fin_maps.v 15.2 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1 2 3 4
Require Export prelude.

Class FinMap K M `{ A, Empty (M A)} `{Lookup K M} `{FMap M} 
    `{PartialAlter K M} `{ A, Dom K (M A)} `{Merge M} := {
5 6 7 8 9 10 11 12 13 14 15 16
  finmap_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;
  elem_of_dom C {A} `{Collection K C} (m : M A) i :
    i  dom C m  is_Some (m !! i);
Robbert Krebbers's avatar
Robbert Krebbers committed
17 18 19 20
  merge_spec {A} f `{!PropHolds (f None None = None)} 
    (m1 m2 : M A) i : merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
}.

21 22 23 24 25 26 27 28
Instance finmap_alter `{PartialAlter K M} : Alter K M := λ A f,
  partial_alter (fmap f).
Instance finmap_insert `{PartialAlter K M} : Insert K M := λ A k x,
  partial_alter (λ _, Some x) k.
Instance finmap_delete `{PartialAlter K M} {A} : Delete K (M A) :=
  partial_alter (λ _, None).
Instance finmap_singleton `{PartialAlter K M} {A} 
  `{Empty (M A)} : Singleton (K * A) (M A) := λ p, <[fst p:=snd p]>.
Robbert Krebbers's avatar
Robbert Krebbers committed
29

30 31
Definition list_to_map `{Insert K M} {A} `{Empty (M A)} (l : list (K * A)) : M A :=
  insert_list l .
Robbert Krebbers's avatar
Robbert Krebbers committed
32

33 34 35 36 37 38
Instance finmap_union `{Merge M} : UnionWith M := λ A f,
  merge (union_with f).
Instance finmap_intersection `{Merge M} : IntersectionWith M := λ A f,
  merge (intersection_with f).
Instance finmap_difference `{Merge M} : DifferenceWith M := λ A f,
  merge (difference_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
39 40

Section finmap.
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227
Context `{FinMap K M} `{ i j : K, Decision (i = j)} {A : Type}.

Global Instance finmap_subseteq: SubsetEq (M A) := λ m n,
   i x, m !! i = Some x  n !! i = Some x.
Global Instance: BoundedPreOrder (M A).
Proof. split. firstorder. intros m i x. rewrite lookup_empty. discriminate. Qed.

Lemma lookup_subseteq_Some (m1 m2 : M A) i x :
  m1  m2  m1 !! i = Some x  m2 !! i = Some x.
Proof. auto. Qed.
Lemma lookup_subseteq_None (m1 m2 : M A) i :
  m1  m2  m2 !! i = None  m1 !! i = None.
Proof. rewrite !eq_None_not_Some. firstorder. Qed.
Lemma lookup_ne (m : M A) i j : m !! i  m !! j  i  j.
Proof. congruence. Qed.

Lemma not_elem_of_dom C `{Collection K C} (m : M A) i :
  i  dom C m  m !! i = None.
Proof. now rewrite (elem_of_dom C), eq_None_not_Some. Qed.

Lemma finmap_empty (m : M A) : ( i, m !! i = None)  m = .
Proof. intros Hm. apply finmap_eq. intros. now rewrite Hm, lookup_empty. Qed.
Lemma dom_empty C `{Collection K C} : dom C ( : M A)  .
Proof.
  split; intro.
  * rewrite (elem_of_dom C), lookup_empty. simplify_is_Some.
  * simplify_elem_of.
Qed.
Lemma dom_empty_inv C `{Collection K C} (m : M A) : dom C m    m = .
Proof.
  intros E. apply finmap_empty. intros. apply (not_elem_of_dom C).
  rewrite E. simplify_elem_of.
Qed.

Lemma lookup_empty_not i : ¬is_Some (( : M A) !! i).
Proof. rewrite lookup_empty. simplify_is_Some. Qed.
Lemma lookup_empty_Some i (x : A) : ¬ !! i = Some x.
Proof. rewrite lookup_empty. discriminate. Qed.

