fin_maps.v 68.5 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
Require Export prelude.relations prelude.vector prelude.orders.
9

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
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
(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
79
Definition map_Forall `{Lookup K A M} (P : K  A  Prop) : M  Prop :=
Robbert Krebbers's avatar
Robbert Krebbers committed
80
  λ m,  i x, m !! i = Some x  P i x.
Robbert Krebbers's avatar
Robbert Krebbers committed
81 82 83
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).
84
Definition map_included `{ A, Lookup K A (M A)} {A}
Robbert Krebbers's avatar
Robbert Krebbers committed
85 86 87 88 89 90 91
  (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 (_  _) => symmetry; eassumption.
Notation "( m ⊥ₘ.)" := (map_disjoint m) (only parsing) : C_scope.
Notation "(.⊥ₘ m )" := (λ m2, m2  m) (only parsing) : C_scope.
92
Instance map_subseteq `{ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
93
  map_included (=).
Robbert Krebbers's avatar
Robbert Krebbers committed
94 95 96 97 98

(** 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. *)
99
Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x).
100 101 102
Instance map_intersection `{Merge M} {A} : Intersection (M A) :=
  intersection_with (λ x _, Some x).

103 104
(** The difference operation removes all values from the first map whose
index contains a value in the second map as well. *)
105
Instance map_difference `{Merge M} {A} : Difference (M A) :=
106
  difference_with (λ _ _, None).
Robbert Krebbers's avatar
Robbert Krebbers committed
107

108 109 110 111 112 113
(** 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)).

114 115 116 117
(** * Theorems *)
Section theorems.
Context `{FinMap K M}.

Robbert Krebbers's avatar
Robbert Krebbers committed
118 119
(** ** 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
133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152
  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.
  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 
153
    (() ==> () ==> ())%signature (merge (M:=M) f) (merge g).
Robbert Krebbers's avatar
Robbert Krebbers committed
154 155 156 157
  Proof.
    by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
  Qed.
  Global Instance union_with_proper :
158
    Proper ((() ==> () ==> ()) ==> () ==> () ==>()) (union_with (M:=M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
159 160 161 162 163 164 165 166 167 168
  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.
    intros m1 m2; split.
    * by intros Hm; apply map_eq; intros i; unfold_leibniz; apply lookup_proper.
    * by intros <-; intros i; fold_leibniz.
  Qed.
169 170 171 172 173 174 175 176 177 178
  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
179 180 181
End setoid.

(** ** General properties *)
182 183 184 185 186
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
187
  unfold subseteq, map_subseteq, map_relation. split; intros Hm i;
188 189
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
190
Global Instance: EmptySpec (M A).
191
Proof.
192 193
  intros A m. rewrite !map_subseteq_spec.
  intros i x. by rewrite lookup_empty.
194
Qed.
195 196
Global Instance:  {A} (R : relation A), PreOrder R  PreOrder (map_included R).
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
197
  split; [intros m i; by destruct (m !! i); simpl|].
198
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
Robbert Krebbers's avatar
Robbert Krebbers committed
199 200
  destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_equality';
    done || etransitivity; eauto.
201
Qed.
202
Global Instance: PartialOrder (() : relation (M A)).
203
Proof.
204 205 206
  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
207 208 209
Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
210
Proof. rewrite !map_subseteq_spec. auto. Qed.
211 212 213 214 215 216
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.
217 218
  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
219 220
Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
221 222
  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
223 224 225 226 227 228 229 230 231
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  .
232 233 234
Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
235 236

(** ** Properties of the [partial_alter] operation *)
237 238 239
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.
240 241
  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
242 243
Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
244 245
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
246 247
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
248
Qed.
249
Lemma partial_alter_commute {A} f g (m : M A) i j :
250
  i  j  partial_alter f i (partial_alter g j m) =
251 252
    partial_alter g j (partial_alter f i m).
Proof.
253 254 255 256 257 258 259
  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.
260 261 262 263
Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
264 265
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
266
Qed.
267
Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m.
268
Proof. by apply partial_alter_self_alt. Qed.
269
Lemma partial_alter_subseteq {A} f (m : M A) i :
270
  m !! i = None  m  partial_alter f i m.
271 272 273 274
Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
275
Lemma partial_alter_subset {A} f (m : M A) i :
276
  m !! i = None  is_Some (f (m !! i))  m  partial_alter f i m.
277
Proof.
278 279 280 281
  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.
282 283 284
Qed.

