fin_maps.v 64.8 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
2 3 4
(* 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
5 6
induction principles for finite maps and implements the tactic
[simplify_map_equality] to simplify goals involving finite maps. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
7
Require Import Permutation.
8 9
Require Export ars vector orders.

10 11
(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
12 13 14 15 16
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. *)
17

Robbert Krebbers's avatar
Robbert Krebbers committed
18 19
(** Finiteness is axiomatized by requiring that each map can be translated
to an association list. The translation to association lists is used to
20
prove well founded recursion on finite maps. *)
21

22 23 24
(** 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]. *)
25

26
Class FinMapToList K A M := map_to_list: M  list (K * A).
Robbert Krebbers's avatar
Robbert Krebbers committed
27

28 29 30
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)} := {
31 32
  map_eq {A} (m1 m2 : M A) : ( i, m1 !! i = m2 !! i)  m1 = m2;
  lookup_empty {A} i : ( : M A) !! i = None;
33 34 35 36
  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;
37
  lookup_fmap {A B} (f : A  B) (m : M A) i : (f <$> m) !! i = f <$> m !! i;
38
  NoDup_map_to_list {A} (m : M A) : NoDup (map_to_list m);
39 40
  elem_of_map_to_list {A} (m : M A) i x :
    (i,x)  map_to_list m  m !! i = Some x;
41
  lookup_omap {A B} (f : A  option B) m i : omap f m !! i = m !! i = f;
42 43 44
  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)
Robbert Krebbers's avatar
Robbert Krebbers committed
45 46
}.

47 48 49
(** * 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
50 51
significant performance loss to make including them in the finite map interface
worthwhile. *)
52 53 54 55 56
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 :=
57
  partial_alter (λ _, None).
58 59
Instance map_singleton `{PartialAlter K A M, Empty M} :
  Singleton (K * A) M := λ p, <[p.1:=p.2]> .
Robbert Krebbers's avatar
Robbert Krebbers committed
60

61
Definition map_of_list `{Insert K A M, Empty M} : list (K * A)  M :=
62
  fold_right (λ p, <[p.1:=p.2]>) .
63 64 65
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)).
Robbert Krebbers's avatar
Robbert Krebbers committed
66

67 68 69 70 71 72
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).
Robbert Krebbers's avatar
Robbert Krebbers committed
73

74 75
(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
76
Definition map_Forall `{Lookup K A M} (P : K  A  Prop) : M  Prop :=
Robbert Krebbers's avatar
Robbert Krebbers committed
77
  λ m,  i x, m !! i = Some x  P i x.
78 79 80 81 82 83 84 85 86
Definition map_Forall2 `{ 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,
  match m1 !! i, m2 !! i with
  | Some x, Some y => R x y
  | Some x, None => P x
  | None, Some y => Q y
  | None, None => True
  end.
87 88
Definition map_included `{ A, Lookup K A (M A)} {A}
  (R : relation A) : relation (M A) := map_Forall2 R (λ _, False) (λ _, True).
89 90 91 92
Instance map_disjoint `{ A, Lookup K A (M A)} {A} : Disjoint (M A) :=
  map_Forall2 (λ _ _, False) (λ _, True) (λ _, True).
Instance map_subseteq `{ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
  map_Forall2 (=) (λ _, False) (λ _, True).
Robbert Krebbers's avatar
Robbert Krebbers committed
93 94 95 96 97

(** 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. *)
98
Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x).
99 100 101
Instance map_intersection `{Merge M} {A} : Intersection (M A) :=
  intersection_with (λ x _, Some x).

102 103
(** The difference operation removes all values from the first map whose
index contains a value in the second map as well. *)
104
Instance map_difference `{Merge M} {A} : Difference (M A) :=
105
  difference_with (λ _ _, None).
Robbert Krebbers's avatar
Robbert Krebbers committed
106

107 108 109 110
(** * Theorems *)
Section theorems.
Context `{FinMap K M}.

111 112 113 114 115 116 117 118
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_Forall2. split; intros Hm i;
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
119
Global Instance: EmptySpec (M A).
120
Proof.
121 122
  intros A m. rewrite !map_subseteq_spec.
  intros i x. by rewrite lookup_empty.
123
Qed.
124 125 126 127 128 129
Global Instance:  {A} (R : relation A), PreOrder R  PreOrder (map_included R).
Proof.
  split; [intros m i; by destruct (m !! i)|].
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
  destruct (m1 !! i), (m2 !! i), (m3 !! i); try done; etransitivity; eauto.
Qed.
130
Global Instance: PartialOrder (() : relation (M A)).
131
Proof.
132 133 134
  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
135 136 137
Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
138
Proof. rewrite !map_subseteq_spec. auto. Qed.
139 140 141 142 143 144
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.
145 146
  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
147 148
Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
149 150
  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
151 152 153 154 155 156 157 158 159
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  .
160 161 162
Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
163 164

