fin_maps.v 68.7 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
2
3
4
5
6
7
(* Copyright (c) 2012-2015, 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_equality] to simplify goals involving finite maps. *)
Require Import Permutation.
8
Require Export prelude.relations prelude.vector prelude.orders.
Robbert Krebbers's avatar
Robbert Krebbers committed
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58

(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
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. *)

(** Finiteness is axiomatized by requiring that each map can be translated
to an association list. The translation to association lists is used to
prove well founded recursion on finite maps. *)

(** 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]. *)

Class FinMapToList K A M := map_to_list: M  list (K * A).

Class FinMap K M `{FMap M,  A, Lookup K A (M A),  A, Empty (M A),  A,
    PartialAlter K A (M A), OMap M, Merge M,  A, FinMapToList K A (M A),
     i j : K, Decision (i = j)} := {
  map_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;
  NoDup_map_to_list {A} (m : M A) : NoDup (map_to_list m);
  elem_of_map_to_list {A} (m : M A) i x :
    (i,x)  map_to_list m  m !! i = Some x;
  lookup_omap {A B} (f : A  option B) m i : omap f m !! i = m !! i = f;
  lookup_merge {A B C} (f : option A  option B  option C)
      `{!PropHolds (f None None = None)} m1 m2 i :
    merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
}.

(** * 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
significant performance loss to make including them in the finite map interface
worthwhile. *)
Instance map_insert `{PartialAlter K A M} : Insert K A M :=
  λ i x, partial_alter (λ _, Some x) i.
Instance map_alter `{PartialAlter K A M} : Alter K A M :=
  λ f, partial_alter (fmap f).
Instance map_delete `{PartialAlter K A M} : Delete K M :=
  partial_alter (λ _, None).
Instance map_singleton `{PartialAlter K A M, Empty M} :
59
  SingletonM K A M := λ i x, <[i:=x]> .
Robbert Krebbers's avatar
Robbert Krebbers committed
60
61
62
63
64
65
66
67
68
69
70
71
72
73

Definition map_of_list `{Insert K A M, Empty M} : list (K * A)  M :=
  fold_right (λ p, <[p.1:=p.2]>) .
Definition map_of_collection `{Elements K C, Insert K A M, Empty M}
    (f : K  option A) (X : C) : M :=
  map_of_list (omap (λ i, (i,) <$> f i) (elements X)).

Instance map_union_with `{Merge M} {A} : UnionWith A (M A) :=
  λ f, merge (union_with f).
Instance map_intersection_with `{Merge M} {A} : IntersectionWith A (M A) :=
  λ f, merge (intersection_with f).
Instance map_difference_with `{Merge M} {A} : DifferenceWith A (M A) :=
  λ f, merge (difference_with f).

74
75
Instance map_equiv `{ A, Lookup K A (M A), Equiv A} : Equiv (M A) | 18 :=
  λ m1 m2,  i, m1 !! i  m2 !! i.
Robbert Krebbers's avatar
Robbert Krebbers committed
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119

(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
Definition map_Forall `{Lookup K A M} (P : K  A  Prop) : M  Prop :=
  λ m,  i x, m !! i = Some x  P i x.
Definition map_relation `{ A, Lookup K A (M A)} {A B} (R : A  B  Prop)
    (P : A  Prop) (Q : B  Prop) (m1 : M A) (m2 : M B) : Prop :=  i,
  option_relation R P Q (m1 !! i) (m2 !! i).
Definition map_included `{ A, Lookup K A (M A)} {A}
  (R : relation A) : relation (M A) := map_relation R (λ _, False) (λ _, True).
Definition map_disjoint `{ A, Lookup K A (M A)} {A} : relation (M A) :=
  map_relation (λ _ _, False) (λ _, True) (λ _, True).
Infix "⊥ₘ" := map_disjoint (at level 70) : C_scope.
Hint Extern 0 (_ ⊥ₘ _) => symmetry; eassumption.
Notation "( m ⊥ₘ.)" := (map_disjoint m) (only parsing) : C_scope.
Notation "(.⊥ₘ m )" := (λ m2, m2 ⊥ₘ m) (only parsing) : C_scope.
Instance map_subseteq `{ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
  map_included (=).

(** 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. *)
Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x).
Instance map_intersection `{Merge M} {A} : Intersection (M A) :=
  intersection_with (λ x _, Some x).

(** The difference operation removes all values from the first map whose
index contains a value in the second map as well. *)
Instance map_difference `{Merge M} {A} : Difference (M A) :=
  difference_with (λ _ _, None).

(** A stronger variant of map that allows the mapped function to use the index
of the elements. Implemented by conversion to lists, so not very efficient. *)
Definition map_imap `{ A, Insert K A (M A),  A, Empty (M A),
     A, FinMapToList K A (M A)} {A B} (f : K  A  option B) (m : M A) : M B :=
  map_of_list (omap (λ ix, (fst ix,) <$> curry f ix) (map_to_list m)).

(** * Theorems *)
Section theorems.
Context `{FinMap K M}.

(** ** Setoids *)
Section setoid.
120
121
  Context `{Equiv A} `{!Equivalence (() : relation A)}.
  Global Instance map_equivalence : Equivalence (() : relation (M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
122
123
124
125
126
127
128
129
130
131
  Proof.
    split.
    * by intros m i.
    * by intros m1 m2 ? i.
    * by intros m1 m2 m3 ?? i; transitivity (m2 !! i).
  Qed.
  Global Instance lookup_proper (i : K) :
    Proper (() ==> ()) (lookup (M:=M A) i).
  Proof. by intros m1 m2 Hm. Qed.
  Global Instance partial_alter_proper :
132
    Proper ((() ==> ()) ==> (=) ==> () ==> ()) (partial_alter (M:=M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
133
134
135
136
137
138
139
140
  Proof.
    by intros f1 f2 Hf i ? <- m1 m2 Hm j; destruct (decide (i = j)) as [->|];
      rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne by done;
      try apply Hf; apply lookup_proper.
  Qed.
  Global Instance insert_proper (i : K) :
    Proper (() ==> () ==> ()) (insert (M:=M A) i).
  Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
141
142
143
  Global Instance singleton_proper k :
    Proper (() ==> ()) (singletonM k : A  M A).
  Proof. by intros ???; apply insert_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
144
145
146
147
148
149
150
151
152
153
154
155
  Global Instance delete_proper (i : K) :
    Proper (() ==> ()) (delete (M:=M A) i).
  Proof. by apply partial_alter_proper; [constructor|]. Qed.
  Global Instance alter_proper :
    Proper ((() ==> ()) ==> (=) ==> () ==> ()) (alter (A:=A) (M:=M A)).
  Proof.
    intros ?? Hf; apply partial_alter_proper.
    by destruct 1; constructor; apply Hf.
  Qed.
  Lemma merge_ext f g
      `{!PropHolds (f None None = None), !PropHolds (g None None = None)} :
    (() ==> () ==> ())%signature f g 
156
    (() ==> () ==> ())%signature (merge (M:=M) f) (merge g).
Robbert Krebbers's avatar
Robbert Krebbers committed
157
158
159
160
  Proof.
    by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
  Qed.
  Global Instance union_with_proper :
161
    Proper ((() ==> () ==> ()) ==> () ==> () ==>()) (union_with (M:=M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
162
163
164
165
166
167
  Proof.
    intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
    by do 2 destruct 1; first [apply Hf | constructor].
  Qed.    
  Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
  Proof.
168
169
    intros m1 m2 Hm; apply map_eq; intros i.
    by unfold_leibniz; apply lookup_proper.
Robbert Krebbers's avatar
Robbert Krebbers committed
170
  Qed.
171
172
173
174
175
176
177
178
179
180
  Lemma map_equiv_empty (m : M A) : m    m = .
  Proof.
    split; [intros Hm; apply map_eq; intros i|by intros ->].
    by rewrite lookup_empty, <-equiv_None, Hm, lookup_empty.
  Qed.
  Lemma map_equiv_lookup (m1 m2 : M A) i x :
    m1  m2  m1 !! i = Some x   y, m2 !! i = Some y  x  y.
  Proof.
    intros Hm ?. destruct (equiv_Some (m1 !! i) (m2 !! i) x) as (y&?&?); eauto.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
End setoid.

(** ** General properties *)
Lemma map_eq_iff {A} (m1 m2 : M A) : m1 = m2   i, m1 !! i = m2 !! i.
Proof. split. by intros ->. apply map_eq. Qed.
Lemma map_subseteq_spec {A} (m1 m2 : M A) :
  m1  m2   i x, m1 !! i = Some x  m2 !! i = Some x.
Proof.
  unfold subseteq, map_subseteq, map_relation. split; intros Hm i;
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Global Instance: EmptySpec (M A).
Proof.
  intros A m. rewrite !map_subseteq_spec.
  intros i x. by rewrite lookup_empty.
Qed.
Global Instance:  {A} (R : relation A), PreOrder R  PreOrder (map_included R).
Proof.
  split; [intros m i; by destruct (m !! i); simpl|].
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
  destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_equality';
    done || etransitivity; eauto.
Qed.
Global Instance: PartialOrder (() : relation (M A)).
Proof.
  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
Proof. rewrite !map_subseteq_spec. auto. Qed.
Lemma lookup_weaken_is_Some {A} (m1 m2 : M A) i :
  is_Some (m1 !! i)  m1  m2  is_Some (m2 !! i).
Proof. inversion 1. eauto using lookup_weaken. Qed.
Lemma lookup_weaken_None {A} (m1 m2 : M A) i :
  m2 !! i = None  m1  m2  m1 !! i = None.
Proof.
  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
Lemma lookup_ne {A} (m : M A) i j : m !! i  m !! j  i  j.
Proof. congruence. Qed.
Lemma map_empty {A} (m : M A) : ( i, m !! i = None)  m = .
Proof. intros Hm. apply map_eq. intros. by rewrite Hm, lookup_empty. Qed.
Lemma lookup_empty_is_Some {A} i : ¬is_Some (( : M A) !! i).
Proof. rewrite lookup_empty. by inversion 1. Qed.
Lemma lookup_empty_Some {A} i (x : A) : ¬∅ !! i = Some x.
Proof. by rewrite lookup_empty. Qed.
Lemma map_subset_empty {A} (m : M A) : m  .
Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.

