fin_maps.v 69.8 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1 2 3 4 5 6
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** Finite maps associate data to keys. This file defines an interface for
finite maps and collects some theory on it. Most importantly, it proves useful
induction principles for finite maps and implements the tactic
[simplify_map_equality] to simplify goals involving finite maps. *)
7 8
From Coq Require Import Permutation.
From prelude Require Export relations vector orders.
Robbert Krebbers's avatar
Robbert Krebbers committed
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
course limits the space of finite map implementations, but since we are mainly
interested in finite maps with numbers as indexes, we do not consider this to
be a serious limitation. The main application of finite maps is to implement
the memory, where extensionality of Leibniz equality is very important for a
convenient use in the assertions of our axiomatic semantics. *)

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

(** Finite map implementations are required to implement the [merge] function
which enables us to give a generic implementation of [union_with],
[intersection_with], and [difference_with]. *)

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

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

(** * Derived operations *)
(** All of the following functions are defined in a generic way for arbitrary
finite map implementations. These generic implementations do not cause a
significant performance loss to make including them in the finite map interface
worthwhile. *)
Instance map_insert `{PartialAlter K A M} : Insert K A M :=
  λ i x, partial_alter (λ _, Some x) i.
Instance map_alter `{PartialAlter K A M} : Alter K A M :=
  λ f, partial_alter (fmap f).
Instance map_delete `{PartialAlter K A M} : Delete K M :=
  partial_alter (λ _, None).
Instance map_singleton `{PartialAlter K A M, Empty M} :
59
  SingletonM K A M := λ i x, <[i:=x]> .
Robbert Krebbers's avatar
Robbert Krebbers committed
60 61 62 63 64 65 66 67 68 69 70 71 72 73

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

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

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

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

(** The union of two finite maps only has a meaningful definition for maps
that are disjoint. However, as working with partial functions is inconvenient
in Coq, we define the union as a total function. In case both finite maps
have a value at the same index, we take the value of the first map. *)
Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x).
Instance map_intersection `{Merge M} {A} : Intersection (M A) :=
  intersection_with (λ x _, Some x).

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

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

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

(** ** Setoids *)
Section setoid.
120 121
  Context `{Equiv A} `{!Equivalence (() : relation A)}.
  Global Instance map_equivalence : Equivalence (() : relation (M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
122 123
  Proof.
    split.
124 125
    - by intros m i.
    - by intros m1 m2 ? i.
126
    - by intros m1 m2 m3 ?? i; trans (m2 !! i).
Robbert Krebbers's avatar
Robbert Krebbers committed
127 128 129 130 131
  Qed.
  Global Instance lookup_proper (i : K) :
    Proper (() ==> ()) (lookup (M:=M A) i).
  Proof. by intros m1 m2 Hm. Qed.
  Global Instance partial_alter_proper :
132
    Proper ((() ==> ()) ==> (=) ==> () ==> ()) (partial_alter (M:=M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
133 134 135 136 137 138 139 140
  Proof.
    by intros f1 f2 Hf i ? <- m1 m2 Hm j; destruct (decide (i = j)) as [->|];
      rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne by done;
      try apply Hf; apply lookup_proper.
  Qed.
  Global Instance insert_proper (i : K) :
    Proper (() ==> () ==> ()) (insert (M:=M A) i).
  Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
141 142 143
  Global Instance singleton_proper k :
    Proper (() ==> ()) (singletonM k : A  M A).
  Proof. by intros ???; apply insert_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
144 145 146 147 148 149 150 151 152 153 154 155
  Global Instance delete_proper (i : K) :
    Proper (() ==> ()) (delete (M:=M A) i).
  Proof. by apply partial_alter_proper; [constructor|]. Qed.
  Global Instance alter_proper :
    Proper ((() ==> ()) ==> (=) ==> () ==> ()) (alter (A:=A) (M:=M A)).
  Proof.
    intros ?? Hf; apply partial_alter_proper.
    by destruct 1; constructor; apply Hf.
  Qed.
  Lemma merge_ext f g
      `{!PropHolds (f None None = None), !PropHolds (g None None = None)} :
    (() ==> () ==> ())%signature f g 
156
    (() ==> () ==> ())%signature (merge (M:=M) f) (merge g).
Robbert Krebbers's avatar
Robbert Krebbers committed
157 158 159 160
  Proof.
    by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
  Qed.
  Global Instance union_with_proper :
161
    Proper ((() ==> () ==> ()) ==> () ==> () ==>()) (union_with (M:=M A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
162 163 164
  Proof.
    intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
    by do 2 destruct 1; first [apply Hf | constructor].
165
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
166 167
  Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
  Proof.
168 169
    intros m1 m2 Hm; apply map_eq; intros i.
    by unfold_leibniz; apply lookup_proper.
Robbert Krebbers's avatar
Robbert Krebbers committed
170
  Qed.
171 172 173 174 175 176 177 178 179 180
  Lemma map_equiv_empty (m : M A) : m    m = .
  Proof.
    split; [intros Hm; apply map_eq; intros i|by intros ->].
    by rewrite lookup_empty, <-equiv_None, Hm, lookup_empty.
  Qed.
  Lemma map_equiv_lookup (m1 m2 : M A) i x :
    m1  m2  m1 !! i = Some x   y, m2 !! i = Some y  x  y.
  Proof.
    intros Hm ?. destruct (equiv_Some (m1 !! i) (m2 !! i) x) as (y&?&?); eauto.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
End setoid.

