fin_maps.v 64.8 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
2 3 4
(* This file is distributed under the terms of the BSD license. *)
(** Finite maps associate data to keys. This file defines an interface for
finite maps and collects some theory on it. Most importantly, it proves useful
5 6
induction principles for finite maps and implements the tactic
[simplify_map_equality] to simplify goals involving finite maps. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
7
Require Import Permutation.
8 9
Require Export ars vector orders.

10 11
(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
12 13 14 15 16
course limits the space of finite map implementations, but since we are mainly
interested in finite maps with numbers as indexes, we do not consider this to
be a serious limitation. The main application of finite maps is to implement
the memory, where extensionality of Leibniz equality is very important for a
convenient use in the assertions of our axiomatic semantics. *)
17

Robbert Krebbers's avatar
Robbert Krebbers committed
18 19
(** Finiteness is axiomatized by requiring that each map can be translated
to an association list. The translation to association lists is used to
20
prove well founded recursion on finite maps. *)
21

22 23 24
(** Finite map implementations are required to implement the [merge] function
which enables us to give a generic implementation of [union_with],
[intersection_with], and [difference_with]. *)
25

26
Class FinMapToList K A M := map_to_list: M  list (K * A).
Robbert Krebbers's avatar
Robbert Krebbers committed
27

28 29 30
Class FinMap K M `{FMap M,  A, Lookup K A (M A),  A, Empty (M A),  A,
    PartialAlter K A (M A), OMap M, Merge M,  A, FinMapToList K A (M A),
     i j : K, Decision (i = j)} := {
31 32
  map_eq {A} (m1 m2 : M A) : ( i, m1 !! i = m2 !! i)  m1 = m2;
  lookup_empty {A} i : ( : M A) !! i = None;
33 34 35 36
  lookup_partial_alter {A} f (m : M A) i :
    partial_alter f i m !! i = f (m !! i);
  lookup_partial_alter_ne {A} f (m : M A) i j :
    i  j  partial_alter f i m !! j = m !! j;
37
  lookup_fmap {A B} (f : A  B) (m : M A) i : (f <$> m) !! i = f <$> m !! i;
38
  NoDup_map_to_list {A} (m : M A) : NoDup (map_to_list m);
39 40
  elem_of_map_to_list {A} (m : M A) i x :
    (i,x)  map_to_list m  m !! i = Some x;
41
  lookup_omap {A B} (f : A  option B) m i : omap f m !! i = m !! i = f;
42 43 44
  lookup_merge {A B C} (f : option A  option B  option C)
      `{!PropHolds (f None None = None)} m1 m2 i :
    merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
Robbert Krebbers's avatar
Robbert Krebbers committed
45 46
}.

47 48 49
(** * Derived operations *)
(** All of the following functions are defined in a generic way for arbitrary
finite map implementations. These generic implementations do not cause a
50 51
significant performance loss to make including them in the finite map interface
worthwhile. *)
52 53 54 55 56
Instance map_insert `{PartialAlter K A M} : Insert K A M :=
  λ i x, partial_alter (λ _, Some x) i.
Instance map_alter `{PartialAlter K A M} : Alter K A M :=
  λ f, partial_alter (fmap f).
Instance map_delete `{PartialAlter K A M} : Delete K M :=
57
  partial_alter (λ _, None).
58 59
Instance map_singleton `{PartialAlter K A M, Empty M} :
  Singleton (K * A) M := λ p, <[p.1:=p.2]> .
Robbert Krebbers's avatar
Robbert Krebbers committed
60

61
Definition map_of_list `{Insert K A M, Empty M} : list (K * A)  M :=
62
  fold_right (λ p, <[p.1:=p.2]>) .
63 64 65
Definition map_of_collection `{Elements K C, Insert K A M, Empty M}
    (f : K  option A) (X : C) : M :=
  map_of_list (omap (λ i, (i,) <$> f i) (elements X)).
Robbert Krebbers's avatar
Robbert Krebbers committed
66

