fin_maps.v 70.2 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
induction principles for finite maps and implements the tactic
6
[simplify_map_eq] to simplify goals involving finite maps. *)
7 8
From Coq Require Import Permutation.
From stdpp Require Export relations vector 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
  lookup_merge {A B C} (f: option A  option B  option C) `{!DiagNone f} m1 m2 i :
43
    merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
Robbert Krebbers's avatar
Robbert Krebbers committed
44 45
}.

46 47 48
(** * 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
49 50
significant performance loss to make including them in the finite map interface
worthwhile. *)
51 52 53 54 55
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 :=
56
  partial_alter (λ _, None).
57
Instance map_singleton `{PartialAlter K A M, Empty M} :
58
  SingletonM K A M := λ i x, <[i:=x]> .
Robbert Krebbers's avatar
Robbert Krebbers committed
59

60
Definition map_of_list `{Insert K A M, Empty M} : list (K * A)  M :=
61
  fold_right (λ p, <[p.1:=p.2]>) .
62 63 64
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
65

66 67 68 69 70 71
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
72

73 74
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
75

76 77
(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
78
Definition map_Forall `{Lookup K A M} (P : K  A  Prop) : M  Prop :=
Robbert Krebbers's avatar
Robbert Krebbers committed
79
  λ m,  i x, m !! i = Some x  P i x.
Robbert Krebbers's avatar
Robbert Krebbers committed
80 81 82
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).
83
Definition map_included `{ A, Lookup K A (M A)} {A}
Robbert Krebbers's avatar
Robbert Krebbers committed
84 85 86 87 88 89 90
  (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.
91
Instance map_subseteq `{ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
92
  map_included (=).
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

107 108 109 110 111 112
(** 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)).

113 114 115 116
(** * Theorems *)
Section theorems.
Context `{FinMap K M}.

Robbert Krebbers's avatar
Robbert Krebbers committed
117 118
(** ** Setoids *)
Section setoid.
119 120
  Context `{Equiv A} `{!Equivalence (() : relation A)}.
  Global Instance map_equivalence : Equivalence (() : relation (M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
121 122
  Proof.
    split.
123 124
    - by intros m i.
    - by intros m1 m2 ? i.
125
    - by intros m1 m2 m3 ?? i; trans (m2 !! i).
Robbert Krebbers's avatar
Robbert Krebbers committed
126 127 128 129 130
  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 :
131
    Proper ((() ==> ()) ==> (=) ==> () ==> ()) (partial_alter (M:=M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
132 133 134 135 136 137 138 139
  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.
140 141 142
  Global Instance singleton_proper k :
    Proper (() ==> ()) (singletonM k : A  M A).
  Proof. by intros ???; apply insert_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
143 144 145 146 147 148 149 150 151
  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.
152
  Lemma merge_ext f g `{!DiagNone f, !DiagNone g} :
Robbert Krebbers's avatar
Robbert Krebbers committed
153
    (() ==> () ==> ())%signature f g 
154
    (() ==> () ==> ())%signature (merge (M:=M) f) (merge g).
Robbert Krebbers's avatar
Robbert Krebbers committed
155 156 157 158
  Proof.
    by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
  Qed.
  Global Instance union_with_proper :
159
    Proper ((() ==> () ==> ()) ==> () ==> () ==>()) (union_with (M:=M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
160 161 162
  Proof.
    intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
    by do 2 destruct 1; first [apply Hf | constructor].
163
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
164 165
  Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
  Proof.
166 167
    intros m1 m2 Hm; apply map_eq; intros i.
    by unfold_leibniz; apply lookup_proper.
Robbert Krebbers's avatar
Robbert Krebbers committed
168
  Qed.
169 170 171 172 173
  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.
174
  Lemma map_equiv_lookup_l (m1 m2 : M A) i x :
175
    m1  m2  m1 !! i = Some x   y, m2 !! i = Some y  x  y.
176
  Proof. generalize (equiv_Some_inv_l (m1 !! i) (m2 !! i) x); naive_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
177 178 179
End setoid.