(** ** Properties of the [partial_alter] operation *)
165 166 167
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.
168 169
  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
170 171
Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
172 173
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
174 175
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
176
Qed.
177
Lemma partial_alter_commute {A} f g (m : M A) i j :
178
  i  j  partial_alter f i (partial_alter g j m) =
179 180
    partial_alter g j (partial_alter f i m).
Proof.
181 182 183 184 185 186 187
  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.
188 189 190 191
Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
192 193
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
194
Qed.
195
Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m.
196
Proof. by apply partial_alter_self_alt. Qed.
197
Lemma partial_alter_subseteq {A} f (m : M A) i :
198
  m !! i = None  m  partial_alter f i m.
199 200 201 202
Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
203
Lemma partial_alter_subset {A} f (m : M A) i :
204
  m !! i = None  is_Some (f (m !! i))  m  partial_alter f i m.
205
Proof.
206 207 208 209
  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.
210 211 212
Qed.

(** ** Properties of the [alter] operation *)
213 214
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.
215
Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal'; auto. Qed.
216
Lemma lookup_alter {A} (f : A  A) m i : alter f i m !! i = f <$> m !! i.
217
Proof. unfold alter. apply lookup_partial_alter. Qed.
218
Lemma lookup_alter_ne {A} (f : A  A) m i j : i  j  alter f i m !! j = m !! j.
219
Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
220 221 222 223 224 225 226 227 228
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.
229 230 231 232
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.
233
  destruct (decide (i = j)) as [->|?].
234 235 236 237 238 239
  * 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.
240 241
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
242
Qed.
243
Lemma alter_None {A} (f : A  A) m i : m !! i = None  alter f i m = m.
244
Proof.
245 246
  intros Hi. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?Hi, ?lookup_alter_ne.
247 248 249 250 251 252 253 254 255 256 257
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.
258
  * destruct (decide (i = j)) as [->|?];
259 260 261 262 263 264
      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.
265 266
  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
267 268 269
Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
270
Lemma delete_singleton {A} i (x : A) : delete i {[i, x]} = .
271 272 273 274 275 276 277
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.
278
Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
279
Proof.
280 281
  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298
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.
299
Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
300 301 302
Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
303
Lemma delete_subseteq_compat {A} (m1 m2 : M A) i :
304
  m1  m2  delete i m1  delete i m2.
305 306 307 308
Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
Qed.
309
Lemma delete_subset_alt {A} (m : M A) i x : m !! i = Some x  delete i m  m.
310
Proof.
311 312 313
  split; [apply delete_subseteq|].
  rewrite !map_subseteq_spec. intros Hi. apply (None_ne_Some x).
  by rewrite <-(lookup_delete m i), (Hi i x).
314
Qed.
315
Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
316 317 318 319 320
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.
321
Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
322
Proof. rewrite lookup_insert. congruence. Qed.
323
Lemma lookup_insert_ne {A} (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
324 325 326 327 328 329 330 331
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.
332
  * destruct (decide (i = j)) as [->|?];
333
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
334
  * intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
335 336 337 338
Qed.
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
339 340 341
  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
342
Qed.
343 344 345 346 347 348 349 350 351 352 353 354
Lemma insert_lookup {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); eauto.
  * rewrite lookup_insert_ne by done. by destruct (m !! j).
Qed.
355
Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
356
Proof. apply partial_alter_subseteq. Qed.
357
Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
358 359
Proof. intro. apply partial_alter_subset; eauto. Qed.
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
360
  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
361
Proof.
362 363 364
  rewrite !map_subseteq_spec. intros ?? j ?.
  destruct (decide (j = i)) as [->|?]; [congruence|].
  rewrite lookup_insert_ne; auto.
365 366
Qed.
Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
367
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
368
Proof.
369 370 371 372
  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.
373 374
Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
375
  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
376
Proof.
377 378
  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
379
  * rewrite lookup_insert. congruence.
380
  * rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
381 382
Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
383
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
384
Proof.
385 386 387
  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
388 389
Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
390
  m1 !! i = None  <[i:=x]> m1  m2 
391 392 393
   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
  intros Hi Hm1m2. exists (delete i m2). split_ands.
394
  * rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto.
395 396 397 398
    by rewrite lookup_insert.
  * eauto using insert_delete_subset.
  * by rewrite lookup_delete.
Qed.
399 400 401 402 403 404 405
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.
406 407
Lemma insert_empty {A} i (x : A) : <[i:=x]> = {[i,x]}.
Proof. done. Qed.
408 409 410