(** ** Properties of the [partial_alter] operation *)
Lemma partial_alter_ext {A} (f g : option A  option A) (m : M A) i :
  ( x, m !! i = x  f x = g x)  partial_alter f i m = partial_alter g i m.
Proof.
  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
Qed.
Lemma partial_alter_commute {A} f g (m : M A) i j :
  i  j  partial_alter f i (partial_alter g j m) =
    partial_alter g j (partial_alter f i m).
Proof.
  intros. apply map_eq; intros jj. destruct (decide (jj = j)) as [->|?].
  { by rewrite lookup_partial_alter_ne,
      !lookup_partial_alter, lookup_partial_alter_ne. }
  destruct (decide (jj = i)) as [->|?].
  * 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 {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
Qed.
Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m.
Proof. by apply partial_alter_self_alt. Qed.
Lemma partial_alter_subseteq {A} f (m : M A) i :
  m !! i = None  m  partial_alter f i m.
Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
Lemma partial_alter_subset {A} f (m : M A) i :
  m !! i = None  is_Some (f (m !! i))  m  partial_alter f i m.
Proof.
  intros Hi Hfi. split; [by apply partial_alter_subseteq|].
  rewrite !map_subseteq_spec. inversion Hfi as [x Hx]. intros Hm.
  apply (Some_ne_None x). rewrite <-(Hm i x); [done|].
  by rewrite lookup_partial_alter.
Qed.

(** ** Properties of the [alter] operation *)
Lemma alter_ext {A} (f g : A  A) (m : M A) i :
  ( x, m !! i = Some x  f x = g x)  alter f i m = alter g i m.
Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal'; auto. Qed.
Lemma lookup_alter {A} (f : A  A) m i : alter f i m !! i = f <$> m !! i.
Proof. unfold alter. apply lookup_partial_alter. Qed.
Lemma lookup_alter_ne {A} (f : A  A) m i j : i  j  alter f i m !! j = m !! j.
Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
Lemma alter_compose {A} (f g : A  A) (m : M A) i:
  alter (f  g) i m = alter f i (alter g i m).
Proof.
  unfold alter, map_alter. rewrite <-partial_alter_compose.
  apply partial_alter_ext. by intros [?|].
Qed.
Lemma alter_commute {A} (f g : A  A) (m : M A) i j :
  i  j  alter f i (alter g j m) = alter g j (alter f i m).
Proof. apply partial_alter_commute. Qed.
Lemma lookup_alter_Some {A} (f : A  A) m i j y :
  alter f i m !! j = Some y 
    (i = j   x, m !! j = Some x  y = f x)  (i  j  m !! j = Some y).
Proof.
  destruct (decide (i = j)) as [->|?].
  * rewrite lookup_alter. naive_solver (simplify_option_equality; eauto).
  * rewrite lookup_alter_ne by done. naive_solver.
Qed.
Lemma lookup_alter_None {A} (f : A  A) m i j :
  alter f i m !! j = None  m !! j = None.
Proof.
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
Qed.
Lemma alter_id {A} (f : A  A) m i :
  ( x, m !! i = Some x  f x = x)  alter f i m = m.
Proof.
  intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
  { rewrite lookup_alter; destruct (m !! j); f_equal'; auto. }
  by rewrite lookup_alter_ne by done.
Qed.

(** ** Properties of the [delete] operation *)
Lemma lookup_delete {A} (m : M A) i : delete i m !! i = None.
Proof. apply lookup_partial_alter. Qed.
Lemma lookup_delete_ne {A} (m : M A) i j : i  j  delete i m !! j = m !! j.
Proof. apply lookup_partial_alter_ne. Qed.
Lemma lookup_delete_Some {A} (m : M A) i j y :
  delete i m !! j = Some y  i  j  m !! j = Some y.
Proof.
  split.
  * destruct (decide (i = j)) as [->|?];
      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
  * intros [??]. by rewrite lookup_delete_ne.
Qed.
Lemma lookup_delete_None {A} (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
346
Lemma delete_singleton {A} i (x : A) : delete i {[i  x]} = .
Robbert Krebbers's avatar
Robbert Krebbers committed
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
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
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
Lemma delete_commute {A} (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_commute. Qed.
Lemma delete_insert_ne {A} (m : M A) i j x :
  i  j  delete i (<[j:=x]>m) = <[j:=x]>(delete i m).
Proof. intro. by apply partial_alter_commute. Qed.
Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
Proof.
  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
Qed.
Lemma delete_partial_alter {A} (m : M A) i f :
  m !! i = None  delete i (partial_alter f i m) = m.
Proof.
  intros. unfold delete, map_delete. rewrite <-partial_alter_compose.
  unfold compose. by apply partial_alter_self_alt.
Qed.
Lemma delete_insert {A} (m : M A) i x :
  m !! i = None  delete i (<[i:=x]>m) = m.
Proof. apply delete_partial_alter. Qed.
Lemma insert_delete {A} (m : M A) i x :
  m !! i = Some x  <[i:=x]>(delete i m) = m.
Proof.
  intros Hmi. unfold delete, map_delete, insert, map_insert.
  rewrite <-partial_alter_compose. unfold compose. rewrite <-Hmi.
  by apply partial_alter_self_alt.
Qed.
Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
Lemma delete_subseteq_compat {A} (m1 m2 : M A) i :
  m1  m2  delete i m1  delete i m2.
Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
Qed.
Lemma delete_subset_alt {A} (m : M A) i x : m !! i = Some x  delete i m  m.
Proof.
  split; [apply delete_subseteq|].
  rewrite !map_subseteq_spec. intros Hi. apply (None_ne_Some x).
  by rewrite <-(lookup_delete m i), (Hi i x).
Qed.
Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
Proof. inversion 1. eauto using delete_subset_alt. Qed.

(** ** Properties of the [insert] operation *)
Lemma lookup_insert {A} (m : M A) i x : <[i:=x]>m !! i = Some x.
Proof. unfold insert. apply lookup_partial_alter. Qed.
Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
Proof. rewrite lookup_insert. congruence. Qed.
Lemma lookup_insert_ne {A} (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_commute {A} (m : M A) i j x y :
  i  j  <[i:=x]>(<[j:=y]>m) = <[j:=y]>(<[i:=x]>m).
Proof. apply partial_alter_commute. Qed.
Lemma lookup_insert_Some {A} (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)) as [->|?];
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
  * intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
Qed.
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
Qed.
Lemma insert_id {A} (m : M A) i x : m !! i = Some x  <[i:=x]>m = m.
Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|];
    by rewrite ?lookup_insert, ?lookup_insert_ne by done.
Qed.
Lemma insert_included {A} R `{!Reflexive R} (m : M A) i x :
  ( y, m !! i = Some y  R y x)  map_included R m (<[i:=x]>m).
Proof.
  intros ? j; destruct (decide (i = j)) as [->|].
  * rewrite lookup_insert. destruct (m !! j); simpl; eauto.
  * rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
Qed.
Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
Proof. apply partial_alter_subseteq. Qed.
Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
Proof. intro. apply partial_alter_subset; eauto. Qed.
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
Proof.
  rewrite !map_subseteq_spec. intros ?? j ?.
  destruct (decide (j = i)) as [->|?]; [congruence|].
  rewrite lookup_insert_ne; auto.
Qed.
Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
Proof.
  rewrite !map_subseteq_spec. intros Hi Hix j y Hj.
  destruct (decide (i = j)) as [->|]; [congruence|].
  rewrite lookup_delete_ne by done.
  apply Hix; by rewrite lookup_insert_ne by done.
Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
Proof.
  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
  * rewrite lookup_insert. congruence.
  * rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
Proof.
  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
  m1 !! i = None  <[i:=x]> m1  m2 
   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
  intros Hi Hm1m2. exists (delete i m2). split_ands.
  * rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto.
    by rewrite lookup_insert.
  * eauto using insert_delete_subset.
  * by rewrite lookup_delete.
Qed.
Lemma fmap_insert {A B} (f : A  B) (m : M A) i x :
  f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  * by rewrite lookup_fmap, !lookup_insert.
  * by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
Qed.
482
Lemma insert_empty {A} i (x : A) : <[i:=x]> = {[i  x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
483
484
485
486
Proof. done. Qed.