(** ** Properties of the [alter] operation *)
285 286
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.
287
Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal'; auto. Qed.
288
Lemma lookup_alter {A} (f : A  A) m i : alter f i m !! i = f <$> m !! i.
289
Proof. unfold alter. apply lookup_partial_alter. Qed.
290
Lemma lookup_alter_ne {A} (f : A  A) m i j : i  j  alter f i m !! j = m !! j.
291
Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
292 293 294 295 296 297 298 299 300
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.
301 302 303 304
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.
305
  destruct (decide (i = j)) as [->|?].
306 307 308 309 310 311
  * 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.
312 313
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
314
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
315 316
Lemma alter_id {A} (f : A  A) m i :
  ( x, m !! i = Some x  f x = x)  alter f i m = m.
317
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
318 319 320
  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.
321 322 323 324 325 326 327 328 329 330 331
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.
332
  * destruct (decide (i = j)) as [->|?];
333 334 335 336 337 338
      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.
339 340
  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
341 342 343
Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
344
Lemma delete_singleton {A} i (x : A) : delete i {[i, x]} = .
345 346 347 348 349 350 351
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.
352
Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
353
Proof.
354 355
  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372
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.
373
Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
374 375 376
Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
377
Lemma delete_subseteq_compat {A} (m1 m2 : M A) i :
378
  m1  m2  delete i m1  delete i m2.
379 380 381 382
Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
Qed.
383
Lemma delete_subset_alt {A} (m : M A) i x : m !! i = Some x  delete i m  m.
384
Proof.
385 386 387
  split; [apply delete_subseteq|].
  rewrite !map_subseteq_spec. intros Hi. apply (None_ne_Some x).
  by rewrite <-(lookup_delete m i), (Hi i x).
388
Qed.
389
Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
390 391 392 393 394
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.
395
Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
396
Proof. rewrite lookup_insert. congruence. Qed.
397
Lemma lookup_insert_ne {A} (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
398 399 400 401 402 403 404 405
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.
406
  * destruct (decide (i = j)) as [->|?];
407
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
408
  * intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
409 410 411 412
Qed.
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
413 414 415
  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
416
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
417
Lemma insert_id {A} (m : M A) i x : m !! i = Some x  <[i:=x]>m = m.
418 419 420 421 422 423 424 425
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 [->|].
Robbert Krebbers's avatar
Robbert Krebbers committed
426 427
  * rewrite lookup_insert. destruct (m !! j); simpl; eauto.
  * rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
428
Qed.
429
Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
430
Proof. apply partial_alter_subseteq. Qed.
431
Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
432 433
Proof. intro. apply partial_alter_subset; eauto. Qed.
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
434
  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
435
Proof.
436 437 438
  rewrite !map_subseteq_spec. intros ?? j ?.
  destruct (decide (j = i)) as [->|?]; [congruence|].
  rewrite lookup_insert_ne; auto.
439 440
Qed.
Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
441
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
442
Proof.
443 444 445 446
  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.
447 448
Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
449
  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
450
Proof.
451 452
  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
453
  * rewrite lookup_insert. congruence.
454
  * rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
455 456
Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
457
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
458
Proof.
459 460 461
  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
462 463
Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
464
  m1 !! i = None  <[i:=x]> m1  m2 
465 466 467
   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
  intros Hi Hm1m2. exists (delete i m2). split_ands.
468
  * rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto.
469 470 471 472
    by rewrite lookup_insert.
  * eauto using insert_delete_subset.
  * by rewrite lookup_delete.
Qed.
473 474 475 476 477 478 479
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.
480 481
Lemma insert_empty {A} i (x : A) : <[i:=x]> = {[i,x]}.
Proof. done. Qed.
482 483 484

(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
485
  {[i, x]} !! j = Some y  i = j  x = y.
486 487
Proof.
  unfold singleton, map_singleton.
488
  rewrite lookup_insert_Some, lookup_empty. simpl. intuition congruence.
489
Qed.
490
Lemma lookup_singleton_None {A} i j (x : A) : {[i, x]} !! j = None  i  j.
491 492 493 494
Proof.
  unfold singleton, map_singleton.
  rewrite lookup_insert_None, lookup_empty. simpl. tauto.
Qed.
495
Lemma lookup_singleton {A} i (x : A) : {[i, x]} !! i = Some x.
496
Proof. by rewrite lookup_singleton_Some. Qed.
497
Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[i, x]} !! j = None.
498
Proof. by rewrite lookup_singleton_None. Qed.
499
Lemma map_non_empty_singleton {A} i (x : A) : {[i,x]}  .
500 501 502 503
Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
504
Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i, x]} = {[i, y]}.
505 506 507 508
Proof.
  unfold singleton, map_singleton, insert, map_insert.
  by rewrite <-partial_alter_compose.
Qed.
509
Lemma alter_singleton {A} (f : A  A) i x : alter f i {[i,x]} = {[i, f x]}.
510
Proof.
511
  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
512 513 514 515
  * 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 :
516
  i  j  alter f i {[j,x]} = {[j,x]}.
517
Proof.
518 519
  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
520 521
Qed.