67 68 69 70 71 72
Instance map_union_with `{Merge M} {A} : UnionWith A (M A) :=
  λ f, merge (union_with f).
Instance map_intersection_with `{Merge M} {A} : IntersectionWith A (M A) :=
  λ f, merge (intersection_with f).
Instance map_difference_with `{Merge M} {A} : DifferenceWith A (M A) :=
  λ f, merge (difference_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
73

74 75
(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
76
Definition map_Forall `{Lookup K A M} (P : K  A  Prop) : M  Prop :=
Robbert Krebbers's avatar
Robbert Krebbers committed
77
  λ m,  i x, m !! i = Some x  P i x.
78 79 80 81 82 83 84 85 86
Definition map_Forall2 `{ 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,
  match m1 !! i, m2 !! i with
  | Some x, Some y => R x y
  | Some x, None => P x
  | None, Some y => Q y
  | None, None => True
  end.
87 88
Definition map_included `{ A, Lookup K A (M A)} {A}
  (R : relation A) : relation (M A) := map_Forall2 R (λ _, False) (λ _, True).
89 90 91 92
Instance map_disjoint `{ A, Lookup K A (M A)} {A} : Disjoint (M A) :=
  map_Forall2 (λ _ _, False) (λ _, True) (λ _, True).
Instance map_subseteq `{ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
  map_Forall2 (=) (λ _, False) (λ _, True).
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
(** * Theorems *)
Section theorems.
Context `{FinMap K M}.

111 112 113 114 115 116 117 118
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_Forall2. split; intros Hm i;
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
119
Global Instance: EmptySpec (M A).
120
Proof.
121 122
  intros A m. rewrite !map_subseteq_spec.
  intros i x. by rewrite lookup_empty.
123
Qed.
124 125 126 127 128 129
Global Instance:  {A} (R : relation A), PreOrder R  PreOrder (map_included R).
Proof.
  split; [intros m i; by destruct (m !! i)|].
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
  destruct (m1 !! i), (m2 !! i), (m3 !! i); try done; etransitivity; eauto.
Qed.
130
Global Instance: PartialOrder (() : relation (M A)).
131
Proof.
132 133 134
  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
135 136 137
Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
138
Proof. rewrite !map_subseteq_spec. auto. Qed.
139 140 141 142 143 144
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.
145 146
  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
147 148
Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
149 150
  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
151 152 153 154 155 156 157 158 159
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  .
160 161 162
Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
163 164

(** ** Properties of the [partial_alter] operation *)
165 166 167
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.
168 169
  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
170 171
Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
172 173
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
174 175
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
176
Qed.
177
Lemma partial_alter_commute {A} f g (m : M A) i j :
178
  i  j  partial_alter f i (partial_alter g j m) =
179 180
    partial_alter g j (partial_alter f i m).
Proof.
181 182 183 184 185 186 187
  intros. apply map_eq; intros jj. destruct (decide (jj = j)) as [->|?].
  { by rewrite lookup_partial_alter_ne,
      !lookup_partial_alter, lookup_partial_alter_ne. }
  destruct (decide (jj = i)) as [->|?].
  * by rewrite lookup_partial_alter,
     !lookup_partial_alter_ne, lookup_partial_alter by congruence.
  * by rewrite !lookup_partial_alter_ne by congruence.
188 189 190 191
Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
192 193
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
194
Qed.
195
Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m.
196
Proof. by apply partial_alter_self_alt. Qed.
197
Lemma partial_alter_subseteq {A} f (m : M A) i :
198
  m !! i = None  m  partial_alter f i m.
199 200 201 202
Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
203
Lemma partial_alter_subset {A} f (m : M A) i :
204
  m !! i = None  is_Some (f (m !! i))  m  partial_alter f i m.
205
Proof.
206 207 208 209
  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.
210 211 212
Qed.

