fin_maps.v 92.4 KB
Newer Older
1
(* Copyright (c) 2012-2019, Coq-std++ developers. *)
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
induction principles for finite maps and implements the tactic
6
[simplify_map_eq] to simplify goals involving finite maps. *)
7
From Coq Require Import Permutation.
8
From stdpp Require Export relations orders vector fin_sets.
9
10
(* FIXME: This file needs a 'Proof Using' hint, but the default we use
   everywhere makes for lots of extra ssumptions. *)
11

12
13
(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
14
15
16
17
18
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. *)
19

Robbert Krebbers's avatar
Robbert Krebbers committed
20
21
(** Finiteness is axiomatized by requiring that each map can be translated
to an association list. The translation to association lists is used to
22
prove well founded recursion on finite maps. *)
23

24
25
26
(** 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]. *)
27

28
Class FinMapToList K A M := map_to_list: M  list (K * A).
29
30
Hint Mode FinMapToList ! - - : typeclass_instances.
Hint Mode FinMapToList - - ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
31

32
33
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),
34
    EqDecision K} := {
35
36
  map_eq {A} (m1 m2 : M A) : ( i, m1 !! i = m2 !! i)  m1 = m2;
  lookup_empty {A} i : ( : M A) !! i = None;
37
38
39
40
  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;
41
  lookup_fmap {A B} (f : A  B) (m : M A) i : (f <$> m) !! i = f <$> m !! i;
42
  NoDup_map_to_list {A} (m : M A) : NoDup (map_to_list m);
43
44
  elem_of_map_to_list {A} (m : M A) i x :
    (i,x)  map_to_list m  m !! i = Some x;
45
  lookup_omap {A B} (f : A  option B) m i : omap f m !! i = m !! i = f;
46
  lookup_merge {A B C} (f: option A  option B  option C) `{!DiagNone f} m1 m2 i :
47
    merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
Robbert Krebbers's avatar
Robbert Krebbers committed
48
49
}.

50
51
52
(** * 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
53
54
significant performance loss, which justifies including them in the finite map
interface as primitive operations. *)
55
56
57
58
59
60
61
62
63
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} :
  SingletonM K A M := λ i x, <[i:=x]> .

64
Definition list_to_map `{Insert K A M, Empty M} : list (K * A)  M :=
65
  fold_right (λ p, <[p.1:=p.2]>) .
66

67
68
Instance map_size `{FinMapToList K A M} : Size M := λ m, length (map_to_list m).

69
Definition map_to_set `{FinMapToList K A M,
70
    Singleton B C, Empty C, Union C} (f : K  A  B) (m : M) : C :=
71
72
  list_to_set (curry f <$> map_to_list m).
Definition set_to_map `{Elements B C, Insert K A M, Empty M}
73
    (f : B  K * A) (X : C) : M :=
74
  list_to_map (f <$> elements X).
Robbert Krebbers's avatar
Robbert Krebbers committed
75

76
77
78
79
80
81
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
82

83
84
85
(** Higher precedence to make sure it's not used for other types with a [Lookup]
instance, such as lists. *)
Instance map_equiv `{ A, Lookup K A (M A), Equiv A} : Equiv (M A) | 20 :=
86
  λ m1 m2,  i, m1 !! i  m2 !! i.
Robbert Krebbers's avatar
Robbert Krebbers committed
87

88
Definition map_Forall `{Lookup K A M} (P : K  A  Prop) : M  Prop :=
Robbert Krebbers's avatar
Robbert Krebbers committed
89
  λ m,  i x, m !! i = Some x  P i x.
Ralf Jung's avatar
Ralf Jung committed
90