Lemma partial_alter_compose (m : M A) i f g :
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
  intros. apply finmap_eq. intros ii. case (decide (i = ii)).
  * intros. subst. now rewrite !lookup_partial_alter.
  * intros. now rewrite !lookup_partial_alter_ne.
Qed.
Lemma partial_alter_comm (m : M A) i j f g :
  i  j 
 partial_alter f i (partial_alter g j m) = partial_alter g j (partial_alter f i m).
Proof.
  intros. apply finmap_eq. intros jj.
  destruct (decide (jj = j)).
  * subst. now rewrite lookup_partial_alter_ne,
     !lookup_partial_alter, lookup_partial_alter_ne.
  * destruct (decide (jj = i)).
    + subst. now rewrite lookup_partial_alter,
       !lookup_partial_alter_ne, lookup_partial_alter by congruence.
    + now rewrite !lookup_partial_alter_ne by congruence.
Qed.
Lemma partial_alter_self_alt (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
  intros. apply finmap_eq. intros ii.
  destruct (decide (i = ii)).
  * subst. now rewrite lookup_partial_alter.
  * now rewrite lookup_partial_alter_ne.
Qed.
Lemma partial_alter_self (m : M A) i : partial_alter (λ _, m !! i) i m = m.
Proof. now apply partial_alter_self_alt. Qed.

Lemma lookup_insert (m : M A) i x : <[i:=x]>m !! i = Some x.
Proof. unfold insert. apply lookup_partial_alter. Qed.
Lemma lookup_insert_rev (m : M A) i x y : <[i:= x ]>m !! i = Some y  x = y.
Proof. rewrite lookup_insert. congruence. Qed.
Lemma lookup_insert_ne (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
Proof. unfold insert. apply lookup_partial_alter_ne. Qed.
Lemma insert_comm (m : M A) i j x y :
  i  j  <[i:=x]>(<[j:=y]>m) = <[j:=y]>(<[i:=x]>m).
Proof. apply partial_alter_comm. Qed.

Lemma lookup_insert_Some (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.
  * destruct (decide (i = j)); subst;
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
  * intros [[??]|[??]].
    + subst. apply lookup_insert.
    + now rewrite lookup_insert_ne.
Qed.
Lemma lookup_insert_None (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
  split.
  * destruct (decide (i = j)); subst;
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
  * intros [??]. now rewrite lookup_insert_ne.
Qed.

Lemma lookup_singleton_Some i j (x y : A) :
  {[(i, x)]} !! j = Some y  i = j  x = y.
Proof.
  unfold singleton, finmap_singleton.
  rewrite lookup_insert_Some, lookup_empty. simpl.
  intuition congruence.
Qed.
Lemma lookup_singleton_None i j (x : A) :
  {[(i, x)]} !! j = None  i  j.
Proof.
  unfold singleton, finmap_singleton.
  rewrite lookup_insert_None, lookup_empty. simpl. tauto.
Qed.

Lemma lookup_singleton i (x : A) : {[(i, x)]} !! i = Some x.
Proof. rewrite lookup_singleton_Some. tauto. Qed.
Lemma lookup_singleton_ne i j (x : A) : i  j  {[(i, x)]} !! j = None.
Proof. now rewrite lookup_singleton_None. Qed.

Lemma lookup_delete (m : M A) i : delete i m !! i = None.
Proof. apply lookup_partial_alter. Qed.
Lemma lookup_delete_ne (m : M A) i j : i  j  delete i m !! j = m !! j.
Proof. apply lookup_partial_alter_ne. Qed.

Lemma lookup_delete_Some (m : M A) i j y :
  delete i m !! j = Some y  i  j  m !! j = Some y.
Proof.
  split.
  * destruct (decide (i = j)); subst;
      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
  * intros [??]. now rewrite lookup_delete_ne.
Qed.
Lemma lookup_delete_None (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
  destruct (decide (i = j)).
  * subst. rewrite lookup_delete. tauto.
  * rewrite lookup_delete_ne; tauto.
Qed.