(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
487
  {[i  x]} !! j = Some y  i = j  x = y.
Robbert Krebbers's avatar
Robbert Krebbers committed
488
Proof.
489
  rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence.
Robbert Krebbers's avatar
Robbert Krebbers committed
490
Qed.
491
492
493
Lemma lookup_singleton_None {A} i j (x : A) : {[i  x]} !! j = None  i  j.
Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; tauto. Qed.
Lemma lookup_singleton {A} i (x : A) : {[i  x]} !! i = Some x.
Robbert Krebbers's avatar
Robbert Krebbers committed
494
Proof. by rewrite lookup_singleton_Some. Qed.
495
Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[i  x]} !! j = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
496
Proof. by rewrite lookup_singleton_None. Qed.
497
Lemma map_non_empty_singleton {A} i (x : A) : {[i  x]}  .
Robbert Krebbers's avatar
Robbert Krebbers committed
498
499
500
501
Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
502
Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i  x]} = {[i  y]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
503
Proof.
504
  unfold singletonM, map_singleton, insert, map_insert.
Robbert Krebbers's avatar
Robbert Krebbers committed
505
506
  by rewrite <-partial_alter_compose.
Qed.
507
Lemma alter_singleton {A} (f : A  A) i x : alter f i {[i  x]} = {[i  f x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
508
509
510
511
512
513
Proof.
  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
  * by rewrite lookup_alter, !lookup_singleton.
  * by rewrite lookup_alter_ne, !lookup_singleton_ne.
Qed.
Lemma alter_singleton_ne {A} (f : A  A) i j x :
514
  i  j  alter f i {[j  x]} = {[j  x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
515
516
517
518
519
520
521
522
523
524
525
Proof.
  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
Qed.

(** ** Properties of the map operations *)
Lemma fmap_empty {A B} (f : A  B) : f <$>  = .
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A  option B) : omap f  = .
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
Lemma omap_singleton {A B} (f : A  option B) i x y :
526
  f x = Some y  omap f {[ i  x ]} = {[ i  y ]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
527
528
529
530
531
532
533
534
535
536
Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|].
  * by rewrite lookup_omap, !lookup_singleton.
  * by rewrite lookup_omap, !lookup_singleton_ne.
Qed.
Lemma map_fmap_id {A} (m : M A) : id <$> m = m.
Proof. apply map_eq; intros i; by rewrite lookup_fmap, option_fmap_id. Qed.
Lemma map_fmap_compose {A B C} (f : A  B) (g : B  C) (m : M A) :
  g  f <$> m = g <$> f <$> m.
Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed.
537
538
539
540
541
542
Lemma map_fmap_setoid_ext `{Equiv A, Equiv B} (f1 f2 : A  B) m :
  ( i x, m !! i = Some x  f1 x  f2 x)  f1 <$> m  f2 <$> m.
Proof.
  intros Hi i; rewrite !lookup_fmap.
  destruct (m !! i) eqn:?; constructor; eauto.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
Lemma map_fmap_ext {A B} (f1 f2 : A  B) m :
  ( i x, m !! i = Some x  f1 x = f2 x)  f1 <$> m = f2 <$> m.
Proof.
  intros Hi; apply map_eq; intros i; rewrite !lookup_fmap.
  by destruct (m !! i) eqn:?; simpl; erewrite ?Hi by eauto.
Qed.

(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
Proof. rewrite !elem_of_map_to_list. congruence. Qed.
Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
Lemma elem_of_map_of_list_1_help {A} (l : list (K * A)) i x :
  (i,x)  l  ( y, (i,y)  l  y = x)  map_of_list l !! i = Some x.
Proof.
  induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|].
  setoid_rewrite elem_of_cons.
  intros [?|?] Hdup; simplify_equality; [by rewrite lookup_insert|].
  destruct (decide (i = j)) as [->|].
  * rewrite lookup_insert; f_equal; eauto.
  * rewrite lookup_insert_ne by done; eauto.
Qed.
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
Proof.
  intros ? Hx; apply elem_of_map_of_list_1_help; eauto using NoDup_fmap_fst.
  intros y; revert Hx. rewrite !elem_of_list_lookup; intros [i' Hi'] [j' Hj'].
  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
  map_of_list l !! i = Some x  (i,x)  l.
Proof.
  induction l as [|[j y] l IH]; simpl; [by rewrite lookup_empty|].
  rewrite elem_of_cons. destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
  i  l.*1  map_of_list l !! i = None.
Proof.
  rewrite elem_of_list_fmap, eq_None_not_Some. intros Hi [x ?]; destruct Hi.
  exists (i,x); simpl; auto using elem_of_map_of_list_2.
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
  map_of_list l !! i = None  i  l.*1.
Proof.
  induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
  rewrite elem_of_cons. destruct (decide (i = j)); simplify_equality.
  * by rewrite lookup_insert.
  * by rewrite lookup_insert_ne; intuition.
Qed.
Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
  i  l.*1  map_of_list l !! i = None.
Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
  NoDup (l1.*1)  l1 ≡ₚ l2  map_of_list l1 = map_of_list l2.
Proof.
  intros ? Hperm. apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-!elem_of_map_of_list; rewrite <-?Hperm.
Qed.
Lemma map_of_list_inj {A} (l1 l2 : list (K * A)) :
  NoDup (l1.*1)  NoDup (l2.*1)  map_of_list l1 = map_of_list l2  l1 ≡ₚ l2.
Proof.
  intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m.
Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
    by auto using NoDup_fst_map_to_list.
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
  NoDup (l.*1)  map_to_list (map_of_list l) ≡ₚ l.
Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
Lemma map_to_list_inj {A} (m1 m2 : M A) :
  map_to_list m1 ≡ₚ map_to_list m2  m1 = m2.
Proof.
  intros. rewrite <-(map_of_to_list m1), <-(map_of_to_list m2).
  auto using map_of_list_proper, NoDup_fst_map_to_list.
Qed.
Lemma map_to_of_list_flip {A} (m1 : M A) l2 :
  map_to_list m1 ≡ₚ l2  m1 = map_of_list l2.
Proof.
  intros. rewrite <-(map_of_to_list m1).
  auto using map_of_list_proper, NoDup_fst_map_to_list.
Qed.
Lemma map_to_list_empty {A} : map_to_list  = @nil (K * A).
Proof.
  apply elem_of_nil_inv. intros [i x].
  rewrite elem_of_map_to_list. apply lookup_empty_Some.
Qed.
Lemma map_to_list_insert {A} (m : M A) i x :
  m !! i = None  map_to_list (<[i:=x]>m) ≡ₚ (i,x) :: map_to_list m.
Proof.
  intros. apply map_of_list_inj; csimpl.
  * apply NoDup_fst_map_to_list.
  * constructor; auto using NoDup_fst_map_to_list.
    rewrite elem_of_list_fmap. intros [[??] [? Hlookup]]; subst; simpl in *.
    rewrite elem_of_map_to_list in Hlookup. congruence.
  * by rewrite !map_of_to_list.
Qed.
Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = .
Proof. done. Qed.
Lemma map_of_list_cons {A} (l : list (K * A)) i x :
  map_of_list ((i, x) :: l) = <[i:=x]>(map_of_list l).
Proof. done. Qed.
Lemma map_to_list_empty_inv_alt {A}  (m : M A) : map_to_list m ≡ₚ []  m = .
Proof. rewrite <-map_to_list_empty. apply map_to_list_inj. Qed.
Lemma map_to_list_empty_inv {A} (m : M A) : map_to_list m = []  m = .
Proof. intros Hm. apply map_to_list_empty_inv_alt. by rewrite Hm. Qed.
Lemma map_to_list_insert_inv {A} (m : M A) l i x :
  map_to_list m ≡ₚ (i,x) :: l  m = <[i:=x]>(map_of_list l).
Proof.
  intros Hperm. apply map_to_list_inj.
  assert (i  l.*1  NoDup (l.*1)) as [].
  { rewrite <-NoDup_cons. change (NoDup (((i,x)::l).*1)). rewrite <-Hperm.
    auto using NoDup_fst_map_to_list. }
  rewrite Hperm, map_to_list_insert, map_to_of_list;
    auto using not_elem_of_map_of_list_1.
Qed.
Lemma map_choose {A} (m : M A) : m     i x, m !! i = Some x.
Proof.
  intros Hemp. destruct (map_to_list m) as [|[i x] l] eqn:Hm.
  { destruct Hemp; eauto using map_to_list_empty_inv. }
672
  exists i, x. rewrite <-elem_of_map_to_list, Hm. by left.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
Qed.

(** Properties of the imap function *)
Lemma lookup_imap {A B} (f : K  A  option B) m i :
  map_imap f m !! i = m !! i = f i.
Proof.
  unfold map_imap; destruct (m !! i = f i) as [y|] eqn:Hi; simpl.
  * destruct (m !! i) as [x|] eqn:?; simplify_equality'.
    apply elem_of_map_of_list_1_help.
    { apply elem_of_list_omap; exists (i,x); split;
        [by apply elem_of_map_to_list|by simplify_option_equality]. }
    intros y'; rewrite elem_of_list_omap; intros ([i' x']&Hi'&?).
    by rewrite elem_of_map_to_list in Hi'; simplify_option_equality.
  * apply not_elem_of_map_of_list; rewrite elem_of_list_fmap.
    intros ([i' x]&->&Hi'); simplify_equality'.
    rewrite elem_of_list_omap in Hi'; destruct Hi' as ([j y]&Hj&?).
    rewrite elem_of_map_to_list in Hj; simplify_option_equality.
Qed.