(** ** Properties of the [alter] operation *)
213 214
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.
215
Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal'; auto. Qed.
216
Lemma lookup_alter {A} (f : A  A) m i : alter f i m !! i = f <$> m !! i.
217
Proof. unfold alter. apply lookup_partial_alter. Qed.
218
Lemma lookup_alter_ne {A} (f : A  A) m i j : i  j  alter f i m !! j = m !! j.
219
Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
220 221 222 223 224 225 226 227 228
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.
229 230 231 232
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.
233
  destruct (decide (i = j)) as [->|?].
234 235 236 237 238 239
  * rewrite lookup_alter. naive_solver (simplify_option_equality; eauto).
  * rewrite lookup_alter_ne by done. naive_solver.
Qed.
Lemma lookup_alter_None {A} (f : A  A) m i j :
  alter f i m !! j = None  m !! j = None.
Proof.
240 241
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
242
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
243 244
Lemma alter_id {A} (f : A  A) m i :
  ( x, m !! i = Some x  f x = x)  alter f i m = m.
245
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
246 247 248
  intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
  { rewrite lookup_alter; destruct (m !! j); f_equal'; auto. }
  by rewrite lookup_alter_ne by done.
249 250 251 252 253 254 255 256 257 258 259
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.
260
  * destruct (decide (i = j)) as [->|?];
261 262 263 264 265 266
      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
  * intros [??]. by rewrite lookup_delete_ne.
Qed.
Lemma lookup_delete_None {A} (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
267 268
  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
269 270 271
Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
272
Lemma delete_singleton {A} i (x : A) : delete i {[i, x]} = .
273 274 275 276 277 278 279
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.
280
Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
281
Proof.
282 283
  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
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.
301
Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
302 303 304
Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
305
Lemma delete_subseteq_compat {A} (m1 m2 : M A) i :
306
  m1  m2  delete i m1  delete i m2.
307 308 309 310
Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
Qed.
311
Lemma delete_subset_alt {A} (m : M A) i x : m !! i = Some x  delete i m  m.
312
Proof.
313 314 315
  split; [apply delete_subseteq|].
  rewrite !map_subseteq_spec. intros Hi. apply (None_ne_Some x).
  by rewrite <-(lookup_delete m i), (Hi i x).
316
Qed.
317
Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
318 319 320 321 322
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.
323
Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
324
Proof. rewrite lookup_insert. congruence. Qed.
325
Lemma lookup_insert_ne {A} (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
326 327 328 329 330 331 332 333
Proof. unfold insert. apply lookup_partial_alter_ne. Qed.
Lemma insert_commute {A} (m : M A) i j x y :
  i  j  <[i:=x]>(<[j:=y]>m) = <[j:=y]>(<[i:=x]>m).
Proof. apply partial_alter_commute. Qed.
Lemma lookup_insert_Some {A} (m : M A) i j x y :
  <[i:=x]>m !! j = Some y  (i = j  x = y)  (i  j  m !! j = Some y).
Proof.
  split.
334
  * destruct (decide (i = j)) as [->|?];
335
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
336
  * intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
337 338 339 340
Qed.
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
341 342 343
  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
344
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
345
Lemma insert_id {A} (m : M A) i x : m !! i = Some x  <[i:=x]>m = m.
346 347 348 349 350 351 352 353 354 355 356
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 [->|].
  * rewrite lookup_insert. destruct (m !! j); eauto.
  * rewrite lookup_insert_ne by done. by destruct (m !! j).
Qed.
357
Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
358
Proof. apply partial_alter_subseteq. Qed.
359
Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
360 361
Proof. intro. apply partial_alter_subset; eauto. Qed.
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
362
  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
363
Proof.
364 365 366
  rewrite !map_subseteq_spec. intros ?? j ?.
  destruct (decide (j = i)) as [->|?]; [congruence|].
  rewrite lookup_insert_ne; auto.
367 368
Qed.
Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
369
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
370
Proof.
371 372 373 374
  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.
375 376
Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
377
  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
378
Proof.
379 380
  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
381
  * rewrite lookup_insert. congruence.
382
  * rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
383 384
Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
385
  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
386
Proof.
387 388 389
  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
390 391
Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
392
  m1 !! i = None  <[i:=x]> m1  m2 
393 394 395
   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
  intros Hi Hm1m2. exists (delete i m2). split_ands.
396
  * rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto.
397 398 399 400
    by rewrite lookup_insert.
  * eauto using insert_delete_subset.
  * by rewrite lookup_delete.
Qed.
401 402 403 404 405 406 407
Lemma fmap_insert {A B} (f : A  B) (m : M A) i x :
  f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  * by rewrite lookup_fmap, !lookup_insert.
  * by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
Qed.
408 409
Lemma insert_empty {A} i (x : A) : <[i:=x]> = {[i,x]}.
Proof. done. Qed.
410 411 412