91
Definition map_relation `{ A, Lookup K A (M A)} {A B} (R : A  B  Prop)
Robbert Krebbers's avatar
Robbert Krebbers committed
92
93
    (P : A  Prop) (Q : B  Prop) (m1 : M A) (m2 : M B) : Prop :=  i,
  option_relation R P Q (m1 !! i) (m2 !! i).
94
Definition map_included `{ A, Lookup K A (M A)} {A}
Robbert Krebbers's avatar
Robbert Krebbers committed
95
  (R : relation A) : relation (M A) := map_relation R (λ _, False) (λ _, True).
96
Definition map_disjoint `{ A, Lookup K A (M A)} {A} : relation (M A) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
97
  map_relation (λ _ _, False) (λ _, True) (λ _, True).
98
Infix "##ₘ" := map_disjoint (at level 70) : stdpp_scope.
Tej Chajed's avatar
Tej Chajed committed
99
Hint Extern 0 (_ ## _) => symmetry; eassumption : core.
100
101
Notation "( m ##ₘ.)" := (map_disjoint m) (only parsing) : stdpp_scope.
Notation "(.##ₘ m )" := (λ m2, m2 ## m) (only parsing) : stdpp_scope.
102
Instance map_subseteq `{ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
103
  map_included (=).
Robbert Krebbers's avatar
Robbert Krebbers committed
104
105
106
107
108

(** 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. *)
109
Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x).
110
111
112
Instance map_intersection `{Merge M} {A} : Intersection (M A) :=
  intersection_with (λ x _, Some x).

113
114
(** The difference operation removes all values from the first map whose
index contains a value in the second map as well. *)
115
Instance map_difference `{Merge M} {A} : Difference (M A) :=
116
  difference_with (λ _ _, None).
Robbert Krebbers's avatar
Robbert Krebbers committed
117

118
119
(** 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. *)
120
121
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 :=
122
  list_to_map (omap (λ ix, (fst ix,) <$> curry f ix) (map_to_list m)).
123

124
125
126
127
128
129
130
(* The zip operation on maps combines two maps key-wise. The keys of resulting
map correspond to the keys that are in both maps. *)
Definition map_zip_with `{Merge M} {A B C} (f : A  B  C) : M A  M B  M C :=
  merge (λ mx my,
    match mx, my with Some x, Some y => Some (f x y) | _, _ => None end).
Notation map_zip := (map_zip_with pair).

131
132
133
134
135
(* Folds a function [f] over a map. The order in which the function is called
is unspecified. *)
Definition map_fold `{FinMapToList K A M} {B}
  (f : K  A  B  B) (b : B) : M  B := foldr (curry f) b  map_to_list.

136
Instance map_filter `{FinMapToList K A M, Insert K A M, Empty M} : Filter (K * A) M :=
137
138
  λ P _, map_fold (λ k v m, if decide (P (k,v)) then <[k := v]>m else m) .

139
140
141
142
143
144
Fixpoint map_seq `{Insert nat A M, Empty M} (start : nat) (xs : list A) : M :=
  match xs with
  | [] => 
  | x :: xs => <[start:=x]> (map_seq (S start) xs)
  end.

145
146
147
148
(** * Theorems *)
Section theorems.
Context `{FinMap K M}.

Robbert Krebbers's avatar
Robbert Krebbers committed
149
150
(** ** Setoids *)
Section setoid.
151
  Context `{Equiv A}.
152

153
154
155
156
  Lemma map_equiv_lookup_l (m1 m2 : M A) i x :
    m1  m2  m1 !! i = Some x   y, m2 !! i = Some y  x  y.
  Proof. generalize (equiv_Some_inv_l (m1 !! i) (m2 !! i) x); naive_solver. Qed.