(** ** General properties *)
Lemma map_eq_iff {A} (m1 m2 : M A) : m1 = m2   i, m1 !! i = m2 !! i.
Proof. split. by intros ->. apply map_eq. Qed.
Lemma map_subseteq_spec {A} (m1 m2 : M A) :
  m1  m2   i x, m1 !! i = Some x  m2 !! i = Some x.
Proof.
  unfold subseteq, map_subseteq, map_relation. split; intros Hm i;
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Global Instance: EmptySpec (M A).
Proof.
  intros A m. rewrite !map_subseteq_spec.
  intros i x. by rewrite lookup_empty.
Qed.
Global Instance:  {A} (R : relation A), PreOrder R  PreOrder (map_included R).
Proof.
  split; [intros m i; by destruct (m !! i); simpl|].
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
201
  destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_eq/=;
202
    done || etrans; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236
Qed.
Global Instance: PartialOrder (() : relation (M A)).
Proof.
  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
Proof. rewrite !map_subseteq_spec. auto. Qed.
Lemma lookup_weaken_is_Some {A} (m1 m2 : M A) i :
  is_Some (m1 !! i)  m1  m2  is_Some (m2 !! i).
Proof. inversion 1. eauto using lookup_weaken. Qed.
Lemma lookup_weaken_None {A} (m1 m2 : M A) i :
  m2 !! i = None  m1  m2  m1 !! i = None.
Proof.
  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
Lemma lookup_ne {A} (m : M A) i j : m !! i  m !! j  i  j.
Proof. congruence. Qed.
Lemma map_empty {A} (m : M A) : ( i, m !! i = None)  m = .
Proof. intros Hm. apply map_eq. intros. by rewrite Hm, lookup_empty. Qed.
Lemma lookup_empty_is_Some {A} i : ¬is_Some (( : M A) !! i).
Proof. rewrite lookup_empty. by inversion 1. Qed.
Lemma lookup_empty_Some {A} i (x : A) : ¬∅ !! i = Some x.
Proof. by rewrite lookup_empty. Qed.
Lemma map_subset_empty {A} (m : M A) : m  .
Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
237 238
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
239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260

(** ** Properties of the [partial_alter] operation *)
Lemma partial_alter_ext {A} (f g : option A  option A) (m : M A) i :
  ( x, m !! i = x  f x = g x)  partial_alter f i m = partial_alter g i m.
Proof.
  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
Qed.
Lemma partial_alter_commute {A} f g (m : M A) i j :
  i  j  partial_alter f i (partial_alter g j m) =
    partial_alter g j (partial_alter f i m).
Proof.
  intros. apply map_eq; intros jj. destruct (decide (jj = j)) as [->|?].
  { by rewrite lookup_partial_alter_ne,
      !lookup_partial_alter, lookup_partial_alter_ne. }
  destruct (decide (jj = i)) as [->|?].
261
  - by rewrite lookup_partial_alter,
Robbert Krebbers's avatar
Robbert Krebbers committed
262
     !lookup_partial_alter_ne, lookup_partial_alter by congruence.
263
  - by rewrite !lookup_partial_alter_ne by congruence.
Robbert Krebbers's avatar
Robbert Krebbers committed
264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290
Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
Qed.
Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m.
Proof. by apply partial_alter_self_alt. Qed.
Lemma partial_alter_subseteq {A} f (m : M A) i :
  m !! i = None  m  partial_alter f i m.
Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
Lemma partial_alter_subset {A} f (m : M A) i :
  m !! i = None  is_Some (f (m !! i))  m  partial_alter f i m.
Proof.
  intros Hi Hfi. split; [by apply partial_alter_subseteq|].
  rewrite !map_subseteq_spec. inversion Hfi as [x Hx]. intros Hm.
  apply (Some_ne_None x). rewrite <-(Hm i x); [done|].
  by rewrite lookup_partial_alter.
Qed.

(** ** Properties of the [alter] operation *)
Lemma alter_ext {A} (f g : A  A) (m : M A) i :
  ( x, m !! i = Some x  f x = g x)  alter f i m = alter g i m.
291
Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309
Lemma lookup_alter {A} (f : A  A) m i : alter f i m !! i = f <$> m !! i.
Proof. unfold alter. apply lookup_partial_alter. Qed.
Lemma lookup_alter_ne {A} (f : A  A) m i j : i  j  alter f i m !! j = m !! j.
Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
Lemma alter_compose {A} (f g : A  A) (m : M A) i:
  alter (f  g) i m = alter f i (alter g i m).
Proof.
  unfold alter, map_alter. rewrite <-partial_alter_compose.
  apply partial_alter_ext. by intros [?|].
Qed.
Lemma alter_commute {A} (f g : A  A) (m : M A) i j :
  i  j  alter f i (alter g j m) = alter g j (alter f i m).
Proof. apply partial_alter_commute. Qed.
Lemma lookup_alter_Some {A} (f : A  A) m i j y :
  alter f i m !! j = Some y 
    (i = j   x, m !! j = Some x  y = f x)  (i  j  m !! j = Some y).
Proof.
  destruct (decide (i = j)) as [->|?].
310
  - rewrite lookup_alter. naive_solver (simplify_option_eq; eauto).
311
  - rewrite lookup_alter_ne by done. naive_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
312 313 314 315 316 317 318 319 320 321 322
Qed.
Lemma lookup_alter_None {A} (f : A  A) m i j :
  alter f i m !! j = None  m !! j = None.
Proof.
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
Qed.
Lemma alter_id {A} (f : A  A) m i :
  ( x, m !! i = Some x  f x = x)  alter f i m = m.
Proof.
  intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
323
  { rewrite lookup_alter; destruct (m !! j); f_equal/=; auto. }
Robbert Krebbers's avatar
Robbert Krebbers committed
324 325 326 327 328 329 330 331 332 333 334 335
  by rewrite lookup_alter_ne by done.
Qed.