(** ** Properties of conversion from collections *)
Lemma lookup_map_of_collection {A} `{FinCollection K C}
    (f : K  option A) X i x :
  map_of_collection f X !! i = Some x  i  X  f i = Some x.
Proof.
  assert (NoDup (fst <$> omap (λ i, (i,) <$> f i) (elements X))).
  { induction (NoDup_elements X) as [|i' l]; csimpl; [constructor|].
    destruct (f i') as [x'|]; csimpl; auto; constructor; auto.
    rewrite elem_of_list_fmap. setoid_rewrite elem_of_list_omap.
    by intros (?&?&?&?&?); simplify_option_equality. }
  unfold map_of_collection; rewrite <-elem_of_map_of_list by done.
  rewrite elem_of_list_omap. setoid_rewrite elem_of_elements; split.
  * intros (?&?&?); simplify_option_equality; eauto.
  * intros [??]; exists i; simplify_option_equality; eauto.
Qed.

(** ** Induction principles *)
Lemma map_ind {A} (P : M A  Prop) :
  P   ( i x m, m !! i = None  P m  P (<[i:=x]>m))   m, P m.
Proof.
  intros ? Hins. cut ( l, NoDup (l.*1)   m, map_to_list m ≡ₚ l  P m).
  { intros help m.
    apply (help (map_to_list m)); auto using NoDup_fst_map_to_list. }
  induction l as [|[i x] l IH]; intros Hnodup m Hml.
  { apply map_to_list_empty_inv_alt in Hml. by subst. }
  inversion_clear Hnodup.
  apply map_to_list_insert_inv in Hml; subst m. apply Hins.
  * by apply not_elem_of_map_of_list_1.
  * apply IH; auto using map_to_of_list.
Qed.
Lemma map_to_list_length {A} (m1 m2 : M A) :
  m1  m2  length (map_to_list m1) < length (map_to_list m2).
Proof.
  revert m2. induction m1 as [|i x m ? IH] using map_ind.
  { intros m2 Hm2. rewrite map_to_list_empty. simpl.
    apply neq_0_lt. intros Hlen. symmetry in Hlen.
    apply nil_length_inv, map_to_list_empty_inv in Hlen.
    rewrite Hlen in Hm2. destruct (irreflexivity ()  Hm2). }
  intros m2 Hm2.
  destruct (insert_subset_inv m m2 i x) as (m2'&?&?&?); auto; subst.
  rewrite !map_to_list_insert; simpl; auto with arith.
Qed.
Lemma map_wf {A} : wf (strict (@subseteq (M A) _)).
Proof.
  apply (wf_projected (<) (length  map_to_list)).
  * by apply map_to_list_length.
  * by apply lt_wf.
Qed.

(** ** Properties of the [map_Forall] predicate *)
Section map_Forall.
Context {A} (P : K  A  Prop).

Lemma map_Forall_to_list m : map_Forall P m  Forall (curry P) (map_to_list m).
Proof.
  rewrite Forall_forall. split.
  * intros Hforall [i x]. rewrite elem_of_map_to_list. by apply (Hforall i x).
  * intros Hforall i x. rewrite <-elem_of_map_to_list. by apply (Hforall (i,x)).
Qed.
Lemma map_Forall_empty : map_Forall P .
Proof. intros i x. by rewrite lookup_empty. Qed.
Lemma map_Forall_impl (Q : K  A  Prop) m :
  map_Forall P m  ( i x, P i x  Q i x)  map_Forall Q m.
Proof. unfold map_Forall; naive_solver. Qed.
Lemma map_Forall_insert_11 m i x : map_Forall P (<[i:=x]>m)  P i x.
Proof. intros Hm. by apply Hm; rewrite lookup_insert. Qed.
Lemma map_Forall_insert_12 m i x :
  m !! i = None  map_Forall P (<[i:=x]>m)  map_Forall P m.
Proof.
  intros ? Hm j y ?; apply Hm. by rewrite lookup_insert_ne by congruence.
Qed.
Lemma map_Forall_insert_2 m i x :
  P i x  map_Forall P m  map_Forall P (<[i:=x]>m).
Proof. intros ?? j y; rewrite lookup_insert_Some; naive_solver. Qed.
Lemma map_Forall_insert m i x :
  m !! i = None  map_Forall P (<[i:=x]>m)  P i x  map_Forall P m.
Proof.
  naive_solver eauto using map_Forall_insert_11,
    map_Forall_insert_12, map_Forall_insert_2.
Qed.
Lemma map_Forall_ind (Q : M A  Prop) :
  Q  
  ( m i x, m !! i = None  P i x  map_Forall P m  Q m  Q (<[i:=x]>m)) 
   m, map_Forall P m  Q m.
Proof.
  intros Hnil Hinsert m. induction m using map_ind; auto.
  rewrite map_Forall_insert by done; intros [??]; eauto.
Qed.

Context `{ i x, Decision (P i x)}.
Global Instance map_Forall_dec m : Decision (map_Forall P m).
Proof.
  refine (cast_if (decide (Forall (curry P) (map_to_list m))));
    by rewrite map_Forall_to_list.
Defined.
Lemma map_not_Forall (m : M A) :
  ¬map_Forall P m   i x, m !! i = Some x  ¬P i x.
Proof.
  split; [|intros (i&x&?&?) Hm; specialize (Hm i x); tauto].
  rewrite map_Forall_to_list. intros Hm.
  apply (not_Forall_Exists _), Exists_exists in Hm.
793
  destruct Hm as ([i x]&?&?). exists i, x. by rewrite <-elem_of_map_to_list.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
Qed.
End map_Forall.

(** ** Properties of the [merge] operation *)
Section merge.
Context {A} (f : option A  option A  option A).
Context `{!PropHolds (f None None = None)}.
Global Instance: LeftId (=) None f  LeftId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
  by rewrite !(lookup_merge f), lookup_empty, (left_id_L None f).
Qed.
Global Instance: RightId (=) None f  RightId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
  by rewrite !(lookup_merge f), lookup_empty, (right_id_L None f).
Qed.
Lemma merge_commutative m1 m2 :
  ( i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) 
  merge f m1 m2 = merge f m2 m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Commutative (=) f  Commutative (=) (merge f).
Proof.
  intros ???. apply merge_commutative. intros. by apply (commutative f).
Qed.
Lemma merge_associative 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 map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Associative (=) f  Associative (=) (merge f).
Proof.
  intros ????. apply merge_associative. intros. by apply (associative_L f).
Qed.
Lemma merge_idempotent m1 :
  ( i, f (m1 !! i) (m1 !! i) = m1 !! i)  merge f m1 m1 = m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Idempotent (=) f  Idempotent (=) (merge f).
Proof. intros ??. apply merge_idempotent. intros. by apply (idempotent f). Qed.
End merge.

Section more_merge.
Context {A B C} (f : option A  option B  option C).
Context `{!PropHolds (f None None = None)}.
Lemma merge_Some m1 m2 m :
  ( i, m !! i = f (m1 !! i) (m2 !! i))  merge f m1 m2 = m.
Proof.
  split; [|intros <-; apply (lookup_merge _) ].
  intros Hlookup. apply map_eq; intros. rewrite Hlookup. apply (lookup_merge _).
Qed.
Lemma merge_empty : merge f   = .
Proof. apply map_eq. intros. by rewrite !(lookup_merge f), !lookup_empty. Qed.
Lemma partial_alter_merge g g1 g2 m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (g2 (m2 !! i)) 
  partial_alter g i (merge f m1 m2) =
    merge f (partial_alter g1 i m1) (partial_alter g2 i m2).
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
Lemma partial_alter_merge_l g g1 m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (m2 !! i) 
  partial_alter g i (merge f m1 m2) = merge f (partial_alter g1 i m1) m2.
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
Lemma partial_alter_merge_r g g2 m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (m1 !! i) (g2 (m2 !! i)) 
  partial_alter g i (merge f m1 m2) = merge f m1 (partial_alter g2 i m2).
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
Lemma insert_merge m1 m2 i x y z :
  f (Some y) (Some z) = Some x 
  <[i:=x]>(merge f m1 m2) = merge f (<[i:=y]>m1) (<[i:=z]>m2).
Proof. by intros; apply partial_alter_merge. Qed.
Lemma merge_singleton i x y z :
876
  f (Some y) (Some z) = Some x  merge f {[i  y]} {[i  z]} = {[i  x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
877
Proof.
878
  intros. by erewrite <-!insert_empty, <-insert_merge, merge_empty by eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
Qed.
Lemma insert_merge_l m1 m2 i x y :
  f (Some y) (m2 !! i) = Some x 
  <[i:=x]>(merge f m1 m2) = merge f (<[i:=y]>m1) m2.
Proof. by intros; apply partial_alter_merge_l. Qed.
Lemma insert_merge_r m1 m2 i x z :
  f (m1 !! i) (Some z) = Some x 
  <[i:=x]>(merge f m1 m2) = merge f m1 (<[i:=z]>m2).
Proof. by intros; apply partial_alter_merge_r. Qed.
End more_merge.

(** ** Properties on the [map_relation] relation *)
Section Forall2.
Context {A B} (R : A  B  Prop) (P : A  Prop) (Q : B  Prop).
Context `{ x y, Decision (R x y),  x, Decision (P x),  y, Decision (Q y)}.