157
  Global Instance map_equivalence : Equivalence (@{A})  Equivalence (@{M A}).
Robbert Krebbers's avatar
Robbert Krebbers committed
158
159
  Proof.
    split.
160
161
    - by intros m i.
    - by intros m1 m2 ? i.
162
    - by intros m1 m2 m3 ?? i; trans (m2 !! i).
Robbert Krebbers's avatar
Robbert Krebbers committed
163
  Qed.
164
  Global Instance lookup_proper (i : K) : Proper ((@{M A}) ==> ()) (lookup i).
Robbert Krebbers's avatar
Robbert Krebbers committed
165
166
  Proof. by intros m1 m2 Hm. Qed.
  Global Instance partial_alter_proper :
167
    Proper ((() ==> ()) ==> (=) ==> () ==> (@{M A})) partial_alter.
Robbert Krebbers's avatar
Robbert Krebbers committed
168
169
170
171
172
173
  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) :
174
    Proper (() ==> () ==> (@{M A})) (insert i).
Robbert Krebbers's avatar
Robbert Krebbers committed
175
  Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
176
  Global Instance singleton_proper k : Proper (() ==> (@{M A})) (singletonM k).
177
178
179
180
  Proof.
    intros ???; apply insert_proper; [done|].
    intros ?. rewrite lookup_empty; constructor.
  Qed.
181
  Global Instance delete_proper (i : K) : Proper (() ==> (@{M A})) (delete i).
Robbert Krebbers's avatar
Robbert Krebbers committed
182
183
  Proof. by apply partial_alter_proper; [constructor|]. Qed.
  Global Instance alter_proper :
184
    Proper ((() ==> ()) ==> (=) ==> () ==> (@{M A})) alter.
Robbert Krebbers's avatar
Robbert Krebbers committed
185
186
187
188
  Proof.
    intros ?? Hf; apply partial_alter_proper.
    by destruct 1; constructor; apply Hf.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
189
190
  Lemma merge_ext `{Equiv B, Equiv C} (f g : option A  option B  option C)
      `{!DiagNone f, !DiagNone g} :
Robbert Krebbers's avatar
Robbert Krebbers committed
191
    (() ==> () ==> ())%signature f g 
192
    (() ==> () ==> (@{M _}))%signature (merge f) (merge g).
Robbert Krebbers's avatar
Robbert Krebbers committed
193
194
195
196
  Proof.
    by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
  Qed.
  Global Instance union_with_proper :
197
    Proper ((() ==> () ==> ()) ==> () ==> () ==>(@{M A})) union_with.
Robbert Krebbers's avatar
Robbert Krebbers committed
198
199
200
  Proof.
    intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
    by do 2 destruct 1; first [apply Hf | constructor].
201
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
202
  Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
203
  Proof. intros m1 m2 Hm; apply map_eq; intros i. apply leibniz_equiv, Hm. Qed.
204
205
  Lemma map_equiv_empty (m : M A) : m    m = .
  Proof.
206
207
208
    split; [intros Hm; apply map_eq; intros i|intros ->].
    - generalize (Hm i). by rewrite lookup_empty, equiv_None.
    - intros ?. rewrite lookup_empty; constructor.
209
  Qed.
210
  Global Instance map_fmap_proper `{Equiv B} (f : A  B) :
211
    Proper (() ==> ()) f  Proper (() ==> (@{M _})) (fmap f).
212
213
214
  Proof.
    intros ? m m' ? k; rewrite !lookup_fmap. by apply option_fmap_proper.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
215
216
217
End setoid.

(** ** General properties *)
218
219
220
221
222
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
223
  unfold subseteq, map_subseteq, map_relation. split; intros Hm i;
224
225
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
226
Global Instance map_included_preorder {A} (R : relation A) :
227
  PreOrder R  PreOrder (map_included R : relation (M A)).
228
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
229
  split; [intros m i; by destruct (m !! i); simpl|].
230
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
231
  destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_eq/=;
232
    done || etrans; eauto.
233
Qed.
234
Global Instance map_subseteq_po : PartialOrder (@{M A}).
235
Proof.
236
237
238
  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
239
240
241
Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
242
Proof. rewrite !map_subseteq_spec. auto. Qed.
243
244
245
246
247
248
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.
249
250
  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
251
252
Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
253
254
  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
255
256
257
258
259
260
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.
261
Lemma lookup_empty_Some {A} i (x : A) : ¬( : M A) !! i = Some x.
262
263
Proof. by rewrite lookup_empty. Qed.
Lemma map_subset_empty {A} (m : M A) : m  .
264
265
266
Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
267
268
Lemma map_fmap_empty {A B} (f : A  B) : f <$> ( : M A) = .
Proof. by apply map_eq; intros i; rewrite lookup_fmap, !lookup_empty. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
269
270
271
272
273
Lemma map_fmap_empty_inv {A B} (f : A  B) m : f <$> m =   m = .
Proof.
  intros Hm. apply map_eq; intros i. generalize (f_equal (lookup i) Hm).
  by rewrite lookup_fmap, !lookup_empty, fmap_None.
Qed.
274