(** ** Properties of the [delete] operation *)
Lemma lookup_delete {A} (m : M A) i : delete i m !! i = None.
Proof. apply lookup_partial_alter. Qed.
Lemma lookup_delete_ne {A} (m : M A) i j : i  j  delete i m !! j = m !! j.
Proof. apply lookup_partial_alter_ne. Qed.
Lemma lookup_delete_Some {A} (m : M A) i j y :
  delete i m !! j = Some y  i  j  m !! j = Some y.
Proof.
  split.
336
  - destruct (decide (i = j)) as [->|?];
Robbert Krebbers's avatar
Robbert Krebbers committed
337
      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
338
  - intros [??]. by rewrite lookup_delete_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
339
Qed.
340 341 342
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
343 344 345 346 347 348 349 350
Lemma lookup_delete_None {A} (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
351
Lemma delete_singleton {A} i (x : A) : delete i {[i := x]} = .
Robbert Krebbers's avatar
Robbert Krebbers committed
352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405
Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
Lemma delete_commute {A} (m : M A) i j :
  delete i (delete j m) = delete j (delete i m).
Proof. destruct (decide (i = j)). by subst. by apply partial_alter_commute. Qed.
Lemma delete_insert_ne {A} (m : M A) i j x :
  i  j  delete i (<[j:=x]>m) = <[j:=x]>(delete i m).
Proof. intro. by apply partial_alter_commute. Qed.
Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
Proof.
  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
Qed.
Lemma delete_partial_alter {A} (m : M A) i f :
  m !! i = None  delete i (partial_alter f i m) = m.
Proof.
  intros. unfold delete, map_delete. rewrite <-partial_alter_compose.
  unfold compose. by apply partial_alter_self_alt.
Qed.
Lemma delete_insert {A} (m : M A) i x :
  m !! i = None  delete i (<[i:=x]>m) = m.
Proof. apply delete_partial_alter. Qed.
Lemma insert_delete {A} (m : M A) i x :
  m !! i = Some x  <[i:=x]>(delete i m) = m.
Proof.
  intros Hmi. unfold delete, map_delete, insert, map_insert.
  rewrite <-partial_alter_compose. unfold compose. rewrite <-Hmi.
  by apply partial_alter_self_alt.
Qed.
Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
Lemma delete_subseteq_compat {A} (m1 m2 : M A) i :
  m1  m2  delete i m1  delete i m2.
Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
Qed.
Lemma delete_subset_alt {A} (m : M A) i x : m !! i = Some x  delete i m  m.
Proof.
  split; [apply delete_subseteq|].
  rewrite !map_subseteq_spec. intros Hi. apply (None_ne_Some x).
  by rewrite <-(lookup_delete m i), (Hi i x).
Qed.
Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
Proof. inversion 1. eauto using delete_subset_alt. Qed.

(** ** Properties of the [insert] operation *)
Lemma lookup_insert {A} (m : M A) i x : <[i:=x]>m !! i = Some x.
Proof. unfold insert. apply lookup_partial_alter. Qed.
Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
Proof. rewrite lookup_insert. congruence. Qed.
Lemma lookup_insert_ne {A} (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
Proof. unfold insert. apply lookup_partial_alter_ne. Qed.
406 407
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
408 409 410 411 412 413 414
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.
415
  - destruct (decide (i = j)) as [->|?];
Robbert Krebbers's avatar
Robbert Krebbers committed
416
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
417
  - intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
418
Qed.
419 420 421
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
Qed.
Lemma insert_id {A} (m : M A) i x : m !! i = Some x  <[i:=x]>m = m.
Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|];
    by rewrite ?lookup_insert, ?lookup_insert_ne by done.
Qed.
Lemma insert_included {A} R `{!Reflexive R} (m : M A) i x :
  ( y, m !! i = Some y  R y x)  map_included R m (<[i:=x]>m).
Proof.
  intros ? j; destruct (decide (i = j)) as [->|].
438 439
  - rewrite lookup_insert. destruct (m !! j); simpl; eauto.
  - rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464
Qed.
Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
Proof. apply partial_alter_subseteq. Qed.
Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
Proof. intro. apply partial_alter_subset; eauto. Qed.
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
Proof.
  rewrite !map_subseteq_spec. intros ?? j ?.
  destruct (decide (j = i)) as [->|?]; [congruence|].
  rewrite lookup_insert_ne; auto.
Qed.
Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
Proof.
  rewrite !map_subseteq_spec. intros Hi Hix j y Hj.
  destruct (decide (i = j)) as [->|]; [congruence|].
  rewrite lookup_delete_ne by done.
  apply Hix; by rewrite lookup_insert_ne by done.
Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
Proof.
  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
465 466
  - rewrite lookup_insert. congruence.
  - rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
467 468 469 470 471 472 473 474 475 476 477 478
Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
Proof.
  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
  m1 !! i = None  <[i:=x]> m1  m2 
   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
479
  intros Hi Hm1m2. exists (delete i m2). split_and?.
480
  - rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
481
    by rewrite lookup_insert.
482 483
  - eauto using insert_delete_subset.
  - by rewrite lookup_delete.
Robbert Krebbers's avatar
Robbert Krebbers committed
484
Qed.
485
Lemma insert_empty {A} i (x : A) : <[i:=x]> = {[i := x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
486 487 488 489
Proof. done. Qed.

(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
490
  {[i := x]} !! j = Some y  i = j  x = y.
Robbert Krebbers's avatar
Robbert Krebbers committed
491
Proof.
492
  rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence.
Robbert Krebbers's avatar
Robbert Krebbers committed
493
Qed.
494
Lemma lookup_singleton_None {A} i j (x : A) : {[i := x]} !! j = None  i  j.
495
Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; tauto. Qed.
496
Lemma lookup_singleton {A} i (x : A) : {[i := x]} !! i = Some x.
Robbert Krebbers's avatar
Robbert Krebbers committed
497
Proof. by rewrite lookup_singleton_Some. Qed.
498
Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[i := x]} !! j = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
499
Proof. by rewrite lookup_singleton_None. Qed.
500
Lemma map_non_empty_singleton {A} i (x : A) : {[i := x]}  .
Robbert Krebbers's avatar
Robbert Krebbers committed
501 502 503 504
Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
505
Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i := x]} = {[i := y]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
506
Proof.
507
  unfold singletonM, map_singleton, insert, map_insert.
Robbert Krebbers's avatar
Robbert Krebbers committed
508 509
  by rewrite <-partial_alter_compose.
Qed.
510
Lemma alter_singleton {A} (f : A  A) i x : alter f i {[i := x]} = {[i := f x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
511 512
Proof.
  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
513 514
  - by rewrite lookup_alter, !lookup_singleton.
  - by rewrite lookup_alter_ne, !lookup_singleton_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
515 516
Qed.
Lemma alter_singleton_ne {A} (f : A  A) i j x :
517
  i  j  alter f i {[j := x]} = {[j := x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
518 519 520 521 522 523 524 525 526 527
Proof.
  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
Qed.

