      + apply finmap_disjoint_empty_l.
      + by rewrite finmap_disjoint_union_l.
  Qed.
  Lemma finmap_disjoint_union_list_r (ms : list (M A)) (m : M A) :
    m   ms  Forall ( m) ms.
  Proof. by rewrite (symmetry_iff ()), finmap_disjoint_union_list_l. Qed.

  Lemma finmap_disjoint_union_list_l_2 (ms : list (M A)) (m : M A) :
    Forall ( m) ms   ms  m.
  Proof. by rewrite finmap_disjoint_union_list_l. Qed.
  Lemma finmap_disjoint_union_list_r_2 (ms : list (M A)) (m : M A) :
    Forall ( m) ms  m   ms.
  Proof. by rewrite finmap_disjoint_union_list_r. Qed.

  (** * Properties of the conversion from lists to maps *)
  Lemma finmap_disjoint_list_to_map_l (m : M A) ixs :
    list_to_map ixs  m  Forall (λ ix, m !! fst ix = None) ixs.
  Proof.
    split.
    * induction ixs; simpl; rewrite ?finmap_disjoint_insert_l in *; intuition.
    * induction 1; simpl.
      + apply finmap_disjoint_empty_l.
      + rewrite finmap_disjoint_insert_l. auto.
  Qed.
  Lemma finmap_disjoint_list_to_map_r (m : M A) ixs :
    m  list_to_map ixs  Forall (λ ix, m !! fst ix = None) ixs.
  Proof. by rewrite (symmetry_iff ()), finmap_disjoint_list_to_map_l. Qed.

  Lemma finmap_disjoint_list_to_map_zip_l (m : M A) is xs :
    same_length is xs 
    list_to_map (zip is xs)  m  Forall (λ i, m !! i = None) is.
  Proof.
    intro. rewrite finmap_disjoint_list_to_map_l.
    rewrite <-(zip_fst is xs) at 2 by done.
    by rewrite Forall_fmap.
  Qed.
  Lemma finmap_disjoint_list_to_map_zip_r (m : M A) is xs :
    same_length is xs 
    m  list_to_map (zip is xs)  Forall (λ i, m !! i = None) is.
  Proof.
    intro. by rewrite (symmetry_iff ()), finmap_disjoint_list_to_map_zip_l.
  Qed.
  Lemma finmap_disjoint_list_to_map_zip_l_2 (m : M A) is xs :
    same_length is xs 
    Forall (λ i, m !! i = None) is 
    list_to_map (zip is xs)  m.
  Proof. intro. by rewrite finmap_disjoint_list_to_map_zip_l. Qed.
  Lemma finmap_disjoint_list_to_map_zip_r_2 (m : M A) is xs :
    same_length is xs 
    Forall (λ i, m !! i = None) is 
    m  list_to_map (zip is xs).
  Proof. intro. by rewrite finmap_disjoint_list_to_map_zip_r. Qed.

  (** * Properties with respect to vectors of elements *)
  Lemma finmap_union_delete_vec {n} (ms : vec (M A) n) (i : fin n) :
    list_disjoint ms 
    ms !!! i   delete (fin_to_nat i) (vec_to_list ms) =  ms.
  Proof.
    induction ms as [|m ? ms]; inversion_clear 1;
      inv_fin i; simpl; [done | intros i].
    rewrite (finmap_union_comm m), (associative_eq _ _), IHms.
    * by apply finmap_union_comm, finmap_disjoint_union_list_l.
    * done.
    * by apply finmap_disjoint_union_list_r, Forall_delete.
  Qed.

  Lemma finmap_union_insert_vec {n} (ms : vec (M A) n) (i : fin n) m :
    m   delete (fin_to_nat i) (vec_to_list ms) 
     vinsert i m ms = m   delete (fin_to_nat i) (vec_to_list ms).
  Proof.
    induction ms as [|m' ? ms IH];
      inv_fin i; simpl; [done | intros i Hdisjoint].
    rewrite finmap_disjoint_union_r in Hdisjoint.
    rewrite IH, !(associative_eq ()), (finmap_union_comm m); intuition.
  Qed.