275
276
277
278
279
Lemma map_subset_alt {A} (m1 m2 : M A) :
  m1  m2  m1  m2   i, m1 !! i = None  is_Some (m2 !! i).
Proof.
  rewrite strict_spec_alt. split.
  - intros [? Heq]; split; [done|].
Robbert Krebbers's avatar
Robbert Krebbers committed
280
    destruct (decide (Exists (λ ix, m1 !! ix.1 = None) (map_to_list m2)))
281
282
283
284
285
286
287
288
289
290
      as [[[i x] [?%elem_of_map_to_list ?]]%Exists_exists
         |Hm%(not_Exists_Forall _)]; [eauto|].
    destruct Heq; apply (anti_symm _), map_subseteq_spec; [done|intros i x Hi].
    assert (is_Some (m1 !! i)) as [x' ?].
    { by apply not_eq_None_Some,
        (proj1 (Forall_forall _ _) Hm (i,x)), elem_of_map_to_list. }
    by rewrite <-(lookup_weaken_inv m1 m2 i x' x).
  - intros [? (i&?&x&?)]; split; [done|]. congruence.
Qed.

291
(** ** Properties of the [partial_alter] operation *)
292
293
294
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.
295
296
  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
297
298
Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
299
300
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
301
302
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
303
Qed.
304
Lemma partial_alter_commute {A} f g (m : M A) i j :
305
  i  j  partial_alter f i (partial_alter g j m) =
306
307
    partial_alter g j (partial_alter f i m).
Proof.
308
309
310
311
  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 [->|?].
312
  - by rewrite lookup_partial_alter,
313
     !lookup_partial_alter_ne, lookup_partial_alter by congruence.
314
  - by rewrite !lookup_partial_alter_ne by congruence.
315
316
317
318
Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
319
320
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
321
Qed.
322
Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m.
323
Proof. by apply partial_alter_self_alt. Qed.
324
Lemma partial_alter_subseteq {A} f (m : M A) i :
325
  m !! i = None  m  partial_alter f i m.
326
327
328
329
Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
330
Lemma partial_alter_subset {A} f (m : M A) i :
331
  m !! i = None  is_Some (f (m !! i))  m  partial_alter f i m.
332
Proof.
333
334
  intros Hi Hfi. apply map_subset_alt; split; [by apply partial_alter_subseteq|].
  exists i. by rewrite lookup_partial_alter.
335
336
337
Qed.

(** ** Properties of the [alter] operation *)
338
Lemma lookup_alter {A} (f : A  A) (m : M A) i : alter f i m !! i = f <$> m !! i.
339
Proof. unfold alter. apply lookup_partial_alter. Qed.
340
341
Lemma lookup_alter_ne {A} (f : A  A) (m : M A) i j :
  i  j  alter f i m !! j = m !! j.
342
Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
343
344
345
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.
346
347
348
349
350
351
352
353
354
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.
355
Lemma lookup_alter_Some {A} (f : A  A) (m : M A) i j y :
356
357
358
  alter f i m !! j = Some y 
    (i = j   x, m !! j = Some x  y = f x)  (i  j  m !! j = Some y).
Proof.
359
  destruct (decide (i = j)) as [->|?].
360
  - rewrite lookup_alter. naive_solver (simplify_option_eq; eauto).
361
  - rewrite lookup_alter_ne by done. naive_solver.
362
Qed.
363
Lemma lookup_alter_None {A} (f : A  A) (m : M A) i j :
364
365
  alter f i m !! j = None  m !! j = None.
Proof.
366
367
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
368
Qed.
369
Lemma lookup_alter_is_Some {A} (f : A  A) (m : M A) i j :
370
371
  is_Some (alter f i m !! j)  is_Some (m !! j).
Proof. by rewrite <-!not_eq_None_Some, lookup_alter_None. Qed.
372
Lemma alter_id {A} (f : A  A) (m : M A) i :
Robbert Krebbers's avatar
Robbert Krebbers committed
373
  ( x, m !! i = Some x  f x = x)  alter f i m = m.
374
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
375
  intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
376
  { rewrite lookup_alter; destruct (m !! j); f_equal/=; auto. }
Robbert Krebbers's avatar
Robbert Krebbers committed
377
  by rewrite lookup_alter_ne by done.
378
Qed.
379
380
381
382
383
384
385
386
387
388
389
390
Lemma alter_mono {A} f (m1 m2 : M A) i : m1  m2  alter f i m1  alter f i m2.
Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_alter_Some. naive_solver.
Qed.
Lemma alter_strict_mono {A} f (m1 m2 : M A) i :
  m1  m2  alter f i m1  alter f i m2.
Proof.
  rewrite !map_subset_alt.
  intros [? (j&?&?)]; split; auto using alter_mono.
  exists j. by rewrite lookup_alter_None, lookup_alter_is_Some.
Qed.
391
392
393
394
395
396
397
398
399
400