(** ** Properties of the map operations *)
Lemma fmap_empty {A B} (f : A  B) : f <$>  = .
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A  option B) : omap f  = .
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
528 529 530
Lemma fmap_insert {A B} (f: A  B) m i x: f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
531 532
  - by rewrite lookup_fmap, !lookup_insert.
  - by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
533 534 535 536 537
Qed.
Lemma omap_insert {A B} (f : A  option B) m i x y :
  f x = Some y  omap f (<[i:=x]>m) = <[i:=y]>(omap f m).
Proof.
  intros; apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
538 539
  - by rewrite lookup_omap, !lookup_insert.
  - by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done.
540
Qed.
541
Lemma map_fmap_singleton {A B} (f : A  B) i x : f <$> {[i := x]} = {[i := f x]}.
542 543 544
Proof.
  by unfold singletonM, map_singleton; rewrite fmap_insert, map_fmap_empty.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
545
Lemma omap_singleton {A B} (f : A  option B) i x y :
546
  f x = Some y  omap f {[ i := x ]} = {[ i := y ]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
547
Proof.
548 549
  intros. unfold singletonM, map_singleton.
  by erewrite omap_insert, omap_empty by eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
550 551 552 553 554 555
Qed.
Lemma map_fmap_id {A} (m : M A) : id <$> m = m.
Proof. apply map_eq; intros i; by rewrite lookup_fmap, option_fmap_id. Qed.
Lemma map_fmap_compose {A B C} (f : A  B) (g : B  C) (m : M A) :
  g  f <$> m = g <$> f <$> m.
Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed.
556 557 558 559 560 561
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
562 563 564 565 566 567
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
568 569 570 571 572 573
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
574 575 576 577 578 579 580 581 582 583 584 585

(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
Proof. rewrite !elem_of_map_to_list. congruence. Qed.
Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
Lemma elem_of_map_of_list_1_help {A} (l : list (K * A)) i x :
  (i,x)  l  ( y, (i,y)  l  y = x)  map_of_list l !! i = Some x.
Proof.
  induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|].
  setoid_rewrite elem_of_cons.
586
  intros [?|?] Hdup; simplify_eq; [by rewrite lookup_insert|].
Robbert Krebbers's avatar
Robbert Krebbers committed
587
  destruct (decide (i = j)) as [->|].
588 589
  - rewrite lookup_insert; f_equal; eauto.
  - rewrite lookup_insert_ne by done; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618
Qed.
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
Proof.
  intros ? Hx; apply elem_of_map_of_list_1_help; eauto using NoDup_fmap_fst.
  intros y; revert Hx. rewrite !elem_of_list_lookup; intros [i' Hi'] [j' Hj'].
  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
  map_of_list l !! i = Some x  (i,x)  l.
Proof.
  induction l as [|[j y] l IH]; simpl; [by rewrite lookup_empty|].
  rewrite elem_of_cons. destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
  i  l.*1  map_of_list l !! i = None.
Proof.
  rewrite elem_of_list_fmap, eq_None_not_Some. intros Hi [x ?]; destruct Hi.
  exists (i,x); simpl; auto using elem_of_map_of_list_2.
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
  map_of_list l !! i = None  i  l.*1.
Proof.
  induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
619
  rewrite elem_of_cons. destruct (decide (i = j)); simplify_eq.
620 621
  - by rewrite lookup_insert.
  - by rewrite lookup_insert_ne; intuition.
Robbert Krebbers's avatar
Robbert Krebbers committed
622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667
Qed.
Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
  i  l.*1  map_of_list l !! i = None.
Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
  NoDup (l1.*1)  l1 ≡ₚ l2  map_of_list l1 = map_of_list l2.
Proof.
  intros ? Hperm. apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-!elem_of_map_of_list; rewrite <-?Hperm.
Qed.
Lemma map_of_list_inj {A} (l1 l2 : list (K * A)) :
  NoDup (l1.*1)  NoDup (l2.*1)  map_of_list l1 = map_of_list l2  l1 ≡ₚ l2.
Proof.
  intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m.
Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
    by auto using NoDup_fst_map_to_list.
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
  NoDup (l.*1)  map_to_list (map_of_list l) ≡ₚ l.
Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
Lemma map_to_list_inj {A} (m1 m2 : M A) :
  map_to_list m1 ≡ₚ map_to_list m2  m1 = m2.
Proof.
  intros. rewrite <-(map_of_to_list m1), <-(map_of_to_list m2).
  auto using map_of_list_proper, NoDup_fst_map_to_list.
Qed.
Lemma map_to_of_list_flip {A} (m1 : M A) l2 :
  map_to_list m1 ≡ₚ l2  m1 = map_of_list l2.
Proof.
  intros. rewrite <-(map_of_to_list m1).
  auto using map_of_list_proper, NoDup_fst_map_to_list.
Qed.
Lemma map_to_list_empty {A} : map_to_list  = @nil (K * A).
Proof.
  apply elem_of_nil_inv. intros [i x].
  rewrite elem_of_map_to_list. apply lookup_empty_Some.
Qed.
Lemma map_to_list_insert {A} (m : M A) i x :
  m !! i = None  map_to_list (<[i:=x]>m) ≡ₚ (i,x) :: map_to_list m.
Proof.
  intros. apply map_of_list_inj; csimpl.
668 669
  - apply NoDup_fst_map_to_list.
  - constructor; auto using NoDup_fst_map_to_list.
Robbert Krebbers's avatar
Robbert Krebbers committed
670 671
    rewrite elem_of_list_fmap. intros [[??] [? Hlookup]]; subst; simpl in *.
    rewrite elem_of_map_to_list in Hlookup. congruence.
672
  - by rewrite !map_of_to_list.
Robbert Krebbers's avatar
Robbert Krebbers committed
673
Qed.
674 675 676 677 678 679
Lemma map_to_list_contains {A} (m1 m2 : M A) :
  m1  m2  map_to_list m1 `contains` map_to_list m2.
Proof.
  intros; apply NoDup_contains; auto using NoDup_map_to_list.
  intros [i x]. rewrite !elem_of_map_to_list; eauto using lookup_weaken.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702
Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = .
Proof. done. Qed.
Lemma map_of_list_cons {A} (l : list (K * A)) i x :
  map_of_list ((i, x) :: l) = <[i:=x]>(map_of_list l).
Proof. done. Qed.
Lemma map_to_list_empty_inv_alt {A}  (m : M A) : map_to_list m ≡ₚ []  m = .
Proof. rewrite <-map_to_list_empty. apply map_to_list_inj. Qed.
Lemma map_to_list_empty_inv {A} (m : M A) : map_to_list m = []  m = .
Proof. intros Hm. apply map_to_list_empty_inv_alt. by rewrite Hm. Qed.
Lemma map_to_list_insert_inv {A} (m : M A) l i x :
  map_to_list m ≡ₚ (i,x) :: l  m = <[i:=x]>(map_of_list l).
Proof.
  intros Hperm. apply map_to_list_inj.
  assert (i  l.*1  NoDup (l.*1)) as [].
  { rewrite <-NoDup_cons. change (NoDup (((i,x)::l).*1)). rewrite <-Hperm.
    auto using NoDup_fst_map_to_list. }
  rewrite Hperm, map_to_list_insert, map_to_of_list;
    auto using not_elem_of_map_of_list_1.
Qed.
Lemma map_choose {A} (m : M A) : m     i x, m !! i = Some x.
Proof.
  intros Hemp. destruct (map_to_list m) as [|[i x] l] eqn:Hm.
  { destruct Hemp; eauto using map_to_list_empty_inv. }
703
  exists i, x. rewrite <-elem_of_map_to_list, Hm. by left.
Robbert Krebbers's avatar
Robbert Krebbers committed
704 705 706 707 708 709 710
Qed.