(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
411
  {[i, x]} !! j = Some y  i = j  x = y.
412 413
Proof.
  unfold singleton, map_singleton.
414
  rewrite lookup_insert_Some, lookup_empty. simpl. intuition congruence.
415
Qed.
416
Lemma lookup_singleton_None {A} i j (x : A) : {[i, x]} !! j = None  i  j.
417 418 419 420
Proof.
  unfold singleton, map_singleton.
  rewrite lookup_insert_None, lookup_empty. simpl. tauto.
Qed.
421
Lemma lookup_singleton {A} i (x : A) : {[i, x]} !! i = Some x.
422
Proof. by rewrite lookup_singleton_Some. Qed.
423
Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[i, x]} !! j = None.
424
Proof. by rewrite lookup_singleton_None. Qed.
425
Lemma map_non_empty_singleton {A} i (x : A) : {[i,x]}  .
426 427 428 429
Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
430
Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i, x]} = {[i, y]}.
431 432 433 434
Proof.
  unfold singleton, map_singleton, insert, map_insert.
  by rewrite <-partial_alter_compose.
Qed.
435
Lemma alter_singleton {A} (f : A  A) i x : alter f i {[i,x]} = {[i, f x]}.
436
Proof.
437
  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
438 439 440 441
  * 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 :
442
  i  j  alter f i {[j,x]} = {[j,x]}.
443
Proof.
444 445
  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
446 447
Qed.

448 449 450 451 452
(** ** 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.
453 454 455 456 457 458 459
Lemma omap_singleton {A B} (f : A  option B) i x y :
  f x = Some y  omap f {[ i,x ]} = {[ i,y ]}.
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.
460

461 462
(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
463
  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
464
Proof. rewrite !elem_of_map_to_list. congruence. Qed.
465
Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
466
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
467 468 469 470 471 472 473 474 475 476
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.
477
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
478
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
479
Proof.
480 481
  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'].
482
  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
483
    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
484 485
Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
486
  map_of_list l !! i = Some x  (i,x)  l.
487
Proof.
488 489 490
  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.
491 492
Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
493
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
494
Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
495
Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
496
  i  l.*1  map_of_list l !! i = None.
497
Proof.
498 499
  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.
500 501
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
502
  map_of_list l !! i = None  i  l.*1.
503
Proof.
504
  induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
505 506 507 508 509
  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 :
510
  i  l.*1  map_of_list l !! i = None.
511
Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
512
Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
513
  NoDup (l1.*1)  l1  l2  map_of_list l1 = map_of_list l2.
514 515 516 517 518
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)) :
519
  NoDup (l1.*1)  NoDup (l2.*1)  map_of_list l1 = map_of_list l2  l1  l2.
520
Proof.
521
  intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
522 523
  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
524
Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m.
525 526 527
Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
528
    by auto using NoDup_fst_map_to_list.
529 530
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
531
  NoDup (l.*1)  map_to_list (map_of_list l)  l.
532
Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
533
Lemma map_to_list_inj {A} (m1 m2 : M A) :
534
  map_to_list m1  map_to_list m2  m1 = m2.
535
Proof.
536
  intros. rewrite <-(map_of_to_list m1), <-(map_of_to_list m2).
537
  auto using map_of_list_proper, NoDup_fst_map_to_list.
538
Qed.
539 540 541 542 543 544
Lemma map_to_of_list_flip {A} (m1 : M A) l2 :
  map_to_list m1  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.
545
Lemma map_to_list_empty {A} : map_to_list  = @nil (K * A).
546 547 548 549 550
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 :
551
  m !! i = None  map_to_list (<[i:=x]>m)  (i,x) :: map_to_list m.
552
Proof.
553
  intros. apply map_of_list_inj; csimpl.
554 555
  * apply NoDup_fst_map_to_list.
  * constructor; auto using NoDup_fst_map_to_list.
556
    rewrite elem_of_list_fmap. intros [[??] [? Hlookup]]; subst; simpl in *.
557 558 559
    rewrite elem_of_map_to_list in Hlookup. congruence.
  * by rewrite !map_of_to_list.
Qed.
560
Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = .
561 562 563 564
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.
565
Lemma map_to_list_empty_inv_alt {A}  (m : M A) : map_to_list m  []  m = .
566
Proof. rewrite <-map_to_list_empty. apply map_to_list_inj. Qed.
567
Lemma map_to_list_empty_inv {A} (m : M A) : map_to_list m = []  m = .
568 569
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 :
570
  map_to_list m  (i,x) :: l  m = <[i:=x]>(map_of_list l).
571 572
Proof.
  intros Hperm. apply map_to_list_inj.
573 574 575
  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. }
576 577 578
  rewrite Hperm, map_to_list_insert, map_to_of_list;
    auto using not_elem_of_map_of_list_1.
Qed.
579 580 581 582 583 584
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. }
  exists i x. rewrite <-elem_of_map_to_list, Hm. by left.
Qed.
585