(** ** 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.
401
  - destruct (decide (i = j)) as [->|?];
402
      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
403
  - intros [??]. by rewrite lookup_delete_ne.
404
Qed.
405
406
407
Lemma lookup_delete_is_Some {A} (m : M A) i j :
  is_Some (delete i m !! j)  i  j  is_Some (m !! j).
Proof. unfold is_Some; setoid_rewrite lookup_delete_Some; naive_solver. Qed.
408
409
410
Lemma lookup_delete_None {A} (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
411
412
  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
413
414
415
416
417
418
419
420
421
Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_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.
422
Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
423
Proof.
424
425
  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
426
Qed.
427
428
429
Lemma delete_idemp {A} (m : M A) i :
  delete i (delete i m) = delete i m.
Proof. by setoid_rewrite <-partial_alter_compose. Qed.
430
431
432
433
434
435
436
437
438
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.
439
440
441
Lemma delete_insert_delete {A} (m : M A) i x :
  delete i (<[i:=x]>m) = delete i m.
Proof. by setoid_rewrite <-partial_alter_compose. Qed.
442
443
Lemma insert_delete {A} (m : M A) i x : <[i:=x]>(delete i m) = <[i:=x]> m.
Proof. symmetry; apply (partial_alter_compose (λ _, Some x)). Qed.
444
Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
445
446
447
Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
448
Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
449
Proof.
450
451
  intros [x ?]; apply map_subset_alt; split; [apply delete_subseteq|].
  exists i. rewrite lookup_delete; eauto.
452
Qed.
453
Lemma delete_mono {A} (m1 m2 : M A) i : m1  m2  delete i m1  delete i m2.
454
Proof.
455
456
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
457
458
459
460
461
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.
462
Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
463
Proof. rewrite lookup_insert. congruence. Qed.
464
Lemma lookup_insert_ne {A} (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
465
Proof. unfold insert. apply lookup_partial_alter_ne. Qed.
466
467
Lemma insert_insert {A} (m : M A) i x y : <[i:=x]>(<[i:=y]>m) = <[i:=x]>m.
Proof. unfold insert, map_insert. by rewrite <-partial_alter_compose. Qed.
468
469
470
471
472
473
474
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.
475
  - destruct (decide (i = j)) as [->|?];
476
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
477
  - intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
478
Qed.
479
480
481
Lemma lookup_insert_is_Some {A} (m : M A) i j x :
  is_Some (<[i:=x]>m !! j)  i = j  i  j  is_Some (m !! j).
Proof. unfold is_Some; setoid_rewrite lookup_insert_Some; naive_solver. Qed.
482
483
484
Lemma lookup_insert_is_Some' {A} (m : M A) i j x :
  is_Some (<[i:=x]>m !! j)  i = j  is_Some (m !! j).
Proof. rewrite lookup_insert_is_Some. destruct (decide (i=j)); naive_solver. Qed.
485
486
487
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
488
489
490
  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
491
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
492
Lemma insert_id {A} (m : M A) i x : m !! i = Some x  <[i:=x]>m = m.
493
494
495
496
497
498
499
500
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 [->|].
501
502
  - rewrite lookup_insert. destruct (m !! j); simpl; eauto.
  - rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
503
Qed.
504
Lemma insert_empty {A} i (x : A) : <[i:=x]>( : M A) = {[i := x]}.
505
506
507
508
509
510
Proof. done. Qed.
Lemma insert_non_empty {A} (m : M A) i x : <[i:=x]>m  .
Proof.
  intros Hi%(f_equal (!! i)). by rewrite lookup_insert, lookup_empty in Hi.
Qed.