(** Properties of the imap function *)
Lemma lookup_imap {A B} (f : K  A  option B) m i :
  map_imap f m !! i = m !! i = f i.
Proof.
  unfold map_imap; destruct (m !! i = f i) as [y|] eqn:Hi; simpl.
711
  - destruct (m !! i) as [x|] eqn:?; simplify_eq/=.
Robbert Krebbers's avatar
Robbert Krebbers committed
712 713
    apply elem_of_map_of_list_1_help.
    { apply elem_of_list_omap; exists (i,x); split;
714
        [by apply elem_of_map_to_list|by simplify_option_eq]. }
Robbert Krebbers's avatar
Robbert Krebbers committed
715
    intros y'; rewrite elem_of_list_omap; intros ([i' x']&Hi'&?).
716
    by rewrite elem_of_map_to_list in Hi'; simplify_option_eq.
717
  - apply not_elem_of_map_of_list; rewrite elem_of_list_fmap.
718
    intros ([i' x]&->&Hi'); simplify_eq/=.
Robbert Krebbers's avatar
Robbert Krebbers committed
719
    rewrite elem_of_list_omap in Hi'; destruct Hi' as ([j y]&Hj&?).
720
    rewrite elem_of_map_to_list in Hj; simplify_option_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
721 722 723 724 725 726 727 728 729 730 731
Qed.

(** ** Properties of conversion from collections *)
Lemma lookup_map_of_collection {A} `{FinCollection K C}
    (f : K  option A) X i x :
  map_of_collection f X !! i = Some x  i  X  f i = Some x.
Proof.
  assert (NoDup (fst <$> omap (λ i, (i,) <$> f i) (elements X))).
  { induction (NoDup_elements X) as [|i' l]; csimpl; [constructor|].
    destruct (f i') as [x'|]; csimpl; auto; constructor; auto.
    rewrite elem_of_list_fmap. setoid_rewrite elem_of_list_omap.
732
    by intros (?&?&?&?&?); simplify_option_eq. }
Robbert Krebbers's avatar
Robbert Krebbers committed
733 734
  unfold map_of_collection; rewrite <-elem_of_map_of_list by done.
  rewrite elem_of_list_omap. setoid_rewrite elem_of_elements; split.
735 736
  - intros (?&?&?); simplify_option_eq; eauto.
  - intros [??]; exists i; simplify_option_eq; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
737 738 739 740 741 742 743 744 745 746 747 748 749
Qed.

(** ** Induction principles *)
Lemma map_ind {A} (P : M A  Prop) :
  P   ( i x m, m !! i = None  P m  P (<[i:=x]>m))   m, P m.
Proof.
  intros ? Hins. cut ( l, NoDup (l.*1)   m, map_to_list m ≡ₚ l  P m).
  { intros help m.
    apply (help (map_to_list m)); auto using NoDup_fst_map_to_list. }
  induction l as [|[i x] l IH]; intros Hnodup m Hml.
  { apply map_to_list_empty_inv_alt in Hml. by subst. }
  inversion_clear Hnodup.
  apply map_to_list_insert_inv in Hml; subst m. apply Hins.
750 751
  - by apply not_elem_of_map_of_list_1.
  - apply IH; auto using map_to_of_list.
Robbert Krebbers's avatar
Robbert Krebbers committed
752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767
Qed.
Lemma map_to_list_length {A} (m1 m2 : M A) :
  m1  m2  length (map_to_list m1) < length (map_to_list m2).
Proof.
  revert m2. induction m1 as [|i x m ? IH] using map_ind.
  { intros m2 Hm2. rewrite map_to_list_empty. simpl.
    apply neq_0_lt. intros Hlen. symmetry in Hlen.
    apply nil_length_inv, map_to_list_empty_inv in Hlen.
    rewrite Hlen in Hm2. destruct (irreflexivity ()  Hm2). }
  intros m2 Hm2.
  destruct (insert_subset_inv m m2 i x) as (m2'&?&?&?); auto; subst.
  rewrite !map_to_list_insert; simpl; auto with arith.
Qed.
Lemma map_wf {A} : wf (strict (@subseteq (M A) _)).
Proof.
  apply (wf_projected (<) (length  map_to_list)).
768 769
  - by apply map_to_list_length.
  - by apply lt_wf.
Robbert Krebbers's avatar
Robbert Krebbers committed
770 771 772 773 774 775 776 777 778
Qed.