(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
413
  {[i, x]} !! j = Some y  i = j  x = y.
414 415
Proof.
  unfold singleton, map_singleton.
416
  rewrite lookup_insert_Some, lookup_empty. simpl. intuition congruence.
417
Qed.
418
Lemma lookup_singleton_None {A} i j (x : A) : {[i, x]} !! j = None  i  j.
419 420 421 422
Proof.
  unfold singleton, map_singleton.
  rewrite lookup_insert_None, lookup_empty. simpl. tauto.
Qed.
423
Lemma lookup_singleton {A} i (x : A) : {[i, x]} !! i = Some x.
424
Proof. by rewrite lookup_singleton_Some. Qed.
425
Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[i, x]} !! j = None.
426
Proof. by rewrite lookup_singleton_None. Qed.
427
Lemma map_non_empty_singleton {A} i (x : A) : {[i,x]}  .
428 429 430 431
Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
432
Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i, x]} = {[i, y]}.
433 434 435 436
Proof.
  unfold singleton, map_singleton, insert, map_insert.
  by rewrite <-partial_alter_compose.
Qed.
437
Lemma alter_singleton {A} (f : A  A) i x : alter f i {[i,x]} = {[i, f x]}.
438
Proof.
439
  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
440 441 442 443
  * by rewrite lookup_alter, !lookup_singleton.
  * by rewrite lookup_alter_ne, !lookup_singleton_ne.
Qed.
Lemma alter_singleton_ne {A} (f : A  A) i j x :
444
  i  j  alter f i {[j,x]} = {[j,x]}.
445
Proof.
446 447
  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
448 449
Qed.

450 451 452 453 454
(** ** 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.
455 456 457 458 459 460 461
Lemma omap_singleton {A B} (f : A  option B) i x y :
  f x = Some y  omap f {[ i,x ]} = {[ i,y ]}.
Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|].
  * by rewrite lookup_omap, !lookup_singleton.
  * by rewrite lookup_omap, !lookup_singleton_ne.
Qed.
462