511
Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
512
Proof. apply partial_alter_subseteq. Qed.
513
Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
514
Proof. intro. apply partial_alter_subset; eauto. Qed.
515
516
517
518
519
Lemma insert_mono {A} (m1 m2 : M A) i x : m1  m2  <[i:=x]> m1  <[i:=x]>m2.
Proof.
  rewrite !map_subseteq_spec.
  intros Hm j y. rewrite !lookup_insert_Some. naive_solver.
Qed.
520
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
521
  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
522
Proof.
523
  intros. trans (<[i:=x]> m1); eauto using insert_subseteq, insert_mono.
524
Qed.
525

526
Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
527
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
528
Proof.
529
530
531
532
  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.
533
534
Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
535
  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
536
Proof.
537
538
  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
539
540
  - rewrite lookup_insert. congruence.
  - rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
541
542
Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
543
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
544
Proof.
545
546
547
  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
548
549
Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
550
  m1 !! i = None  <[i:=x]> m1  m2 
551
552
   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
553
  intros Hi Hm1m2. exists (delete i m2). split_and?.
554
555
  - rewrite insert_delete, insert_id. done.
    eapply lookup_weaken, strict_include; eauto. by rewrite lookup_insert.
556
557
  - eauto using insert_delete_subset.
  - by rewrite lookup_delete.
558
559
560
561
Qed.

(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
562
  ({[i := x]} : M A) !! j = Some y  i = j  x = y.
563
Proof.
564
  rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence.
565
Qed.
566
567
Lemma lookup_singleton_None {A} i j (x : A) :
  ({[i := x]} : M A) !! j = None  i  j.
568
Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; tauto. Qed.
569
Lemma lookup_singleton {A} i (x : A) : ({[i := x]} : M A) !! i = Some x.
570
Proof. by rewrite lookup_singleton_Some. Qed.
571
572
Lemma lookup_singleton_ne {A} i j (x : A) :
  i  j  ({[i := x]} : M A) !! j = None.
573
Proof. by rewrite lookup_singleton_None. Qed.
574
Lemma map_non_empty_singleton {A} i (x : A) : {[i := x]}  ( : M A).
575
576
577
578
Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
579
Lemma insert_singleton {A} i (x y : A) : <[i:=y]>({[i := x]} : M A) = {[i := y]}.
580
Proof.
581
  unfold singletonM, map_singleton, insert, map_insert.
582
583
  by rewrite <-partial_alter_compose.
Qed.
584
585
Lemma alter_singleton {A} (f : A  A) i x :
  alter f i ({[i := x]} : M A) = {[i := f x]}.
586
Proof.
587
  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
588
589
  - by rewrite lookup_alter, !lookup_singleton.
  - by rewrite lookup_alter_ne, !lookup_singleton_ne.
590
591
Qed.
Lemma alter_singleton_ne {A} (f : A  A) i j x :
592
  i  j  alter f i ({[j := x]} : M A) = {[j := x]}.
593
Proof.
594
595
  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
596
Qed.
597
Lemma singleton_non_empty {A} i (x : A) : {[i:=x]}  ( : M A).
598
Proof. apply insert_non_empty. Qed.
599
Lemma delete_singleton {A} i (x : A) : delete i {[i := x]} = ( : M A).
600
Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
601
Lemma delete_singleton_ne {A} i j (x : A) :
602
  i  j  delete i ({[j := x]} : M A) = {[j := x]}.
603
Proof. intro. apply delete_notin. by apply lookup_singleton_ne. Qed.
604