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

Lemma map_Forall_to_list m : map_Forall P m  Forall (curry P) (map_to_list m).
Proof.
  rewrite Forall_forall. split.
779 780
  - intros Hforall [i x]. rewrite elem_of_map_to_list. by apply (Hforall i x).
  - intros Hforall i x. rewrite <-elem_of_map_to_list. by apply (Hforall (i,x)).
Robbert Krebbers's avatar
Robbert Krebbers committed
781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823
Qed.
Lemma map_Forall_empty : map_Forall P .
Proof. intros i x. by rewrite lookup_empty. Qed.
Lemma map_Forall_impl (Q : K  A  Prop) m :
  map_Forall P m  ( i x, P i x  Q i x)  map_Forall Q m.
Proof. unfold map_Forall; naive_solver. Qed.
Lemma map_Forall_insert_11 m i x : map_Forall P (<[i:=x]>m)  P i x.
Proof. intros Hm. by apply Hm; rewrite lookup_insert. Qed.
Lemma map_Forall_insert_12 m i x :
  m !! i = None  map_Forall P (<[i:=x]>m)  map_Forall P m.
Proof.
  intros ? Hm j y ?; apply Hm. by rewrite lookup_insert_ne by congruence.
Qed.
Lemma map_Forall_insert_2 m i x :
  P i x  map_Forall P m  map_Forall P (<[i:=x]>m).
Proof. intros ?? j y; rewrite lookup_insert_Some; naive_solver. Qed.
Lemma map_Forall_insert m i x :
  m !! i = None  map_Forall P (<[i:=x]>m)  P i x  map_Forall P m.
Proof.
  naive_solver eauto using map_Forall_insert_11,
    map_Forall_insert_12, map_Forall_insert_2.
Qed.
Lemma map_Forall_ind (Q : M A  Prop) :
  Q  
  ( m i x, m !! i = None  P i x  map_Forall P m  Q m  Q (<[i:=x]>m)) 
   m, map_Forall P m  Q m.
Proof.
  intros Hnil Hinsert m. induction m using map_ind; auto.
  rewrite map_Forall_insert by done; intros [??]; eauto.
Qed.

Context `{ i x, Decision (P i x)}.
Global Instance map_Forall_dec m : Decision (map_Forall P m).
Proof.
  refine (cast_if (decide (Forall (curry P) (map_to_list m))));
    by rewrite map_Forall_to_list.
Defined.
Lemma map_not_Forall (m : M A) :
  ¬map_Forall P m   i x, m !! i = Some x  ¬P i x.
Proof.
  split; [|intros (i&x&?&?) Hm; specialize (Hm i x); tauto].
  rewrite map_Forall_to_list. intros Hm.
  apply (not_Forall_Exists _), Exists_exists in Hm.
824
  destruct Hm as ([i x]&?&?). exists i, x. by rewrite <-elem_of_map_to_list.
Robbert Krebbers's avatar
Robbert Krebbers committed
825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841
Qed.
End map_Forall.

(** ** Properties of the [merge] operation *)
Section merge.
Context {A} (f : option A  option A  option A).
Context `{!PropHolds (f None None = None)}.
Global Instance: LeftId (=) None f  LeftId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
  by rewrite !(lookup_merge f), lookup_empty, (left_id_L None f).
Qed.
Global Instance: RightId (=) None f  RightId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
  by rewrite !(lookup_merge f), lookup_empty, (right_id_L None f).
Qed.
842
Lemma merge_comm m1 m2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
843 844 845
  ( i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) 
  merge f m1 m2 = merge f m2 m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
846
Global Instance: Comm (=) f  Comm (=) (merge f).
Robbert Krebbers's avatar
Robbert Krebbers committed
847
Proof.
848
  intros ???. apply merge_comm. intros. by apply (comm f).
Robbert Krebbers's avatar
Robbert Krebbers committed
849
Qed.
850
Lemma merge_assoc m1 m2 m3 :
Robbert Krebbers's avatar
Robbert Krebbers committed
851 852 853 854
  ( i, f (m1 !! i) (f (m2 !! i) (m3 !! i)) =
        f (f (m1 !! i) (m2 !! i)) (m3 !! i)) 
  merge f m1 (merge f m2 m3) = merge f (merge f m1 m2) m3.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
855
Global Instance: Assoc (=) f  Assoc (=) (merge f).
Robbert Krebbers's avatar
Robbert Krebbers committed
856
Proof.
857
  intros ????. apply merge_assoc. intros. by apply (assoc_L f).
Robbert Krebbers's avatar
Robbert Krebbers committed
858
Qed.
859
Lemma merge_idemp m1 :
Robbert Krebbers's avatar
Robbert Krebbers committed
860 861
  ( i, f (m1 !! i) (m1 !! i) = m1 !! i)  merge f m1 m1 = m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
862 863
Global Instance: IdemP (=) f  IdemP (=) (merge f).
Proof. intros ??. apply merge_idemp. intros. by apply (idemp f). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882
End merge.