586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602
(** ** 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 *)
603
Lemma map_ind {A} (P : M A  Prop) :
604
  P   ( i x m, m !! i = None  P m  P (<[i:=x]>m))   m, P m.
605
Proof.
606
  intros ? Hins. cut ( l, NoDup (l.*1)   m, map_to_list m  l  P m).
607
  { intros help m.
608
    apply (help (map_to_list m)); auto using NoDup_fst_map_to_list. }
609 610 611
  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.
612
  apply map_to_list_insert_inv in Hml; subst m. apply Hins.
613 614 615 616
  * 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) :
617
  m1  m2  length (map_to_list m1) < length (map_to_list m2).
618 619 620 621
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.
622
    apply nil_length_inv, map_to_list_empty_inv in Hlen.
623 624 625 626 627
    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.
628
Lemma map_wf {A} : wf (strict (@subseteq (M A) _)).
629 630 631 632 633 634
Proof.
  apply (wf_projected (<) (length  map_to_list)).
  * by apply map_to_list_length.
  * by apply lt_wf.
Qed.

635
(** ** Properties of the [map_Forall] predicate *)
636
Section map_Forall.
637 638
Context {A} (P : K  A  Prop).

639
Lemma map_Forall_to_list m : map_Forall P m  Forall (curry P) (map_to_list m).
640 641
Proof.
  rewrite Forall_forall. split.
642 643
  * 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)).
644
Qed.
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
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.
674 675

Context `{ i x, Decision (P i x)}.
676
Global Instance map_Forall_dec m : Decision (map_Forall P m).
677 678
Proof.
  refine (cast_if (decide (Forall (curry P) (map_to_list m))));
679
    by rewrite map_Forall_to_list.
680
Defined.
681 682
Lemma map_not_Forall (m : M A) :
  ¬map_Forall P m   i x, m !! i = Some x  ¬P i x.
683
Proof.
684 685 686 687
  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.
  destruct Hm as ([i x]&?&?). exists i x. by rewrite <-elem_of_map_to_list.
688
Qed.
689
End map_Forall.
690 691 692 693

(** ** Properties of the [merge] operation *)
Section merge.
Context {A} (f : option A  option A  option A).
694
Context `{!PropHolds (f None None = None)}.
695 696 697
Global Instance: LeftId (=) None f  LeftId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
698
  by rewrite !(lookup_merge f), lookup_empty, (left_id_L None f).
699 700 701 702
Qed.
Global Instance: RightId (=) None f  RightId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
703
  by rewrite !(lookup_merge f), lookup_empty, (right_id_L None f).
704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719
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.
720
  intros ????. apply merge_associative. intros. by apply (associative_L f).
721 722
Qed.
Lemma merge_idempotent m1 :
723
  ( i, f (m1 !! i) (m1 !! i) = m1 !! i)  merge f m1 m1 = m1.
724 725
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Idempotent (=) f  Idempotent (=) (merge f).
726
Proof. intros ??. apply merge_idempotent. intros. by apply (idempotent f). Qed.
727
End merge.
728

729 730 731 732 733 734 735 736 737 738 739 740
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 :
741 742 743 744 745 746 747 748
  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.
749
Lemma partial_alter_merge_l g g1 m1 m2 i :
750 751 752 753 754 755 756
  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.
757
Lemma partial_alter_merge_r g g2 m1 m2 i :
758 759 760 761 762 763 764
  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.
765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783
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 :
  f (Some y) (Some z) = Some x  merge f {[i,y]} {[i,z]} = {[i,x]}.
Proof.
  intros. unfold singleton, map_singleton; simpl.
  by erewrite <-insert_merge, merge_empty by eauto.
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.
784

785 786 787 788 789 790 791 792 793 794 795 796 797
(** ** Properties on the [map_Forall2] 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_Forall2_alt (m1 : M A) (m2 : M B) :
798
  map_Forall2 R P Q m1 m2  map_Forall (λ _, Is_true) (merge f m1 m2).
799 800
Proof.
  split.
801 802
  * intros Hm i P'; rewrite lookup_merge by done; intros.
    specialize (Hm i). destruct (m1 !! i), (m2 !! i);
803
      simplify_equality'; auto using bool_decide_pack.
804 805 806 807 808 809 810
  * 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_Forall2_dec `{ x y, Decision (R x y),  x, Decision (P x),
   y, Decision (Q y)} m1 m2 : Decision (map_Forall2 R P Q m1 m2).
Proof.
811
  refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2))));
812 813
    abstract by rewrite map_Forall2_alt.
Defined.