collections.v 47.6 KB
Newer Older
1
(* Copyright (c) 2012-2017, Coq-std++ developers. *)
2 3 4 5
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on collections. Most
importantly, it implements some tactics to automatically solve goals involving
collections. *)
6
From stdpp Require Export orders list.
7 8
(* FIXME: This file needs a 'Proof Using' hint, but the default we use
   everywhere makes for lots of extra ssumptions. *)
9

10 11 12
(* Higher precedence to make sure these instances are not used for other types
with an [ElemOf] instance, such as lists. *)
Instance collection_equiv `{ElemOf A C} : Equiv C | 20 := λ X Y,
13
   x, x  X  x  Y.
14
Instance collection_subseteq `{ElemOf A C} : SubsetEq C | 20 := λ X Y,
15
   x, x  X  x  Y.
16
Instance collection_disjoint `{ElemOf A C} : Disjoint C | 20 := λ X Y,
17 18
   x, x  X  x  Y  False.
Typeclasses Opaque collection_equiv collection_subseteq collection_disjoint.
19

20 21
(** * Setoids *)
Section setoids_simple.
22
  Context `{SimpleCollection A C}.
Robbert Krebbers's avatar
Robbert Krebbers committed
23

24
  Global Instance collection_equivalence : Equivalence (@{C}).
25
  Proof.
26 27 28 29
    split.
    - done.
    - intros X Y ? x. by symmetry.
    - intros X Y Z ?? x; by trans (x  Y).
30
  Qed.
31
  Global Instance singleton_proper : Proper ((=) ==> (@{C})) singleton.
32
  Proof. apply _. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
33
  Global Instance elem_of_proper : Proper ((=) ==> () ==> iff) (@{C}) | 5.
34
  Proof. by intros x ? <- X Y. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
35
  Global Instance disjoint_proper: Proper (() ==> () ==> iff) (##@{C}).
36
  Proof.
37
    intros X1 X2 HX Y1 Y2 HY; apply forall_proper; intros x. by rewrite HX, HY.
38
  Qed.
39
  Global Instance union_proper : Proper (() ==> () ==> (@{C})) union.
40
  Proof. intros X1 X2 HX Y1 Y2 HY x. rewrite !elem_of_union. f_equiv; auto. Qed.
41
  Global Instance union_list_proper: Proper (() ==> (@{C})) union_list.
42
  Proof. by induction 1; simpl; try apply union_proper. Qed.
43
  Global Instance subseteq_proper : Proper ((@{C}) ==> (@{C}) ==> iff) ().
44 45 46 47 48 49 50
  Proof.
    intros X1 X2 HX Y1 Y2 HY. apply forall_proper; intros x. by rewrite HX, HY.
  Qed.
End setoids_simple.

Section setoids.
  Context `{Collection A C}.
51

52 53
  (** * Setoids *)
  Global Instance intersection_proper :
54
    Proper (() ==> () ==> (@{C})) intersection.
55
  Proof.
56
    intros X1 X2 HX Y1 Y2 HY x. by rewrite !elem_of_intersection, HX, HY.
57
  Qed.
58
  Global Instance difference_proper :
59
     Proper (() ==> () ==> (@{C})) difference.
60
  Proof.
61
    intros X1 X2 HX Y1 Y2 HY x. by rewrite !elem_of_difference, HX, HY.
62
  Qed.
63
End setoids.
Robbert Krebbers's avatar
Robbert Krebbers committed
64

65 66 67 68 69
Section setoids_monad.
  Context `{CollectionMonad M}.