Let f (mx : option A) (my : option B) : option bool :=
  match mx, my with
  | Some x, Some y => Some (bool_decide (R x y))
  | Some x, None => Some (bool_decide (P x))
  | None, Some y => Some (bool_decide (Q y))
  | None, None => None
  end.
Lemma map_relation_alt (m1 : M A) (m2 : M B) :
  map_relation R P Q m1 m2  map_Forall (λ _, Is_true) (merge f m1 m2).
Proof.
  split.
  * intros Hm i P'; rewrite lookup_merge by done; intros.
    specialize (Hm i). destruct (m1 !! i), (m2 !! i);
      simplify_equality'; auto using bool_decide_pack.
  * intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done.
    destruct (m1 !! i), (m2 !! i); simplify_equality'; auto;
      by eapply bool_decide_unpack, Hm.
Qed.
Global Instance map_relation_dec `{ x y, Decision (R x y),  x, Decision (P x),
   y, Decision (Q y)} m1 m2 : Decision (map_relation R P Q m1 m2).
Proof.
  refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2))));
    abstract by rewrite map_relation_alt.
Defined.
(** Due to the finiteness of finite maps, we can extract a witness if the
relation does not hold. *)
Lemma map_not_Forall2 (m1 : M A) (m2 : M B) :
  ¬map_relation R P Q m1 m2   i,
    ( x y, m1 !! i = Some x  m2 !! i = Some y  ¬R x y)
     ( x, m1 !! i = Some x  m2 !! i = None  ¬P x)
     ( y, m1 !! i = None  m2 !! i = Some y  ¬Q y).
Proof.
  split.
  * rewrite map_relation_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i.
    rewrite lookup_merge in Hm by done.
    destruct (m1 !! i), (m2 !! i); naive_solver auto 2 using bool_decide_pack.
  * unfold map_relation, option_relation.
    by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
      specialize (Hm i); simplify_option_equality.
Qed.
End Forall2.

(** ** Properties on the disjoint maps *)
Lemma map_disjoint_spec {A} (m1 m2 : M A) :
  m1 ⊥ₘ m2   i x y, m1 !! i = Some x  m2 !! i = Some y  False.
Proof.
  split; intros Hm i; specialize (Hm i);
    destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_disjoint_alt {A} (m1 m2 : M A) :
  m1 ⊥ₘ 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 map_not_disjoint {A} (m1 m2 : M A) :
  ¬m1 ⊥ₘ m2   i x1 x2, m1 !! i = Some x1  m2 !! i = Some x2.
Proof.
  unfold disjoint, map_disjoint. rewrite map_not_Forall2 by solve_decision.
  split; [|naive_solver].
  intros [i[(x&y&?&?&?)|[(x&?&?&[])|(y&?&?&[])]]]; naive_solver.
Qed.
Global Instance: Symmetric (map_disjoint : relation (M A)).
Proof. intros A m1 m2. rewrite !map_disjoint_spec. naive_solver. Qed.
Lemma map_disjoint_empty_l {A} (m : M A) :  ⊥ₘ m.
Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
Lemma map_disjoint_empty_r {A} (m : M A) : m ⊥ₘ .
Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
Lemma map_disjoint_weaken {A} (m1 m1' m2 m2' : M A) :
  m1' ⊥ₘ m2'  m1  m1'  m2  m2'  m1 ⊥ₘ m2.
Proof. rewrite !map_subseteq_spec, !map_disjoint_spec. eauto. Qed.
Lemma map_disjoint_weaken_l {A} (m1 m1' m2  : M A) :
  m1' ⊥ₘ m2  m1  m1'  m1 ⊥ₘ m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_weaken_r {A} (m1 m2 m2' : M A) :
  m1 ⊥ₘ m2'  m2  m2'  m1 ⊥ₘ m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_Some_l {A} (m1 m2 : M A) i x:
  m1 ⊥ₘ m2  m1 !! i = Some x  m2 !! i = None.
Proof. rewrite map_disjoint_spec, eq_None_not_Some. intros ?? [??]; eauto. Qed.
Lemma map_disjoint_Some_r {A} (m1 m2 : M A) i x:
  m1 ⊥ₘ m2  m2 !! i = Some x  m1 !! i = None.
Proof. rewrite (symmetry_iff map_disjoint). apply map_disjoint_Some_l. Qed.
978
Lemma map_disjoint_singleton_l {A} (m: M A) i x : {[ix]} ⊥ₘ m  m !! i = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
979
980
Proof.
  split; [|rewrite !map_disjoint_spec].
981
  * intro. apply (map_disjoint_Some_l {[i  x]} _ _ x);
Robbert Krebbers's avatar
Robbert Krebbers committed
982
983
984
985
986
987
      auto using lookup_singleton.
  * intros ? j y1 y2. destruct (decide (i = j)) as [->|].
    + rewrite lookup_singleton. intuition congruence.
    + by rewrite lookup_singleton_ne.
Qed.
Lemma map_disjoint_singleton_r {A} (m : M A) i x :
988
  m ⊥ₘ {[i  x]}  m !! i = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
989
990
Proof. by rewrite (symmetry_iff map_disjoint), map_disjoint_singleton_l. Qed.
Lemma map_disjoint_singleton_l_2 {A} (m : M A) i x :
991
  m !! i = None  {[i  x]} ⊥ₘ m.
Robbert Krebbers's avatar
Robbert Krebbers committed
992
993
Proof. by rewrite map_disjoint_singleton_l. Qed.
Lemma map_disjoint_singleton_r_2 {A} (m : M A) i x :
994
  m !! i = None  m ⊥ₘ {[i  x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
Proof. by rewrite map_disjoint_singleton_r. Qed.
Lemma map_disjoint_delete_l {A} (m1 m2 : M A) i : m1 ⊥ₘ m2  delete i m1 ⊥ₘ m2.
Proof.
  rewrite !map_disjoint_alt. intros Hdisjoint j. destruct (Hdisjoint j); auto.
  rewrite lookup_delete_None. tauto.
Qed.
Lemma map_disjoint_delete_r {A} (m1 m2 : M A) i : m1 ⊥ₘ m2  m1 ⊥ₘ delete i m2.
Proof. symmetry. by apply map_disjoint_delete_l. Qed.

(** ** Properties of the [union_with] operation *)
Section union_with.
Context {A} (f : A  A  option A).

Lemma lookup_union_with m1 m2 i :
  union_with f m1 m2 !! i = union_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-(lookup_merge _). Qed.
Lemma lookup_union_with_Some m1 m2 i z :
  union_with f m1 m2 !! i = Some z 
    (m1 !! i = Some z  m2 !! i = None) 
    (m1 !! i = None  m2 !! i = Some z) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  f x y = Some z).
Proof.
  rewrite lookup_union_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Global Instance: LeftId (@eq (M A))  (union_with f).
Proof. unfold union_with, map_union_with. apply _. Qed.
Global Instance: RightId (@eq (M A))  (union_with f).
Proof. unfold union_with, map_union_with. apply _. Qed.
Lemma union_with_commutative m1 m2 :
  ( i x y, m1 !! i = Some x  m2 !! i = Some y  f x y = f y x) 
  union_with f m1 m2 = union_with f m2 m1.
Proof.
  intros. apply (merge_commutative _). intros i.
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
Global Instance: Commutative (=) f  Commutative (@eq (M A)) (union_with f).
Proof. intros ???. apply union_with_commutative. eauto. Qed.
Lemma union_with_idempotent m :
  ( i x, m !! i = Some x  f x x = Some x)  union_with f m m = m.
Proof.
  intros. apply (merge_idempotent _). intros i.
  destruct (m !! i) eqn:?; simpl; eauto.
Qed.
Lemma alter_union_with (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f (g x) (g y)) 
  alter g i (union_with f m1 m2) =
    union_with f (alter g i m1) (alter g i m2).
Proof.
  intros. apply (partial_alter_merge _).
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
Lemma alter_union_with_l (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f (g x) y) 
  ( y, m1 !! i = None  m2 !! i = Some y  g y = y) 
  alter g i (union_with f m1 m2) = union_with f (alter g i m1) m2.
Proof.
  intros. apply (partial_alter_merge_l _).
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; f_equal'; auto.
Qed.
Lemma alter_union_with_r (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f x (g y)) 
  ( x, m1 !! i = Some x  m2 !! i = None  g x = x) 
  alter g i (union_with f m1 m2) = union_with f m1 (alter g i m2).
Proof.
  intros. apply (partial_alter_merge_r _).
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; f_equal'; auto.
Qed.
Lemma delete_union_with m1 m2 i :
  delete i (union_with f m1 m2) = union_with f (delete i m1) (delete i m2).
Proof. by apply (partial_alter_merge _). Qed.
Lemma foldr_delete_union_with (m1 m2 : M A) is :
  foldr delete (union_with f m1 m2) is =
    union_with f (foldr delete m1 is) (foldr delete m2 is).
Proof. induction is; simpl. done. by rewrite IHis, delete_union_with. Qed.
Lemma insert_union_with m1 m2 i x y z :
  f x y = Some z 
  <[i:=z]>(union_with f m1 m2) = union_with f (<[i:=x]>m1) (<[i:=y]>m2).
Proof. by intros; apply (partial_alter_merge _). Qed.
Lemma insert_union_with_l m1 m2 i x :
  m2 !! i = None  <[i:=x]>(union_with f m1 m2) = union_with f (<[i:=x]>m1) m2.
Proof.
  intros Hm2. unfold union_with, map_union_with.
  by erewrite (insert_merge_l _) by (by rewrite Hm2).
Qed.
Lemma insert_union_with_r m1 m2 i x :
  m1 !! i = None  <[i:=x]>(union_with f m1 m2) = union_with f m1 (<[i:=x]>m2).
Proof.
  intros Hm1. unfold union_with, map_union_with.
  by erewrite (insert_merge_r _) by (by rewrite Hm1).
Qed.
End union_with.