(** ** General properties *)
180 181 182 183 184
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
185
  unfold subseteq, map_subseteq, map_relation. split; intros Hm i;
186 187
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
188
Global Instance: EmptySpec (M A).
189
Proof.
190 191
  intros A m. rewrite !map_subseteq_spec.
  intros i x. by rewrite lookup_empty.
192
Qed.
193 194
Global Instance:  {A} (R : relation A), PreOrder R  PreOrder (map_included R).
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
195
  split; [intros m i; by destruct (m !! i); simpl|].
196
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
197
  destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_eq/=;
198
    done || etrans; eauto.
199
Qed.
200
Global Instance: PartialOrder (() : relation (M A)).
201
Proof.
202 203 204
  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
205 206 207
Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
208
Proof. rewrite !map_subseteq_spec. auto. Qed.
209 210 211 212 213 214
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.
215 216
  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
217 218
Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
219 220
  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
221 222 223 224 225 226 227 228 229
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  .
230 231 232
Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
233 234
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.
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
  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 [->|?].
257
  - by rewrite lookup_partial_alter,
258
     !lookup_partial_alter_ne, lookup_partial_alter by congruence.
259
  - 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
  - rewrite lookup_alter. naive_solver (simplify_option_eq; eauto).
307
  - rewrite lookup_alter_ne by done. naive_solver.
308 309 310 311
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
  intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
319
  { rewrite lookup_alter; destruct (m !! j); f_equal/=; auto. }
Robbert Krebbers's avatar
Robbert Krebbers committed
320
  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
      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
334
  - intros [??]. by rewrite lookup_delete_ne.
335
Qed.
336 337 338
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.
339 340 341
Lemma lookup_delete_None {A} (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
342 343
  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
344 345 346
Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
347
Lemma delete_singleton {A} i (x : A) : delete i {[i := x]} = .
348 349 350 351 352 353 354
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.
355
Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
356
Proof.
357 358
  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
359 360 361 362 363 364 365 366 367 368
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.
369 370
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.
371
Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
372 373 374
Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
375
Lemma delete_subseteq_compat {A} (m1 m2 : M A) i :
376
  m1  m2  delete i m1  delete i m2.
377 378 379 380
Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
Qed.
381
Lemma delete_subset_alt {A} (m : M A) i x : m !! i = Some x  delete i m  m.
382
Proof.
383 384 385
  split; [apply delete_subseteq|].
  rewrite !map_subseteq_spec. intros Hi. apply (None_ne_Some x).
  by rewrite <-(lookup_delete m i), (Hi i x).
386
Qed.
387
Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
388 389 390 391 392
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.
393
Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
394
Proof. rewrite lookup_insert. congruence. Qed.
395
Lemma lookup_insert_ne {A} (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
396
Proof. unfold insert. apply lookup_partial_alter_ne. Qed.
397 398
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.
399 400 401 402 403 404 405
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
Qed.
410 411 412
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.
413 414 415
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
416 417 418
  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
419
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
420
Lemma insert_id {A} (m : M A) i x : m !! i = Some x  <[i:=x]>m = m.
421 422 423 424 425 426 427 428
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 [->|].
429 430
  - rewrite lookup_insert. destruct (m !! j); simpl; eauto.
  - rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
431
Qed.
432
Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
433
Proof. apply partial_alter_subseteq. Qed.
434
Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
435 436
Proof. intro. apply partial_alter_subset; eauto. Qed.
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
437
  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
438
Proof.
439 440 441
  rewrite !map_subseteq_spec. intros ?? j ?.
  destruct (decide (j = i)) as [->|?]; [congruence|].
  rewrite lookup_insert_ne; auto.
442 443
Qed.
Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
444
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
445
Proof.
446 447 448 449
  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.
450 451
Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
452
  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
453
Proof.
454 455
  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
456 457
  - rewrite lookup_insert. congruence.
  - rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
458 459
Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
460
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
461
Proof.
462 463 464
  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
465 466
Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
467
  m1 !! i = None  <[i:=x]> m1  m2 
468 469
   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
470
  intros Hi Hm1m2. exists (delete i m2). split_and?.
471 472
  - rewrite insert_delete, insert_id. done.
    eapply lookup_weaken, strict_include; eauto. by rewrite lookup_insert.
473 474
  - eauto using insert_delete_subset.
  - by rewrite lookup_delete.
475
Qed.
476
Lemma insert_empty {A} i (x : A) : <[i:=x]> = {[i := x]}.
477
Proof. done. Qed.
478 479 480

(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
481
  {[i := x]} !! j = Some y  i = j  x = y.
482
Proof.
483
  rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence.
484
Qed.
485
Lemma lookup_singleton_None {A} i j (x : A) : {[i := x]} !! j = None  i  j.
486
Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; tauto. Qed.
487
Lemma lookup_singleton {A} i (x : A) : {[i := x]} !! i = Some x.
488
Proof. by rewrite lookup_singleton_Some. Qed.
489
Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[i := x]} !! j = None.
490
Proof. by rewrite lookup_singleton_None. Qed.
491
Lemma map_non_empty_singleton {A} i (x : A) : {[i := x]}  .
492 493 494 495
Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
496
Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i := x]} = {[i := y]}.
497
Proof.
498
  unfold singletonM, map_singleton, insert, map_insert.
499 500
  by rewrite <-partial_alter_compose.
Qed.
501
Lemma alter_singleton {A} (f : A  A) i x : alter f i {[i := x]} = {[i := f x]}.
502
Proof.
503
  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
504 505
  - by rewrite lookup_alter, !lookup_singleton.
  - by rewrite lookup_alter_ne, !lookup_singleton_ne.
506 507
Qed.
Lemma alter_singleton_ne {A} (f : A  A) i j x :
508
  i  j  alter f i {[j := x]} = {[j := x]}.
509
Proof.
510 511
  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
512 513
Qed.