Section more_merge.
Context {A B C} (f : option A  option B  option C).
Context `{!PropHolds (f None None = None)}.
Lemma merge_Some m1 m2 m :
  ( i, m !! i = f (m1 !! i) (m2 !! i))  merge f m1 m2 = m.
Proof.
  split; [|intros <-; apply (lookup_merge _) ].
  intros Hlookup. apply map_eq; intros. rewrite Hlookup. apply (lookup_merge _).
Qed.
Lemma merge_empty : merge f   = .
Proof. apply map_eq. intros. by rewrite !(lookup_merge f), !lookup_empty. Qed.
Lemma partial_alter_merge g g1 g2 m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (g2 (m2 !! i)) 
  partial_alter g i (merge f m1 m2) =
    merge f (partial_alter g1 i m1) (partial_alter g2 i m2).
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
883 884
  - by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  - by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Robbert Krebbers's avatar
Robbert Krebbers committed
885 886 887 888 889 890
Qed.
Lemma partial_alter_merge_l g g1 m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (m2 !! i) 
  partial_alter g i (merge f m1 m2) = merge f (partial_alter g1 i m1) m2.
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
891 892
  - by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  - by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Robbert Krebbers's avatar
Robbert Krebbers committed
893 894 895 896 897 898
Qed.
Lemma partial_alter_merge_r g g2 m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (m1 !! i) (g2 (m2 !! i)) 
  partial_alter g i (merge f m1 m2) = merge f m1 (partial_alter g2 i m2).
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
899 900
  - by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  - by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Robbert Krebbers's avatar
Robbert Krebbers committed
901 902 903 904 905 906
Qed.
Lemma insert_merge m1 m2 i x y z :
  f (Some y) (Some z) = Some x 
  <[i:=x]>(merge f m1 m2) = merge f (<[i:=y]>m1) (<[i:=z]>m2).
Proof. by intros; apply partial_alter_merge. Qed.
Lemma merge_singleton i x y z :
907
  f (Some y) (Some z) = Some x  merge f {[i := y]} {[i := z]} = {[i := x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
908
Proof.
909
  intros. by erewrite <-!insert_empty, <-insert_merge, merge_empty by eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936
Qed.
Lemma insert_merge_l m1 m2 i x y :
  f (Some y) (m2 !! i) = Some x 
  <[i:=x]>(merge f m1 m2) = merge f (<[i:=y]>m1) m2.
Proof. by intros; apply partial_alter_merge_l. Qed.
Lemma insert_merge_r m1 m2 i x z :
  f (m1 !! i) (Some z) = Some x 
  <[i:=x]>(merge f m1 m2) = merge f m1 (<[i:=z]>m2).
Proof. by intros; apply partial_alter_merge_r. Qed.
End more_merge.

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

Let f (mx : option A) (my : option B) : option bool :=
  match mx, my with
  | Some x, Some y => Some (bool_decide (R x y))
  | Some x, None => Some (bool_decide (P x))
  | None, Some y => Some (bool_decide (Q y))
  | None, None => None
  end.
Lemma map_relation_alt (m1 : M A) (m2 : M B) :
  map_relation R P Q m1 m2  map_Forall (λ _, Is_true) (merge f m1 m2).
Proof.
  split.
937
  - intros Hm i P'; rewrite lookup_merge by done; intros.
Robbert Krebbers's avatar
Robbert Krebbers committed
938
    specialize (Hm i). destruct (m1 !! i), (m2 !! i);
939
      simplify_eq/=; auto using bool_decide_pack.
940
  - intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done.
941
    destruct (m1 !! i), (m2 !! i); simplify_eq/=; auto;
Robbert Krebbers's avatar
Robbert Krebbers committed
942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958
      by eapply bool_decide_unpack, Hm.
Qed.
Global Instance map_relation_dec `{ x y, Decision (R x y),  x, Decision (P x),
   y, Decision (Q y)} m1 m2 : Decision (map_relation R P Q m1 m2).
Proof.
  refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2))));
    abstract by rewrite map_relation_alt.
Defined.
(** Due to the finiteness of finite maps, we can extract a witness if the
relation does not hold. *)
Lemma map_not_Forall2 (m1 : M A) (m2 : M B) :
  ¬map_relation R P Q m1 m2   i,
    ( x y, m1 !! i = Some x  m2 !! i = Some y  ¬R x y)
     ( x, m1 !! i = Some x  m2 !! i = None  ¬P x)
     ( y, m1 !! i = None  m2 !! i = Some y  ¬Q y).
Proof.
  split.
959
  - rewrite map_relation_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i.
Robbert Krebbers's avatar
Robbert Krebbers committed
960 961
    rewrite lookup_merge in Hm by done.
    destruct (m1 !! i), (m2 !! i); naive_solver auto 2 using bool_decide_pack.
962
  - unfold map_relation, option_relation.
Robbert Krebbers's avatar
Robbert Krebbers committed
963
    by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
964
      specialize (Hm i); simplify_option_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008
Qed.
End Forall2.