(** ** Properties of the [union] operation *)
Global Instance: LeftId (@eq (M A))  () := _.
Global Instance: RightId (@eq (M A))  () := _.
Global Instance: Associative (@eq (M A)) ().
Proof.
  intros A m1 m2 m3. unfold union, map_union, union_with, map_union_with.
  apply (merge_associative _). intros i.
  by destruct (m1 !! i), (m2 !! i), (m3 !! i).
Qed.
Global Instance: Idempotent (@eq (M A)) ().
Proof. intros A ?. by apply union_with_idempotent. Qed.
Lemma lookup_union_Some_raw {A} (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, map_union, union_with, map_union_with. rewrite (lookup_merge _).
  destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Qed.
Lemma lookup_union_None {A} (m1 m2 : M A) i :
  (m1  m2) !! i = None  m1 !! i = None  m2 !! i = None.
Proof.
  unfold union, map_union, union_with, map_union_with. rewrite (lookup_merge _).
  destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Qed.
Lemma map_positive_l {A} (m1 m2 : M A) : m1  m2 =   m1 = .
Proof.
  intros Hm. apply map_empty. intros i. apply (f_equal (!! i)) in Hm.
  rewrite lookup_empty, lookup_union_None in Hm; tauto.
Qed.
Lemma map_positive_l_alt {A} (m1 m2 : M A) : m1    m1  m2  .
Proof. eauto using map_positive_l. Qed.
Lemma lookup_union_Some {A} (m1 m2 : M A) i x :
  m1 ⊥ₘ m2  (m1  m2) !! i = Some x  m1 !! i = Some x  m2 !! i = Some x.
Proof.
  intros Hdisjoint. rewrite lookup_union_Some_raw.
  intuition eauto using map_disjoint_Some_r.
Qed.
Lemma lookup_union_Some_l {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  (m1  m2) !! i = Some x.
Proof. intro. rewrite lookup_union_Some_raw; intuition. Qed.
Lemma lookup_union_Some_r {A} (m1 m2 : M A) i x :
  m1 ⊥ₘ m2  m2 !! i = Some x  (m1  m2) !! i = Some x.
Proof. intro. rewrite lookup_union_Some; intuition. Qed.
Lemma map_union_commutative {A} (m1 m2 : M A) : m1 ⊥ₘ m2  m1  m2 = m2  m1.
Proof.
  intros Hdisjoint. apply (merge_commutative (union_with (λ x _, Some x))).
  intros i. specialize (Hdisjoint i).
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Lemma map_subseteq_union {A} (m1 m2 : M A) : m1  m2  m1  m2 = m2.
Proof.
  rewrite map_subseteq_spec.
  intros Hm1m2. apply map_eq. intros i. apply option_eq. intros x.
  rewrite lookup_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 map_union_subseteq_l {A} (m1 m2 : M A) : m1  m1  m2.
Proof.
  rewrite map_subseteq_spec. intros ? i x. rewrite lookup_union_Some_raw. tauto.
Qed.
Lemma map_union_subseteq_r {A} (m1 m2 : M A) : m1 ⊥ₘ m2  m2  m1  m2.
Proof.
  intros. rewrite map_union_commutative by done. by apply map_union_subseteq_l.
Qed.
Lemma map_union_subseteq_l_alt {A} (m1 m2 m3 : M A) : m1  m2  m1  m2  m3.
Proof. intros. transitivity m2; auto using map_union_subseteq_l. Qed.
Lemma map_union_subseteq_r_alt {A} (m1 m2 m3 : M A) :
  m2 ⊥ₘ m3  m1  m3  m1  m2  m3.
Proof. intros. transitivity m3; auto using map_union_subseteq_r. Qed.
Lemma map_union_preserving_l {A} (m1 m2 m3 : M A) : m1  m2  m3  m1  m3  m2.
Proof.
  rewrite !map_subseteq_spec. intros ???.
  rewrite !lookup_union_Some_raw. naive_solver.
Qed.
Lemma map_union_preserving_r {A} (m1 m2 m3 : M A) :
  m2 ⊥ₘ m3  m1  m2  m1  m3  m2  m3.
Proof.
  intros. rewrite !(map_union_commutative _ m3)
    by eauto using map_disjoint_weaken_l.
  by apply map_union_preserving_l.
Qed.
Lemma map_union_reflecting_l {A} (m1 m2 m3 : M A) :
  m3 ⊥ₘ m1  m3 ⊥ₘ m2  m3  m1  m3  m2  m1  m2.
Proof.
  rewrite !map_subseteq_spec. intros Hm31 Hm32 Hm i x ?. specialize (Hm i x).
  rewrite !lookup_union_Some in Hm by done. destruct Hm; auto.
  by rewrite map_disjoint_spec in Hm31; destruct (Hm31 i x x).
Qed.
Lemma map_union_reflecting_r {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3  m2 ⊥ₘ m3  m1  m3  m2  m3  m1  m2.
Proof.
  intros ??. rewrite !(map_union_commutative _ m3) by done.
  by apply map_union_reflecting_l.
Qed.
Lemma map_union_cancel_l {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3  m2 ⊥ₘ m3  m3  m1 = m3  m2  m1 = m2.
Proof.
  intros. apply (anti_symmetric ());
    apply map_union_reflecting_l with m3; auto using (reflexive_eq (R:=())).
Qed.
Lemma map_union_cancel_r {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3  m2 ⊥ₘ m3  m1  m3 = m2  m3  m1 = m2.
Proof.
  intros. apply (anti_symmetric ());
    apply map_union_reflecting_r with m3; auto using (reflexive_eq (R:=())).
Qed.
Lemma map_disjoint_union_l {A} (m1 m2 m3 : M A) :
  m1  m2 ⊥ₘ m3  m1 ⊥ₘ m3  m2 ⊥ₘ m3.
Proof.
  rewrite !map_disjoint_alt. setoid_rewrite lookup_union_None. naive_solver.
Qed.
Lemma map_disjoint_union_r {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m2  m3  m1 ⊥ₘ m2  m1 ⊥ₘ m3.
Proof.
  rewrite !map_disjoint_alt. setoid_rewrite lookup_union_None. naive_solver.
Qed.
Lemma map_disjoint_union_l_2 {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3  m2 ⊥ₘ m3  m1  m2 ⊥ₘ m3.
Proof. by rewrite map_disjoint_union_l. Qed.
Lemma map_disjoint_union_r_2 {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m2  m1 ⊥ₘ m3  m1 ⊥ₘ m2  m3.
Proof. by rewrite map_disjoint_union_r. Qed.
1211
Lemma insert_union_singleton_l {A} (m : M A) i x : <[i:=x]>m = {[i  x]}  m.
Robbert Krebbers's avatar
Robbert Krebbers committed
1212
1213
1214
1215
1216
1217
1218
1219
Proof.
  apply map_eq. intros j. apply option_eq. intros y.
  rewrite lookup_union_Some_raw.
  destruct (decide (i = j)); subst.
  * rewrite !lookup_singleton, lookup_insert. intuition congruence.
  * rewrite !lookup_singleton_ne, lookup_insert_ne; intuition congruence.
Qed.
Lemma insert_union_singleton_r {A} (m : M A) i x :
1220
  m !! i = None  <[i:=x]>m = m  {[i  x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
Proof.
  intro. rewrite insert_union_singleton_l, map_union_commutative; [done |].
  by apply map_disjoint_singleton_l.
Qed.
Lemma map_disjoint_insert_l {A} (m1 m2 : M A) i x :
  <[i:=x]>m1 ⊥ₘ m2  m2 !! i = None  m1 ⊥ₘ m2.
Proof.
  rewrite insert_union_singleton_l.
  by rewrite map_disjoint_union_l, map_disjoint_singleton_l.
Qed.
Lemma map_disjoint_insert_r {A} (m1 m2 : M A) i x :
  m1 ⊥ₘ <[i:=x]>m2  m1 !! i = None  m1 ⊥ₘ m2.
Proof.
  rewrite insert_union_singleton_l.
  by rewrite map_disjoint_union_r, map_disjoint_singleton_r.
Qed.
Lemma map_disjoint_insert_l_2 {A} (m1 m2 : M A) i x :
  m2 !! i = None  m1 ⊥ₘ m2  <[i:=x]>m1 ⊥ₘ m2.
Proof. by rewrite map_disjoint_insert_l. Qed.
Lemma map_disjoint_insert_r_2 {A} (m1 m2 : M A) i x :
  m1 !! i = None  m1 ⊥ₘ m2  m1 ⊥ₘ <[i:=x]>m2.
Proof. by rewrite map_disjoint_insert_r. Qed.
Lemma insert_union_l {A} (m1 m2 : M A) i x :
  <[i:=x]>(m1  m2) = <[i:=x]>m1  m2.
Proof. by rewrite !insert_union_singleton_l, (associative_L ()). Qed.
Lemma insert_union_r {A} (m1 m2 : M A) i x :
  m1 !! i = None  <[i:=x]>(m1  m2) = m1  <[i:=x]>m2.
Proof.
  intro. rewrite !insert_union_singleton_l, !(associative_L ()).
  rewrite (map_union_commutative m1); [done |].
  by apply map_disjoint_singleton_r.
Qed.
Lemma foldr_insert_union {A} (m : M A) l :
  foldr (λ p, <[p.1:=p.2]>) m l = map_of_list l  m.
Proof.
  induction l as [|i l IH]; simpl; [by rewrite (left_id_L _ _)|].
  by rewrite IH, insert_union_l.
Qed.
Lemma delete_union {A} (m1 m2 : M A) i :
  delete i (m1  m2) = delete i m1  delete i m2.
Proof. apply delete_union_with. Qed.