Lemma delete_empty i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. now rewrite lookup_empty. Qed.
Lemma delete_singleton i (x : A) : delete i {[(i, x)]} = .
Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
Lemma delete_comm (m : M A) i j : delete i (delete j m) = delete j (delete i m).
Proof. destruct (decide (i = j)). now subst. now apply partial_alter_comm. Qed.
Lemma delete_insert_comm (m : M A) i j x :
  i  j  delete i (<[j:=x]>m) = <[j:=x]>(delete i m).
Proof. intro. now apply partial_alter_comm. Qed.

Lemma delete_notin (m : M A) i : m !! i = None  delete i m = m.
Proof.
  intros. apply finmap_eq. intros j.
  destruct (decide (i = j)).
  * subst. now rewrite lookup_delete.
  * now apply lookup_delete_ne.
Qed.

Lemma delete_partial_alter (m : M A) i f :
  m !! i = None  delete i (partial_alter f i m) = m.
Proof.
  intros. unfold delete, finmap_delete. rewrite <-partial_alter_compose. 
  rapply partial_alter_self_alt. congruence.
Qed.
Lemma delete_partial_alter_dom C `{Collection K C} (m : M A) i f : 
  i  dom C m  delete i (partial_alter f i m) = m.
Proof. rewrite (not_elem_of_dom C). apply delete_partial_alter. Qed.
Lemma delete_insert (m : M A) i x : m !! i = None  delete i (<[i:=x]>m) = m.
Proof. apply delete_partial_alter. Qed.
Lemma delete_insert_dom C `{Collection K C} (m : M A) i x :
  i  dom C m  delete i (<[i:=x]>m) = m.
Proof. rewrite (not_elem_of_dom C). apply delete_partial_alter. Qed.
Lemma insert_delete (m : M A) i x : m !! i = Some x  <[i:=x]>(delete i m) = m.
Proof.
  intros Hmi. unfold delete, finmap_delete, insert, finmap_insert.
  rewrite <-partial_alter_compose. unfold compose. rewrite <-Hmi.
  now apply partial_alter_self_alt.
Qed.

Lemma elem_of_dom_delete C `{Collection K C} (m : M A) i j :
  i  dom C (delete j m)  i  j  i  dom C m.
Proof.
  rewrite !(elem_of_dom C). unfold is_Some.
  setoid_rewrite lookup_delete_Some. firstorder auto.
Qed.
Lemma not_elem_of_dom_delete C `{Collection K C} (m : M A) i :
  i  dom C (delete i m).
Proof. apply (not_elem_of_dom C), lookup_delete. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
228

229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286
Lemma lookup_delete_list (m : M A) is j :
  In j is  delete_list is m !! j = None.
Proof.
  induction is as [|i is]; simpl; [easy |].
  intros [?|?].
  * subst. now rewrite lookup_delete.
  * destruct (decide (i = j)).
    + subst. now rewrite lookup_delete.
    + rewrite lookup_delete_ne; auto.
Qed.
Lemma lookup_delete_list_notin (m : M A) is j :
  ¬In j is  delete_list is m !! j = m !! j.
Proof.
  induction is; simpl; [easy |].
  intros. rewrite lookup_delete_ne; tauto.
Qed.

Lemma delete_list_notin (m : M A) is :
  Forall (λ i, m !! i = None) is  delete_list is m = m.
Proof.
  induction 1; simpl; [easy |].
  rewrite delete_notin; congruence.
Qed.
Lemma delete_list_insert_comm (m : M A) is j x :
  ¬In j is  delete_list is (<[j:=x]>m) = <[j:=x]>(delete_list is m).
Proof.
  induction is; simpl; [easy |].
  intros. rewrite IHis, delete_insert_comm; tauto.
Qed.