522 523 524 525 526
(** ** Properties of the map operations *)
Lemma fmap_empty {A B} (f : A  B) : f <$>  = .
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A  option B) : omap f  = .
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
527 528 529 530 531 532 533
Lemma omap_singleton {A B} (f : A  option B) i x y :
  f x = Some y  omap f {[ i,x ]} = {[ i,y ]}.
Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|].
  * by rewrite lookup_omap, !lookup_singleton.
  * by rewrite lookup_omap, !lookup_singleton_ne.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
534 535 536 537 538
Lemma map_fmap_id {A} (m : M A) : id <$> 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 <$> m.
Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed.
539 540 541 542 543 544
Lemma map_fmap_setoid_ext `{Equiv A, Equiv B} (f1 f2 : A  B) m :
  ( i x, m !! i = Some x  f1 x  f2 x)  f1 <$> m  f2 <$> m.
Proof.
  intros Hi i; rewrite !lookup_fmap.
  destruct (m !! i) eqn:?; constructor; eauto.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
545 546 547 548 549 550
Lemma map_fmap_ext {A B} (f1 f2 : A  B) m :
  ( i x, m !! i = Some x  f1 x = f2 x)  f1 <$> 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.
551

552 553
(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
554
  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
555
Proof. rewrite !elem_of_map_to_list. congruence. Qed.
556
Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
557
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
558 559 560 561 562 563 564 565 566 567
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.
568
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
569
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
570
Proof.
571 572
  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'].
573
  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
574
    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
575 576
Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
577
  map_of_list l !! i = Some x  (i,x)  l.
578
Proof.
579 580 581
  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.
582 583
Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
584
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
585
Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
586
Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
587
  i  l.*1  map_of_list l !! i = None.
588
Proof.
589 590
  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.
591 592
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
593
  map_of_list l !! i = None  i  l.*1.
594
Proof.
595
  induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
596 597 598 599 600
  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 :
601
  i  l.*1  map_of_list l !! i = None.
602
Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
603
Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
604
  NoDup (l1.*1)  l1  l2  map_of_list l1 = map_of_list l2.
605 606 607 608 609
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)) :
610
  NoDup (l1.*1)  NoDup (l2.*1)  map_of_list l1 = map_of_list l2  l1  l2.
611
Proof.
612
  intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
613 614
  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
615
Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m.
616 617 618
Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
619
    by auto using NoDup_fst_map_to_list.
620 621
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
622
  NoDup (l.*1)  map_to_list (map_of_list l)  l.
623
Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
624
Lemma map_to_list_inj {A} (m1 m2 : M A) :
625
  map_to_list m1  map_to_list m2  m1 = m2.
626
Proof.
627
  intros. rewrite <-(map_of_to_list m1), <-(map_of_to_list m2).
628
  auto using map_of_list_proper, NoDup_fst_map_to_list.
629
Qed.
630 631 632 633 634 635
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.
636
Lemma map_to_list_empty {A} : map_to_list  = @nil (K * A).
637 638 639 640 641
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 :
642
  m !! i = None  map_to_list (<[i:=x]>m)  (i,x) :: map_to_list m.
643
Proof.
644
  intros. apply map_of_list_inj; csimpl.
645 646
  * apply NoDup_fst_map_to_list.
  * constructor; auto using NoDup_fst_map_to_list.
647
    rewrite elem_of_list_fmap. intros [[??] [? Hlookup]]; subst; simpl in *.
648 649 650
    rewrite elem_of_map_to_list in Hlookup. congruence.
  * by rewrite !map_of_to_list.
Qed.
651
Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = .
652 653 654 655
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.
656
Lemma map_to_list_empty_inv_alt {A}  (m : M A) : map_to_list m  []  m = .
657
Proof. rewrite <-map_to_list_empty. apply map_to_list_inj. Qed.
658
Lemma map_to_list_empty_inv {A} (m : M A) : map_to_list m = []  m = .
659 660
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 :
661
  map_to_list m  (i,x) :: l  m = <[i:=x]>(map_of_list l).
662 663
Proof.
  intros Hperm. apply map_to_list_inj.
664 665 666
  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. }
667 668 669
  rewrite Hperm, map_to_list_insert, map_to_of_list;
    auto using not_elem_of_map_of_list_1.
Qed.
670 671 672 673
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. }
674
  exists i, x. rewrite <-elem_of_map_to_list, Hm. by left.
675
Qed.
676