(** ** Properties of the [union_list] operation *)
Lemma map_disjoint_union_list_l {A} (ms : list (M A)) (m : M A) :
   ms ⊥ₘ m  Forall (.⊥ₘ m) ms.
Proof.
  split.
  * induction ms; simpl; rewrite ?map_disjoint_union_l; intuition.
  * induction 1; simpl; [apply map_disjoint_empty_l |].
    by rewrite map_disjoint_union_l.
Qed.
Lemma map_disjoint_union_list_r {A} (ms : list (M A)) (m : M A) :
  m ⊥ₘ  ms  Forall (.⊥ₘ m) ms.
Proof. by rewrite (symmetry_iff map_disjoint), map_disjoint_union_list_l. Qed.
Lemma map_disjoint_union_list_l_2 {A} (ms : list (M A)) (m : M A) :
  Forall (.⊥ₘ m) ms   ms ⊥ₘ m.
Proof. by rewrite map_disjoint_union_list_l. Qed.
Lemma map_disjoint_union_list_r_2 {A} (ms : list (M A)) (m : M A) :
  Forall (.⊥ₘ m) ms  m ⊥ₘ  ms.
Proof. by rewrite map_disjoint_union_list_r. Qed.

(** ** Properties of the folding the [delete] function *)
Lemma lookup_foldr_delete {A} (m : M A) is j :
  j  is  foldr delete m is !! j = None.
Proof.
  induction 1 as [|i j is]; simpl; [by rewrite lookup_delete|].
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne by done.
Qed.
Lemma lookup_foldr_delete_not_elem_of {A} (m : M A) is j :
  j  is  foldr delete m is !! j = m !! j.
Proof.
  induction is; simpl; [done |]. rewrite elem_of_cons; intros.
  rewrite lookup_delete_ne; intuition.
Qed.
Lemma foldr_delete_notin {A} (m : M A) is :
  Forall (λ i, m !! i = None) is  foldr delete m is = m.
Proof. induction 1; simpl; [done |]. rewrite delete_notin; congruence. Qed.
Lemma foldr_delete_insert_ne {A} (m : M A) is j x :
  j  is  foldr delete (<[j:=x]>m) is = <[j:=x]>(foldr delete m is).
Proof.
  induction is; simpl; [done |]. rewrite elem_of_cons. intros.
  rewrite IHis, delete_insert_ne; intuition.
Qed.
Lemma map_disjoint_foldr_delete_l {A} (m1 m2 : M A) is :
  m1 ⊥ₘ m2  foldr delete m1 is ⊥ₘ m2.
Proof. induction is; simpl; auto using map_disjoint_delete_l. Qed.
Lemma map_disjoint_foldr_delete_r {A} (m1 m2 : M A) is :
  m1 ⊥ₘ m2  m1 ⊥ₘ foldr delete m2 is.
Proof. induction is; simpl; auto using map_disjoint_delete_r. Qed.
Lemma foldr_delete_union {A} (m1 m2 : M A) is :
  foldr delete (m1  m2) is = foldr delete m1 is  foldr delete m2 is.
Proof. apply foldr_delete_union_with. Qed.

(** ** Properties on disjointness of conversion to lists *)
Lemma map_disjoint_of_list_l {A} (m : M A) ixs :
  map_of_list ixs ⊥ₘ m  Forall (λ ix, m !! ix.1 = None) ixs.
Proof.
  split.
  * induction ixs; simpl; rewrite ?map_disjoint_insert_l in *; intuition.
  * induction 1; simpl; [apply map_disjoint_empty_l|].
    rewrite map_disjoint_insert_l. auto.
Qed.
Lemma map_disjoint_of_list_r {A} (m : M A) ixs :
  m ⊥ₘ map_of_list ixs  Forall (λ ix, m !! ix.1 = None) ixs.
Proof. by rewrite (symmetry_iff map_disjoint), map_disjoint_of_list_l. Qed.
Lemma map_disjoint_of_list_zip_l {A} (m : M A) is xs :
  length is = length xs 
  map_of_list (zip is xs) ⊥ₘ m  Forall (λ i, m !! i = None) is.
Proof.
  intro. rewrite map_disjoint_of_list_l.
  rewrite <-(fst_zip is xs) at 2 by lia. by rewrite Forall_fmap.
Qed.
Lemma map_disjoint_of_list_zip_r {A} (m : M A) is xs :
  length is = length xs 
  m ⊥ₘ map_of_list (zip is xs)  Forall (λ i, m !! i = None) is.
Proof.
  intro. by rewrite (symmetry_iff map_disjoint), map_disjoint_of_list_zip_l.
Qed.
Lemma map_disjoint_of_list_zip_l_2 {A} (m : M A) is xs :
  length is = length xs  Forall (λ i, m !! i = None) is 
  map_of_list (zip is xs) ⊥ₘ m.
Proof. intro. by rewrite map_disjoint_of_list_zip_l. Qed.
Lemma map_disjoint_of_list_zip_r_2 {A} (m : M A) is xs :
  length is = length xs  Forall (λ i, m !! i = None) is 
  m ⊥ₘ map_of_list (zip is xs).
Proof. intro. by rewrite map_disjoint_of_list_zip_r. Qed.

(** ** Properties of the [intersection_with] operation *)
Lemma lookup_intersection_with {A} (f : A  A  option A) m1 m2 i :
  intersection_with f m1 m2 !! i = intersection_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-(lookup_merge _). Qed.
Lemma lookup_intersection_with_Some {A} (f : A  A  option A) m1 m2 i z :
  intersection_with f m1 m2 !! i = Some z 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  f x y = Some z).
Proof.
  rewrite lookup_intersection_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.

(** ** Properties of the [intersection] operation *)
Lemma lookup_intersection_Some {A} (m1 m2 : M A) i x :
  (m1  m2) !! i = Some x  m1 !! i = Some x  is_Some (m2 !! i).
Proof.
  unfold intersection, map_intersection. rewrite lookup_intersection_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Lemma lookup_intersection_None {A} (m1 m2 : M A) i :
  (m1  m2) !! i = None  m1 !! i = None  m2 !! i = None.
Proof.
  unfold intersection, map_intersection. rewrite lookup_intersection_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.

(** ** Properties of the [difference_with] operation *)
Lemma lookup_difference_with {A} (f : A  A  option A) m1 m2 i :
  difference_with f m1 m2 !! i = difference_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-lookup_merge by done. Qed.
Lemma lookup_difference_with_Some {A} (f : A  A  option A) m1 m2 i z :
  difference_with f m1 m2 !! i = Some z 
    (m1 !! i = Some z  m2 !! i = None) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  f x y = Some z).
Proof.
  rewrite lookup_difference_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.

(** ** Properties of the [difference] operation *)
Lemma lookup_difference_Some {A} (m1 m2 : M A) i x :
  (m1  m2) !! i = Some x  m1 !! i = Some x  m2 !! i = None.
Proof.
  unfold difference, map_difference; rewrite lookup_difference_with.
  destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Qed.
Lemma lookup_difference_None {A} (m1 m2 : M A) i :
  (m1  m2) !! i = None  m1 !! i = None  is_Some (m2 !! i).
Proof.
  unfold difference, map_difference; rewrite lookup_difference_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Lemma map_disjoint_difference_l {A} (m1 m2 : M A) : m1  m2  m2  m1 ⊥ₘ m1.
Proof.
  intros Hm i; specialize (Hm i).
  unfold difference, map_difference; rewrite lookup_difference_with.
  by destruct (m1 !! i), (m2 !! i).
Qed.
Lemma map_disjoint_difference_r {A} (m1 m2 : M A) : m1  m2  m1 ⊥ₘ m2  m1.
Proof. intros. symmetry. by apply map_disjoint_difference_l. Qed.
Lemma map_difference_union {A} (m1 m2 : M A) :
  m1  m2  m1  m2  m1 = m2.
Proof.
  rewrite map_subseteq_spec. intro Hm1m2. apply map_eq. intros i.
  apply option_eq. intros v. specialize (Hm1m2 i).
  unfold difference, map_difference, difference_with, map_difference_with.
  rewrite lookup_union_Some_raw, (lookup_merge _).
  destruct (m1 !! i) as [x'|], (m2 !! i);
    try specialize (Hm1m2 x'); compute; intuition congruence.
Qed.
End theorems.