514 515 516 517 518
(** ** Properties of the map operations *)
Lemma fmap_empty {A B} (f : A  B) : f <$>  = .
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A  option B) : omap f  = .
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
519 520 521
Lemma fmap_insert {A B} (f: A  B) m i x: f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
522 523
  - by rewrite lookup_fmap, !lookup_insert.
  - by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
524 525 526 527 528
Qed.
Lemma omap_insert {A B} (f : A  option B) m i x y :
  f x = Some y  omap f (<[i:=x]>m) = <[i:=y]>(omap f m).
Proof.
  intros; apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
529 530
  - by rewrite lookup_omap, !lookup_insert.
  - by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done.
531
Qed.
532
Lemma map_fmap_singleton {A B} (f : A  B) i x : f <$> {[i := x]} = {[i := f x]}.
533 534 535
Proof.
  by unfold singletonM, map_singleton; rewrite fmap_insert, map_fmap_empty.
Qed.
536
Lemma omap_singleton {A B} (f : A  option B) i x y :
537
  f x = Some y  omap f {[ i := x ]} = {[ i := y ]}.
538
Proof.
539 540
  intros. unfold singletonM, map_singleton.
  by erewrite omap_insert, omap_empty by eauto.
541
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
542 543 544 545 546
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.
547 548 549 550 551 552
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
553 554 555 556 557 558
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
559 560 561 562 563 564
Lemma omap_ext {A B} (f1 f2 : A  option B) m :
  ( 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.
565

566 567
(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
568
  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
569
Proof. rewrite !elem_of_map_to_list. congruence. Qed.
570
Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
571
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
572 573 574 575 576
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.
577
  intros [?|?] Hdup; simplify_eq; [by rewrite lookup_insert|].
578
  destruct (decide (i = j)) as [->|].
579 580
  - rewrite lookup_insert; f_equal; eauto.
  - rewrite lookup_insert_ne by done; eauto.
581
Qed.
582
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
583
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
584
Proof.
585 586
  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'].
587
  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
588
    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
589 590
Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
591
  map_of_list l !! i = Some x  (i,x)  l.
592
Proof.
593 594 595
  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.
596 597
Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
598
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
599
Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
600
Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
601
  i  l.*1  map_of_list l !! i = None.
602
Proof.
603 604
  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.
605 606
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
607
  map_of_list l !! i = None  i  l.*1.
608
Proof.
609
  induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
610
  rewrite elem_of_cons. destruct (decide (i = j)); simplify_eq.
611 612
  - by rewrite lookup_insert.
  - by rewrite lookup_insert_ne; intuition.
613 614
Qed.
Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
615
  i  l.*1  map_of_list l !! i = None.
616
Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
617
Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
618
  NoDup (l1.*1)  l1  l2  map_of_list l1 = map_of_list l2.
619 620 621 622 623
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)) :
624
  NoDup (l1.*1)  NoDup (l2.*1)  map_of_list l1 = map_of_list l2  l1  l2.
625
Proof.
626
  intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
627 628
  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
629
Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m.
630 631 632
Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
633
    by auto using NoDup_fst_map_to_list.
634 635
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
636
  NoDup (l.*1)  map_to_list (map_of_list l)  l.
637
Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
638
Lemma map_to_list_inj {A} (m1 m2 : M A) :
639
  map_to_list m1  map_to_list m2  m1 = m2.
640
Proof.
641
  intros. rewrite <-(map_of_to_list m1), <-(map_of_to_list m2).
642
  auto using map_of_list_proper, NoDup_fst_map_to_list.
643
Qed.
644 645 646 647 648 649
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.
650 651 652 653 654 655 656 657 658 659 660 661 662

Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = .
Proof. done. Qed.
Lemma map_of_list_cons {A} (l : list (K * A)) i x :
  map_of_list ((i, x) :: l) = <[i:=x]>(map_of_list l).
Proof. done. Qed.
Lemma map_of_list_fmap {A B} (f : A  B) l :
  map_of_list (prod_map id f <$> l) = f <$> map_of_list l.
Proof.
  induction l as [|[i x] l IH]; csimpl; rewrite ?fmap_empty; auto.
  rewrite <-map_of_list_cons; simpl. by rewrite IH, <-fmap_insert.
Qed.

663
Lemma map_to_list_empty {A} : map_to_list  = @nil (K * A).
664 665 666 667 668
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 :
669
  m !! i = None  map_to_list (<[i:=x]>m)  (i,x) :: map_to_list m.
670
Proof.
671
  intros. apply map_of_list_inj; csimpl.
</