463 464
(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
465
  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
466
Proof. rewrite !elem_of_map_to_list. congruence. Qed.
467
Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
468
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
469 470 471 472 473 474 475 476 477 478
Lemma elem_of_map_of_list_1_help {A} (l : list (K * A)) i x :
  (i,x)  l  ( y, (i,y)  l  y = x)  map_of_list l !! i = Some x.
Proof.
  induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|].
  setoid_rewrite elem_of_cons.
  intros [?|?] Hdup; simplify_equality; [by rewrite lookup_insert|].
  destruct (decide (i = j)) as [->|].
  * rewrite lookup_insert; f_equal; eauto.
  * rewrite lookup_insert_ne by done; eauto.
Qed.
479
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
480
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
481
Proof.
482 483
  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'].
484
  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
485
    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
486 487
Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
488
  map_of_list l !! i = Some x  (i,x)  l.
489
Proof.
490 491 492
  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.
493 494
Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
495
  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
496
Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
497
Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
498
  i  l.*1  map_of_list l !! i = None.
499
Proof.
500 501
  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.
502 503
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
504
  map_of_list l !! i = None  i  l.*1.
505
Proof.
506
  induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
507 508 509 510 511
  rewrite elem_of_cons. destruct (decide (i = j)); simplify_equality.
  * by rewrite lookup_insert.
  * by rewrite lookup_insert_ne; intuition.
Qed.
Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
512
  i  l.*1  map_of_list l !! i = None.
513
Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
514
Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
515
  NoDup (l1.*1)  l1  l2  map_of_list l1 = map_of_list l2.
516 517 518 519 520
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)) :
521
  NoDup (l1.*1)  NoDup (l2.*1)  map_of_list l1 = map_of_list l2  l1  l2.
522
Proof.
523
  intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
524 525
  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
526
Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m.
527 528 529
Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
530
    by auto using NoDup_fst_map_to_list.
531 532
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
533
  NoDup (l.*1)  map_to_list (map_of_list l)  l.
534
Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
535
Lemma map_to_list_inj {A} (m1 m2 : M A) :
536
  map_to_list m1  map_to_list m2  m1 = m2.
537
Proof.
538
  intros. rewrite <-(map_of_to_list m1), <-(map_of_to_list m2).
539
  auto using map_of_list_proper, NoDup_fst_map_to_list.
540
Qed.
541 542 543 544 545 546
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.
547
Lemma map_to_list_empty {A} : map_to_list  = @nil (K * A).
548 549 550 551 552
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 :
553
  m !! i = None  map_to_list (<[i:=x]>m)  (i,x) :: map_to_list m.
554
Proof.
555
  intros. apply map_of_list_inj; csimpl.
556 557
  * apply NoDup_fst_map_to_list.
  * constructor; auto using NoDup_fst_map_to_list.
558
    rewrite elem_of_list_fmap. intros [[??] [? Hlookup]]; subst; simpl in *.
559 560 561
    rewrite elem_of_map_to_list in Hlookup. congruence.
  * by rewrite !map_of_to_list.
Qed.
562
Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = .
563 564 565 566
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.
567
Lemma map_to_list_empty_inv_alt {A}  (m : M A) : map_to_list m  []  m = .
568
Proof. rewrite <-map_to_list_empty. apply map_to_list_inj. Qed.
569
Lemma map_to_list_empty_inv {A} (m : M A) : map_to_list m = []  m = .
570 571
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 :
572
  map_to_list m  (i,x) :: l  m = <[i:=x]>(map_of_list l).
573 574
Proof.
  intros Hperm. apply map_to_list_inj.
575 576 577
  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. }
578 579 580
  rewrite Hperm, map_to_list_insert, map_to_of_list;
    auto using not_elem_of_map_of_list_1.
Qed.
581 582 583 584 585 586
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. }
  exists i x. rewrite <-elem_of_map_to_list, Hm. by left.
Qed.
587

588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604
(** ** 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.
    by intros (?&?&?&?&?); simplify_option_equality. }
  unfold map_of_collection; rewrite <-elem_of_map_of_list by done.
  rewrite elem_of_list_omap. setoid_rewrite elem_of_elements; split.
  * intros (?&?&?); simplify_option_equality; eauto.
  * intros [??]; exists i; simplify_option_equality; eauto.
Qed.

(** ** Induction principles *)
605
Lemma map_ind {A} (P : M A  Prop) :
606
  P   ( i x m, m !! i = None  P m  P (<[i:=x]>m))   m, P m.
607
Proof.
608
  intros ? Hins. cut ( l, NoDup (l.*1)   m, map_to_list m  l  P m).
609
  { intros help m.
610
    apply (help (map_to_list m)); auto using NoDup_fst_map_to_list. }
611 612 613
  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.
614
  apply map_to_list_insert_inv in Hml; subst m. apply Hins.
615 616 617 618
  * by apply not_elem_of_map_of_list_1.
  * apply IH; auto using map_to_of_list.
Qed.
Lemma map_to_list_length {A} (m1 m2 : M A) :
619
  m1  m2  length (map_to_list m1) < length (map_to_list m2).
620 621 622 623
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.
624
    apply nil_length_inv, map_to_list_empty_inv in Hlen.
625 626 627 628 629
    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.
630
Lemma map_wf {A} : wf (strict (@subseteq (M A) _)).
631 632 633 634 635 636
Proof.
  apply (wf_projected (<) (length  map_to_list)).
  * by apply map_to_list_length.
  * by apply lt_wf.
Qed.