Lemma finmap_ind C (P : M A  Prop) `{FinCollection K C} :
  P   ( i x m, i  dom C m  P m  P (<[i:=x]>m))   m, P m.
Proof.
  intros Hemp Hinsert.
  intros m. apply (collection_ind (λ X,  m, dom C m  X  P m)) with (dom C m).
  * solve_proper.
  * clear m. intros m Hm. rewrite finmap_empty.
    + easy.
    + intros. rewrite <-(not_elem_of_dom C), Hm.
      now simplify_elem_of.
  * clear m. intros i X Hi IH m Hdom.
    assert (is_Some (m !! i)) as [x Hx].
    { apply (elem_of_dom C).
      rewrite Hdom. clear Hdom.
      now simplify_elem_of. }
    rewrite <-(insert_delete m i x) by easy.
    apply Hinsert.
    { now apply (not_elem_of_dom_delete C). }
    apply IH. apply elem_of_equiv. intros.
    rewrite (elem_of_dom_delete C). rewrite Hdom.
    clear Hdom. simplify_elem_of.
  * easy.
Qed.

Section merge.
  Context (f : option A  option A  option A).

  Global Instance: LeftId (=) None f  LeftId (=)  (merge f).
Robbert Krebbers's avatar
Robbert Krebbers committed
287
  Proof.
288 289
    intros ??. apply finmap_eq. intros.
    now rewrite !(merge_spec f), lookup_empty, (left_id None f).
Robbert Krebbers's avatar
Robbert Krebbers committed
290
  Qed.
291
  Global Instance: RightId (=) None f  RightId (=)  (merge f).
Robbert Krebbers's avatar
Robbert Krebbers committed
292
  Proof.
293 294
    intros ??. apply finmap_eq. intros.
    now rewrite !(merge_spec f), lookup_empty, (right_id None f).
Robbert Krebbers's avatar
Robbert Krebbers committed
295
  Qed.
296 297 298 299 300 301 302
  Global Instance: Idempotent (=) f  Idempotent (=) (merge f).
  Proof. intros ??. apply finmap_eq. intros. now rewrite !(merge_spec f). Qed.

  Context `{!PropHolds (f None None = None)}.

  Lemma merge_spec_alt m1 m2 m :
    ( i, m !! i = f (m1 !! i) (m2 !! i))  merge f m1 m2 = m.
Robbert Krebbers's avatar
Robbert Krebbers committed
303
  Proof.
304 305 306
    split; [| intro; subst; apply (merge_spec _) ].
    intros Hlookup. apply finmap_eq. intros. rewrite Hlookup.
    apply (merge_spec _).
Robbert Krebbers's avatar
Robbert Krebbers committed
307
  Qed.
308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328

  Lemma merge_comm m1 m2 :
    ( i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) 
    merge f m1 m2 = merge f m2 m1.
  Proof. intros. apply finmap_eq. intros. now rewrite !(merge_spec f). Qed.
  Global Instance: Commutative (=) f  Commutative (=) (merge f).
  Proof. intros ???. apply merge_comm. intros. now apply (commutative f). Qed.

  Lemma merge_assoc m1 m2 m3 :
    ( i, f (m1 !! i) (f (m2 !! i) (m3 !! i)) = f (f (m1 !! i) (m2 !! i)) (m3 !! i))  
    merge f m1 (merge f m2 m3) = merge f (merge f m1 m2) m3.
  Proof. intros. apply finmap_eq. intros. now rewrite !(merge_spec f). Qed.
  Global Instance: Associative (=) f  Associative (=) (merge f).
  Proof. intros ????. apply merge_assoc. intros. now apply (associative f). Qed.
End merge.

Section union_intersection.
  Context (f : A  A  A).