605
606
607
608
609
(** ** Properties of the map operations *)
Lemma fmap_empty {A B} (f : A  B) : f <$>  = .
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A  option B) : omap f  = .
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
610
611
612
Lemma fmap_insert {A B} (f: A  B) m i x: f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
613
614
  - by rewrite lookup_fmap, !lookup_insert.
  - by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
615
Qed.
616
617
618
619
620
621
Lemma fmap_delete {A B} (f: A  B) m i: f <$> delete i m = delete i (f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  - by rewrite lookup_fmap, !lookup_delete.
  - by rewrite lookup_fmap, !lookup_delete_ne, lookup_fmap by done.
Qed.
622
623
624
625
Lemma omap_insert {A B} (f : A  option B) m i x y :
  f x = Some y  omap f (<[i:=x]>m) = <[i:=y]>(omap f m).
Proof.
  intros; apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
626
627
  - by rewrite lookup_omap, !lookup_insert.
  - by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done.
628
Qed.
629
Lemma map_fmap_singleton {A B} (f : A  B) i x : f <$> {[i := x]} = {[i := f x]}.
630
631
632
Proof.
  by unfold singletonM, map_singleton; rewrite fmap_insert, map_fmap_empty.
Qed.
633
Lemma omap_singleton {A B} (f : A  option B) i x y :
634
  f x = Some y  omap f {[ i := x ]} = {[ i := y ]}.
635
Proof.
636
637
  intros. unfold singletonM, map_singleton.
  by erewrite omap_insert, omap_empty by eauto.
638
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
639
640
641
Lemma map_fmap_id {A} (m : M A) : id <$> m = m.
Proof. apply map_eq; intros i; by rewrite lookup_fmap, option_fmap_id. Qed.
Lemma map_fmap_compose {A B C} (f : A  B) (g : B  C) (m : M A) :
642
  g  f <$> m = g <$> (f <$> m).
Robbert Krebbers's avatar
Robbert Krebbers committed
643
Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed.
644
Lemma map_fmap_equiv_ext `{Equiv A, Equiv B} (f1 f2 : A  B) (m : M A) :
645
646
647
648
649
  ( 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.
650
Lemma map_fmap_ext {A B} (f1 f2 : A  B) (m : M A) :
Robbert Krebbers's avatar
Robbert Krebbers committed
651
652
653
654
655
  ( 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.
656
Lemma omap_ext {A B} (f1 f2 : A  option B) (m : M A) :
Robbert Krebbers's avatar
Robbert Krebbers committed
657
658
659
660
661
  ( i x, m !! i = Some x  f1 x = f2 x)  omap f1 m = omap f2 m.
Proof.
  intros Hi; apply map_eq; intros i; rewrite !lookup_omap.
  by destruct (m !! i) eqn:?; simpl; erewrite ?Hi by eauto.
Qed.
662

663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
Lemma map_fmap_mono {A B} (f : A  B) (m1 m2 : M A) :
  m1  m2  f <$> m1  f <$> m2.
Proof.
  rewrite !map_subseteq_spec; intros Hm i x.
  rewrite !lookup_fmap, !fmap_Some. naive_solver.
Qed.
Lemma map_fmap_strict_mono {A B} (f : A  B) (m1 m2 : M A) :
  m1  m2  f <$> m1  f <$> m2.
Proof.
  rewrite !map_subset_alt.
  intros [? (j&?&?)]; split; auto using map_fmap_mono.
  exists j. by rewrite !lookup_fmap, fmap_None, fmap_is_Some.
Qed.
Lemma map_omap_mono {A B} (f : A  option B) (m1 m2 : M A) :
  m1  m2  omap f m1  omap f m2.
Proof.
  rewrite !map_subseteq_spec; intros Hm i x.
  rewrite !lookup_omap, !bind_Some. naive_solver.
Qed.

683
(** ** Properties of conversion to lists *)
684
685
686
Lemma elem_of_map_to_list' {A} (m : M A) ix :
  ix  map_to_list m  m !! ix.1 = Some (ix.2).
Proof. destruct ix as [i x]. apply elem_of_map_to_list. Qed.
687
Lemma map_to_list_unique {A} (m : M A) i x y :
688
  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
689
Proof. rewrite !elem_of_map_to_list. congruence. Qed.
690
Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
691
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
692
693
Lemma elem_of_list_to_map_1' {A} (l : list (K * A)) i x :
  ( y, (i,y)  l  x = y)  (i,x)  l  (list_to_map l : M A) !! i = Some x.
694
695
696
Proof.
  induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|].
  setoid_rewrite elem_of_cons.
697
  intros Hdup [?|?]; simplify_eq; [by rewrite lookup_insert|].
698
  destruct (decide (i = j)) as [->|].