(** * Tactics *)
(** The tactic [decompose_map_disjoint] simplifies occurrences of [disjoint]
in the hypotheses that involve the empty map [], the union [()] or insert
1424
[<[_:=_]>] operation, the singleton [{[_ _]}] map, and disjointness of lists of
Robbert Krebbers's avatar
Robbert Krebbers committed
1425
1426
1427
1428
1429
1430
maps. This tactic does not yield any information loss as all simplifications
performed are reversible. *)
Ltac decompose_map_disjoint := repeat
  match goal with
  | H : _  _ ⊥ₘ _ |- _ => apply map_disjoint_union_l in H; destruct H
  | H : _ ⊥ₘ _  _ |- _ => apply map_disjoint_union_r in H; destruct H
1431
1432
  | H : {[ _  _ ]} ⊥ₘ _ |- _ => apply map_disjoint_singleton_l in H
  | H : _ ⊥ₘ {[ _  _ ]} |- _ =>  apply map_disjoint_singleton_r in H
Robbert Krebbers's avatar
Robbert Krebbers committed
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
  | H : <[_:=_]>_ ⊥ₘ _ |- _ => apply map_disjoint_insert_l in H; destruct H
  | H : _ ⊥ₘ <[_:=_]>_ |- _ => apply map_disjoint_insert_r in H; destruct H
  | H :  _ ⊥ₘ _ |- _ => apply map_disjoint_union_list_l in H
  | H : _ ⊥ₘ  _ |- _ => apply map_disjoint_union_list_r in H
  | H :  ⊥ₘ _ |- _ => clear H
  | H : _ ⊥ₘ  |- _ => clear H
  | H : Forall (.⊥ₘ _) _ |- _ => rewrite Forall_vlookup in H
  | H : Forall (.⊥ₘ _) [] |- _ => clear H
  | H : Forall (.⊥ₘ _) (_ :: _) |- _ => rewrite Forall_cons in H; destruct H
  | H : Forall (.⊥ₘ _) (_ :: _) |- _ => rewrite Forall_app in H; destruct H
  end.

(** To prove a disjointness property, we first decompose all hypotheses, and
then use an auto database to prove the required property. *)
Create HintDb map_disjoint.
Ltac solve_map_disjoint :=
  solve [decompose_map_disjoint; auto with map_disjoint].

(** We declare these hints using [Hint Extern] instead of [Hint Resolve] as
[eauto] works badly with hints parametrized by type class constraints. *)
Hint Extern 1 (_ ⊥ₘ _) => done : map_disjoint.
Hint Extern 2 ( ⊥ₘ _) => apply map_disjoint_empty_l : map_disjoint.
Hint Extern 2 (_ ⊥ₘ ) => apply map_disjoint_empty_r : map_disjoint.
1456
Hint Extern 2 ({[ _  _ ]} ⊥ₘ _) =>
Robbert Krebbers's avatar
Robbert Krebbers committed
1457
  apply map_disjoint_singleton_l_2 : map_disjoint.
1458
Hint Extern 2 (_ ⊥ₘ {[ _  _ ]}) =>
Robbert Krebbers's avatar
Robbert Krebbers committed
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
  apply map_disjoint_singleton_r_2 : map_disjoint.
Hint Extern 2 (_  _ ⊥ₘ _) => apply map_disjoint_union_l_2 : map_disjoint.
Hint Extern 2 (_ ⊥ₘ _  _) => apply map_disjoint_union_r_2 : map_disjoint.
Hint Extern 2 (<[_:=_]>_ ⊥ₘ _) => apply map_disjoint_insert_l_2 : map_disjoint.
Hint Extern 2 (_ ⊥ₘ <[_:=_]>_) => apply map_disjoint_insert_r_2 : map_disjoint.
Hint Extern 2 (delete _ _ ⊥ₘ _) => apply map_disjoint_delete_l : map_disjoint.
Hint Extern 2 (_ ⊥ₘ delete _ _) => apply map_disjoint_delete_r : map_disjoint.
Hint Extern 2 (map_of_list _ ⊥ₘ _) =>
  apply map_disjoint_of_list_zip_l_2 : mem_disjoint.
Hint Extern 2 (_ ⊥ₘ map_of_list _) =>
  apply map_disjoint_of_list_zip_r_2 : mem_disjoint.
Hint Extern 2 ( _ ⊥ₘ _) => apply map_disjoint_union_list_l_2 : mem_disjoint.
Hint Extern 2 (_ ⊥ₘ  _) => apply map_disjoint_union_list_r_2 : mem_disjoint.
Hint Extern 2 (foldr delete _ _ ⊥ₘ _) =>
  apply map_disjoint_foldr_delete_l : map_disjoint.
Hint Extern 2 (_ ⊥ₘ foldr delete _ _) =>
  apply map_disjoint_foldr_delete_r : map_disjoint.

(** The tactic [simpl_map by tac] simplifies occurrences of finite map look
ups. It uses [tac] to discharge generated inequalities. Look ups in unions do
not have nice equational properties, hence it invokes [tac] to prove that such
look ups yield [Some]. *)
Tactic Notation "simpl_map" "by" tactic3(tac) := repeat
  match goal with
  | H : context[  !! _ ] |- _ => rewrite lookup_empty in H
  | H : context[ (<[_:=_]>_) !! _ ] |- _ =>
    rewrite lookup_insert in H || rewrite lookup_insert_ne in H by tac
  | H : context[ (alter _ _ _) !! _] |- _ =>
    rewrite lookup_alter in H || rewrite lookup_alter_ne in H by tac
  | H : context[ (delete _ _) !! _] |- _ =>
    rewrite lookup_delete in H || rewrite lookup_delete_ne in H by tac
1490
  | H : context[ {[ _  _ ]} !! _ ] |- _ =>
Robbert Krebbers's avatar
Robbert Krebbers committed
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
    rewrite lookup_singleton in H || rewrite lookup_singleton_ne in H by tac
  | H : context[ (_ <$> _) !! _ ] |- _ => rewrite lookup_fmap in H
  | H : context[ (omap _ _) !! _ ] |- _ => rewrite lookup_omap in H
  | H : context[ lookup (A:=?A) ?i (?m1  ?m2) ] |- _ =>
    let x := fresh in evar (x:A);
    let x' := eval unfold x in x in clear x;
    let E := fresh in
    assert ((m1  m2) !! i = Some x') as E by (clear H; by tac);
    rewrite E in H; clear E
  | |- context[  !! _ ] => rewrite lookup_empty
  | |- context[ (<[_:=_]>_) !! _ ] =>
    rewrite lookup_insert || rewrite lookup_insert_ne by tac
  | |- context[ (alter _ _ _) !! _ ] =>
    rewrite lookup_alter || rewrite lookup_alter_ne by tac
  | |- context[ (delete _ _) !! _ ] =>
    rewrite lookup_delete || rewrite lookup_delete_ne by tac
1507
  | |- context[ {[ _  _ ]} !! _ ] =>
Robbert Krebbers's avatar
Robbert Krebbers committed
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
    rewrite lookup_singleton || rewrite lookup_singleton_ne by tac
  | |- context[ (_ <$> _) !! _ ] => rewrite lookup_fmap
  | |- context[ (omap _ _) !! _ ] => rewrite lookup_omap
  | |- context [ lookup (A:=?A) ?i ?m ] =>
    let x := fresh in evar (x:A);
    let x' := eval unfold x in x in clear x;
    let E := fresh in
    assert (m !! i = Some x') as E by tac;
    rewrite E; clear E
  end.

Create HintDb simpl_map.
Tactic Notation "simpl_map" := simpl_map by eauto with simpl_map map_disjoint.

Hint Extern 80 ((_  _) !! _ = Some _) => apply lookup_union_Some_l : simpl_map.
Hint Extern 81 ((_  _) !! _ = Some _) => apply lookup_union_Some_r : simpl_map.
1524
Hint Extern 80 ({[ __ ]} !! _ = Some _) => apply lookup_singleton : simpl_map.
Robbert Krebbers's avatar
Robbert Krebbers committed
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
Hint Extern 80 (<[_:=_]> _ !! _ = Some _) => apply lookup_insert : simpl_map.

(** Now we take everything together and also discharge conflicting look ups,
simplify overlapping look ups, and perform cancellations of equalities
involving unions. *)
Tactic Notation "simplify_map_equality" "by" tactic3(tac) :=
  decompose_map_disjoint;
  repeat match goal with
  | _ => progress simpl_map by tac
  | _ => progress simplify_equality
  | _ => progress simpl_option_monad by tac
1536
1537
  | H : {[ _  _ ]} !! _ = None |- _ => rewrite lookup_singleton_None in H
  | H : {[ _  _ ]} !! _ = Some _ |- _ =>
Robbert Krebbers's avatar
Robbert Krebbers committed
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
    rewrite lookup_singleton_Some in H; destruct H
  | 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
  | H1 : ?m1 !! ?i = Some ?x, H2 : ?m2 !! ?i = None |- _ =>
    let H3 := fresh in
    apply (lookup_weaken _ m2) in H1; [congruence|by tac]
  | H : ?m  _ = ?m  _ |- _ =>
    apply map_union_cancel_l in H; [|by tac|by tac]
  | H : _  ?m = _  ?m |- _ =>
    apply map_union_cancel_r in H; [|by tac|by tac]
1550
1551
  | H : {[?i  ?x]} =  |- _ => by destruct (map_non_empty_singleton i x)
  | H :  = {[?i  ?x]} |- _ => by destruct (map_non_empty_singleton i x)
Robbert Krebbers's avatar
Robbert Krebbers committed
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
  | H : ?m !! ?i = Some _, H2 : ?m !! ?j = None |- _ =>
     unless (i  j) by done;
     assert (i  j) by (by intros ?; simplify_equality)
  end.
Tactic Notation "simplify_map_equality'" "by" tactic3(tac) :=
  repeat (progress csimpl in * || simplify_map_equality by tac).
Tactic Notation "simplify_map_equality" :=
  simplify_map_equality by eauto with simpl_map map_disjoint.
Tactic Notation "simplify_map_equality'" :=
  simplify_map_equality' by eauto with simpl_map map_disjoint.