(** ** Properties on the disjoint maps *)
Lemma map_disjoint_spec {A} (m1 m2 : M A) :
  m1 ⊥ₘ m2   i x y, m1 !! i = Some x  m2 !! i = Some y  False.
Proof.
  split; intros Hm i; specialize (Hm i);
    destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_disjoint_alt {A} (m1 m2 : M A) :
  m1 ⊥ₘ m2   i, m1 !! i = None  m2 !! i = None.
Proof.
  split; intros Hm1m2 i; specialize (Hm1m2 i);
    destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_not_disjoint {A} (m1 m2 : M A) :
  ¬m1 ⊥ₘ m2   i x1 x2, m1 !! i = Some x1  m2 !! i = Some x2.
Proof.
  unfold disjoint, map_disjoint. rewrite map_not_Forall2 by solve_decision.
  split; [|naive_solver].
  intros [i[(x&y&?&?&?)|[(x&?&?&[])|(y&?&?&[])]]]; naive_solver.
Qed.
Global Instance: Symmetric (map_disjoint : relation (M A)).
Proof. intros A m1 m2. rewrite !map_disjoint_spec. naive_solver. Qed.
Lemma map_disjoint_empty_l {A} (m : M A) :  ⊥ₘ m.
Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
Lemma map_disjoint_empty_r {A} (m : M A) : m ⊥ₘ .
Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
Lemma map_disjoint_weaken {A} (m1 m1' m2 m2' : M A) :
  m1' ⊥ₘ m2'  m1  m1'  m2  m2'  m1 ⊥ₘ m2.
Proof. rewrite !map_subseteq_spec, !map_disjoint_spec. eauto. Qed.
Lemma map_disjoint_weaken_l {A} (m1 m1' m2  : M A) :
  m1' ⊥ₘ m2  m1  m1'  m1 ⊥ₘ m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_weaken_r {A} (m1 m2 m2' : M A) :
  m1 ⊥ₘ m2'  m2  m2'  m1 ⊥ₘ m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_Some_l {A} (m1 m2 : M A) i x:
  m1 ⊥ₘ m2  m1 !! i = Some x  m2 !! i = None.
Proof. rewrite map_disjoint_spec, eq_None_not_Some. intros ?? [??]; eauto. Qed.
Lemma map_disjoint_Some_r {A} (m1 m2 : M A) i x:
  m1 ⊥ₘ m2  m2 !! i = Some x  m1 !! i = None.
Proof. rewrite (symmetry_iff map_disjoint). apply map_disjoint_Some_l. Qed.
1009
Lemma map_disjoint_singleton_l {A} (m: M A) i x : {[i:=x]} ⊥ₘ m  m !! i = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
1010 1011
Proof.
  split; [|rewrite !map_disjoint_spec].
1012
  - intro. apply (map_disjoint_Some_l {[i := x]} _ _ x);
Robbert Krebbers's avatar
Robbert Krebbers committed
1013
      auto using lookup_singleton.
1014
  - intros ? j y1 y2. destruct (decide (i = j)) as [->|].
Robbert Krebbers's avatar
Robbert Krebbers committed
1015 1016 1017 1018
    + rewrite lookup_singleton. intuition congruence.
    + by rewrite lookup_singleton_ne.
Qed.
Lemma map_disjoint_singleton_r {A} (m : M A) i x :
1019
  m ⊥ₘ {[i := x]}  m !! i = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
1020 1021
Proof. by rewrite (symmetry_iff map_disjoint), map_disjoint_singleton_l. Qed.
Lemma map_disjoint_singleton_l_2 {A} (m : M A) i x :
1022
  m !! i = None  {[i := x]} ⊥ₘ m.
Robbert Krebbers's avatar
Robbert Krebbers committed
1023 1024
Proof. by rewrite map_disjoint_singleton_l. Qed.
Lemma map_disjoint_singleton_r_2 {A} (m : M A) i x :
1025
  m !! i = None  m ⊥ₘ {[i := x]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054
Proof. by rewrite map_disjoint_singleton_r. Qed.
Lemma map_disjoint_delete_l {A} (m1 m2 : M A) i : m1 ⊥ₘ m2  delete i m1 ⊥ₘ m2.
Proof.
  rewrite !map_disjoint_alt. intros Hdisjoint j. destruct (Hdisjoint j); auto.
  rewrite lookup_delete_None. tauto.
Qed.
Lemma map_disjoint_delete_r {A} (m1 m2 : M A) i : m1 ⊥ₘ m2  m1 ⊥ₘ delete i m2.
Proof. symmetry. by apply map_disjoint_delete_l. Qed.

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

Lemma lookup_union_with m1 m2 i :
  union_with f m1 m2 !! i = union_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-(lookup_merge _). Qed.
Lemma lookup_union_with_Some m1 m2 i z :
  union_with f m1 m2 !! i = Some z 
    (m1 !! i = Some z  m2 !! i = None) 
    (m1 !! i = None  m2 !! i = Some z) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  f x y = Some z).
Proof.
  rewrite lookup_union_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Global Instance: LeftId (@eq (M A))  (union_with f).
Proof. unfold union_with, map_union_with. apply _. Qed.
Global Instance: RightId (@eq (M A))  (union_with f).
Proof. unfold union_with, map_union_with. apply _. Qed.
1055
Lemma union_with_comm m1 m2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
1056 1057 1058
  ( i x y, m1 !! i = Some x  m2 !! i = Some y  f x y = f y x) 
  union_with f m1 m2 = union_with f m2 m1.
Proof.
1059
  intros. apply (merge_comm _). intros i.
Robbert Krebbers's avatar
Robbert Krebbers committed
1060 1061
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
1062 1063 1064
Global Instance: Comm (=) f  Comm (@eq (M A)) (union_with f).
Proof. intros ???. apply union_with_comm. eauto. Qed.
Lemma union_with_idemp m :
Robbert Krebbers's avatar
Robbert Krebbers committed
1065 1066
  ( i x, m !! i = Some x  f x x = Some x)  union_with f m m = m.
Proof.
1067
  intros. apply (merge_idemp _). intros i.
Robbert Krebbers's avatar
Robbert Krebbers committed
1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083
  destruct (m !! i) eqn:?; simpl; eauto.
Qed.
Lemma alter_union_with (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f (g x) (g y)) 
  alter g i (union_with f m1 m2) =
    union_with f (alter g i m1) (alter g i m2).
Proof.
  intros. apply (partial_alter_merge _).
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
Lemma alter_union_with_l (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f (g x) y) 
  ( y, m1 !! i = None  m2 !! i = Some y  g y = y) 
  alter g i (union_with f m1 m2) = union_with f (alter g i m1) m2.
Proof.
  intros. apply (partial_alter_merge_l _).
1084
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; f_equal/=; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
1085 1086 1087 1088 1089 1090 1091
Qed.
Lemma alter_union_with_r (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f x (g y)) 
  ( x, m1 !! i = Some x  m2 !! i = None  g x = x) 
  alter g i (union_with f m1 m2) = union_with f m1 (alter g i m2).
Proof.
  intros. apply (partial_alter_merge_r _).
1092
  destruct (m1 !! i)