  Lemma finmap_union_merge m1 m2 i x y :
    m1 !! i = Some x  m2 !! i = Some y  union_with f m1 m2 !! i = Some (f x y).
329
  Proof.
330 331 332 333 334
    intros Hx Hy. unfold union_with, finmap_union.
    now rewrite (merge_spec _), Hx, Hy.
  Qed.   
  Lemma finmap_union_l m1 m2 i x :
    m1 !! i = Some x  m2 !! i = None  union_with f m1 m2 !! i = Some x.
335
  Proof.
336 337
    intros Hx Hy. unfold union_with, finmap_union.
    now rewrite (merge_spec _), Hx, Hy.
338
  Qed.
339 340
  Lemma finmap_union_r m1 m2 i y :
    m1 !! i = None  m2 !! i = Some y  union_with f m1 m2 !! i = Some y.
Robbert Krebbers's avatar
Robbert Krebbers committed
341
  Proof.
342 343
    intros Hx Hy. unfold union_with, finmap_union.
    now rewrite (merge_spec _), Hx, Hy.
Robbert Krebbers's avatar
Robbert Krebbers committed
344
  Qed.
345 346
  Lemma finmap_union_None m1 m2 i :
    union_with f m1 m2 !! i = None  m1 !! i = None  m2 !! i = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
347
  Proof.
348 349
    unfold union_with, finmap_union. rewrite (merge_spec _).
    destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Robbert Krebbers's avatar
Robbert Krebbers committed
350 351
  Qed.

352 353 354 355 356 357 358 359 360
  Global Instance: LeftId (=)  (union_with f : M A  M A  M A) := _.
  Global Instance: RightId (=)  (union_with f : M A  M A  M A) := _.
  Global Instance:
    Commutative (=) f  Commutative (=) (union_with f : M A  M A  M A) := _.
  Global Instance:
    Associative (=) f  Associative (=) (union_with f : M A  M A  M A) := _.
  Global Instance:
    Idempotent (=) f  Idempotent (=) (union_with f : M A  M A  M A) := _.
End union_intersection.
Robbert Krebbers's avatar
Robbert Krebbers committed
361

362 363 364 365 366 367 368 369 370
Lemma lookup_insert_list (m : M A) l1 l2 i x :
  (y, ¬In (i,y) l1)  insert_list (l1 ++ (i,x) :: l2) m !! i = Some x.
Proof.
  induction l1 as [|[j y] l1 IH]; simpl.
   intros. now rewrite lookup_insert.
  intros Hy. rewrite lookup_insert_ne. apply IH.
   firstorder.
  intro. apply (Hy y). left. congruence.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
371

372 373 374 375 376 377 378 379 380
Lemma lookup_insert_list_not_in (m : M A) l i :
  (y, ¬In (i,y) l)  insert_list l m !! i = m !! i.
Proof.
  induction l as [|[j y] l IH]; simpl.
   easy.
  intros Hy. rewrite lookup_insert_ne.
   firstorder.
  intro. apply (Hy y). left. congruence.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
381
End finmap.
382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401

Tactic Notation "simplify_map" "by" tactic(T) := repeat
  match goal with
  | _ => progress simplify_eqs
  | H : context[  !! _ ] |- _ => rewrite lookup_empty in H
  | H : context[ (<[_:=_]>_) !! _ ] |- _ => rewrite lookup_insert in H
  | H : context[ (<[_:=_]>_) !! _ ] |- _ => rewrite lookup_insert_ne in H by T
  | H : context[ (delete _ _) !! _ ] |- _ => rewrite lookup_delete in H
  | H : context[ (delete _ _) !! _ ] |- _ => rewrite lookup_delete_ne in H by T
  | H : context[ {[ _ ]} !! _ ] |- _ => rewrite lookup_singleton in H
  | H : context[ {[ _ ]} !! _ ] |- _ => rewrite lookup_singleton_ne in H by T
  | |- context[  !! _ ] => rewrite lookup_empty
  | |- context[ (<[_:=_]>_) !! _ ] => rewrite lookup_insert
  | |- context[ (<[_:=_]>_) !! _ ] => rewrite lookup_insert_ne by T
  | |- context[ (delete _ _) !! _ ] => rewrite lookup_delete
  | |- context[ (delete _ _) !! _ ] => rewrite lookup_delete_ne by T
  | |- context[ {[ _ ]} !! _ ] => rewrite lookup_singleton
  | |- context[ {[ _ ]} !! _ ] => rewrite lookup_singleton_ne by T
  end.
Tactic Notation "simplify_map" := simplify_map by auto.