  Lemma finmap_list_disjoint_delete_vec {n} (ms : vec (M A) n) (i : fin n) :
    list_disjoint ms 
    Forall ( ms !!! i) (delete (fin_to_nat i) (vec_to_list ms)).
  Proof.
    induction ms; inversion_clear 1; inv_fin i; simpl.
    * done.
    * constructor. symmetry. by apply Forall_vlookup. auto.
  Qed.

  Lemma finmap_list_disjoint_insert_vec {n} (ms : vec (M A) n) (i : fin n) m :
    list_disjoint ms 
    Forall ( m) (delete (fin_to_nat i) (vec_to_list ms)) 
    list_disjoint (vinsert i m ms).
  Proof.
    induction ms as [|m' ? ms]; inversion_clear 1; inv_fin i; simpl.
    { constructor; auto. }
    intros i. inversion_clear 1. constructor; [| by auto].
    apply Forall_vlookup_2. intros j.
    destruct (decide (i = j)); subst;
      rewrite ?vlookup_insert, ?vlookup_insert_ne by done.
    * done.
    * by apply Forall_vlookup.
  Qed.

  (** * Properties of the difference operation *)
  Lemma finmap_difference_Some (m1 m2 : M A) i x :
    (m1  m2) !! i = Some x  m1 !! i = Some x  m2 !! i = None.
  Proof.
    unfold difference, finmap_difference, difference_with, finmap_difference_with.
    rewrite (merge_spec _).
    destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
  Qed.

  Lemma finmap_disjoint_difference_l (m1 m2 m3 : M A) :
    m2  m3  m1  m3  m2.
  Proof.
    intros E i. specialize (E i).
    unfold difference, finmap_difference, difference_with, finmap_difference_with.
    rewrite (merge_spec _).
    destruct (m1 !! i), (m2 !! i), (m3 !! i); compute; try intuition congruence.
    ediscriminate E; eauto.
  Qed.
  Lemma finmap_disjoint_difference_r (m1 m2 m3 : M A) :
    m2  m3  m2  m1  m3.
  Proof. intros. symmetry. by apply finmap_disjoint_difference_l. Qed.

  Lemma finmap_union_difference (m1 m2 : M A) :
    m1  m2  m2 = m1  m2  m1.
  Proof.
    intro Hm1m2. apply finmap_eq. intros i.
    apply option_eq. intros v. specialize (Hm1m2 i).
    unfold difference, finmap_difference, difference_with, finmap_difference_with.
    rewrite finmap_union_Some_raw, (merge_spec _).
    destruct (m1 !! i) as [v'|], (m2 !! i);
      try specialize (Hm1m2 v'); compute; intuition congruence.
  Qed.
End finmap_more.

(** The tactic [simplify_finmap_disjoint] simplifies occurences of [disjoint]
in the conclusion and hypotheses that involve the union, insert, or singleton
operation. *)
Ltac decompose_finmap_disjoint := repeat
  match goal with
  | H : _  _  _ |- _ =>
    apply finmap_disjoint_union_l in H; destruct H
  | H : _  _  _ |- _ =>
    apply finmap_disjoint_union_r in H; destruct H
  | H : {[ _ ]}  _ |- _ => apply finmap_disjoint_singleton_l in H
  | H : _  {[ _ ]} |- _ =>  apply finmap_disjoint_singleton_r in H
  | H : <[_:=_]>_  _ |- _ =>
    apply finmap_disjoint_insert_l in H; destruct H
  | H : _  <[_:=_]>_ |- _ =>
    apply finmap_disjoint_insert_r in H; destruct H
  | H :  _  _ |- _ => apply finmap_disjoint_union_list_l in H
  | H : _   _ |- _ => apply finmap_disjoint_union_list_r in H
  | H :  ⊥_ |- _ => clear H
  | H : _   |- _ => clear H
  | H : list_disjoint [] |- _ => clear H
  | H : list_disjoint [_] |- _ => clear H
  | H : list_disjoint (_ :: _) |- _ =>
    apply list_disjoint_cons_inv in H; destruct H
  | H : Forall ( _) _ |- _ => rewrite Forall_vlookup in H
  | H : Forall ( _) [] |- _ => clear H
  | H : Forall ( _) (_ :: _) |- _ =>
    apply Forall_inv in H; destruct H
  end.