637
(** ** Properties of the [map_Forall] predicate *)
638
Section map_Forall.
639 640
Context {A} (P : K  A  Prop).

641
Lemma map_Forall_to_list m : map_Forall P m  Forall (curry P) (map_to_list m).
642 643
Proof.
  rewrite Forall_forall. split.
644 645
  * 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)).
646
Qed.
647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675
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.
676 677

Context `{ i x, Decision (P i x)}.
678
Global Instance map_Forall_dec m : Decision (map_Forall P m).
679 680
Proof.
  refine (cast_if (decide (Forall (curry P) (map_to_list m))));
681
    by rewrite map_Forall_to_list.
682
Defined.
683 684
Lemma map_not_Forall (m : M A) :
  ¬map_Forall P m   i x, m !! i = Some x  ¬P i x.
685
Proof.
686 687 688 689
  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.
  destruct Hm as ([i x]&?&?). exists i x. by rewrite <-elem_of_map_to_list.
690
Qed.
691
End map_Forall.
692 693 694 695

(** ** Properties of the [merge] operation *)
Section merge.
Context {A} (f : option A  option A  option A).
696
Context `{!PropHolds (f None None = None)}.
697 698 699
Global Instance: LeftId (=) None f  LeftId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
700
  by rewrite !(lookup_merge f), lookup_empty, (left_id_L None f).
701 702 703 704
Qed.
Global Instance: RightId (=) None f  RightId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
705
  by rewrite !(lookup_merge f), lookup_empty, (right_id_L None f).
706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721
Qed.
Lemma merge_commutative m1 m2 :
  ( 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.
Global Instance: Commutative (=) f  Commutative (=) (merge f).
Proof.
  intros ???. apply merge_commutative. intros. by apply (commutative f).
Qed.
Lemma merge_associative 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 map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Associative (=) f  Associative (=) (merge f).
Proof.
722
  intros ????. apply merge_associative. intros. by apply (associative_L f).
723 724
Qed.
Lemma merge_idempotent m1 :
725
  ( i, f (m1 !! i) (m1 !! i) = m1 !! i)  merge f m1 m1 = m1.
726 727
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Idempotent (=) f  Idempotent (=) (merge f).
728
Proof. intros ??. apply merge_idempotent. intros. by apply (idempotent f). Qed.
729
End merge.
730

731 732 733 734 735 736 737 738 739 740 741 742
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 :
743 744 745 746 747 748 749 750
  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.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
751
Lemma partial_alter_merge_l g g1 m1 m2 i :
752 753 754 755 756 757 758
  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.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
759
Lemma partial_alter_merge_r g g2 m1 m2 i :
760 761 762 763 764 765 766
  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.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785
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 :
  f (Some y) (Some z) = Some x  merge f {[i,y]} {[i,z]} = {[i,x]}.
Proof.
  intros. unfold singleton, map_singleton; simpl.
  by erewrite <-insert_merge, merge_empty by eauto.
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.
786

787 788 789 790 791 792 793 794 795 796 797 798 799
(** ** Properties on the [map_Forall2] 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_Forall2_alt (m1 : M A) (m2 : M B) :
800
  map_Forall2 R P Q m1 m2  map_Forall (λ _, Is_true) (merge f m1 m2).
801 802
Proof.
  split.
803 804
  * intros Hm i P'; rewrite lookup_merge by done; intros.
    specialize (Hm i). destruct (m1 !! i), (m2 !! i);
805
      simplify_equality'; auto using bool_decide_pack.
806 807 808 809 810 811 812
  * intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done.
    destruct (m1 !! i), (m2 !! i); simplify_equality'; auto;
      by eapply bool_decide_unpack, Hm.
Qed.
Global Instance map_Forall2_dec `{ x y, Decision (R x y),  x, Decision (P x),
   y, Decision (Q y)} m1 m2 : Decision (map_Forall2 R P Q m1 m2).
Proof.
813
  refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2))));
Robbert Krebbers's avatar