  Global Instance collection_fmap_proper {A B} :
    Proper (pointwise_relation _ (=) ==> () ==> ()) (@fmap M _ A B).
70
  Proof.
71 72
    intros f1 f2 Hf X1 X2 HX x. rewrite !elem_of_fmap. f_equiv; intros z.
    by rewrite HX, Hf.
73
  Qed.
74
  Global Instance collection_bind_proper {A B} :
75
    Proper (pointwise_relation _ () ==> () ==> ()) (@mbind M _ A B).
76 77
  Proof.
    intros f1 f2 Hf X1 X2 HX x. rewrite !elem_of_bind. f_equiv; intros z.
78
    by rewrite HX, (Hf z).
79 80 81 82 83 84 85
  Qed.
  Global Instance collection_join_proper {A} :
    Proper (() ==> ()) (@mjoin M _ A).
  Proof.
    intros X1 X2 HX x. rewrite !elem_of_join. f_equiv; intros z. by rewrite HX.
  Qed.
End setoids_monad.
86

87 88 89 90 91
(** * Tactics *)
(** The tactic [set_unfold] transforms all occurrences of [(∪)], [(∩)], [(∖)],
[(<$>)], [∅], [{[_]}], [(≡)], and [(⊆)] into logically equivalent propositions
involving just [∈]. For example, [A → x ∈ X ∪ ∅] becomes [A → x ∈ X ∨ False].

92 93 94
This transformation is implemented using type classes instead of setoid
rewriting to ensure that we traverse each term at most once and to be able to
deal with occurences of the set operations under binders. *)
95
Class SetUnfold (P Q : Prop) := { set_unfold : P  Q }.
96
Arguments set_unfold _ _ {_} : assert.
97 98 99 100 101
Hint Mode SetUnfold + - : typeclass_instances.

Class SetUnfoldSimpl (P Q : Prop) := { set_unfold_simpl : SetUnfold P Q }.
Hint Extern 0 (SetUnfoldSimpl _ _) => csimpl; constructor : typeclass_instances.

102
Instance set_unfold_default P : SetUnfold P P | 1000. done. Qed.
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
Definition set_unfold_1 `{SetUnfold P Q} : P  Q := proj1 (set_unfold P Q).
Definition set_unfold_2 `{SetUnfold P Q} : Q  P := proj2 (set_unfold P Q).

Lemma set_unfold_impl P Q P' Q' :
  SetUnfold P P'  SetUnfold Q Q'  SetUnfold (P  Q) (P'  Q').
Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed.
Lemma set_unfold_and P Q P' Q' :
  SetUnfold P P'  SetUnfold Q Q'  SetUnfold (P  Q) (P'  Q').
Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed.
Lemma set_unfold_or P Q P' Q' :
  SetUnfold P P'  SetUnfold Q Q'  SetUnfold (P  Q) (P'  Q').
Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed.
Lemma set_unfold_iff P Q P' Q' :
  SetUnfold P P'  SetUnfold Q Q'  SetUnfold (P  Q) (P'  Q').
Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed.
Lemma set_unfold_not P P' : SetUnfold P P'  SetUnfold (¬P) (¬P').
Proof. constructor. by rewrite (set_unfold P P'). Qed.
Lemma set_unfold_forall {A} (P P' : A  Prop) :
  ( x, SetUnfold (P x) (P' x))  SetUnfold ( x, P x) ( x, P' x).
Proof. constructor. naive_solver. Qed.
Lemma set_unfold_exist {A} (P P' : A  Prop) :
  ( x, SetUnfold (P x) (P' x))  SetUnfold ( x, P x) ( x, P' x).
Proof. constructor. naive_solver. Qed.

(* Avoid too eager application of the above instances (and thus too eager
unfolding of type class transparent definitions). *)
Hint Extern 0 (SetUnfold (_  _) _) =>
  class_apply set_unfold_impl : typeclass_instances.
Hint Extern 0 (SetUnfold (_  _) _) =>
  class_apply set_unfold_and : typeclass_instances.
Hint Extern 0 (SetUnfold (_  _) _) =>
  class_apply set_unfold_or : typeclass_instances.
Hint Extern 0 (SetUnfold (_  _) _) =>
  class_apply set_unfold_iff : typeclass_instances.
Hint Extern 0 (SetUnfold (¬ _) _) =>
  class_apply set_unfold_not : typeclass_instances.
Hint Extern 1 (SetUnfold ( _, _) _) =>
  class_apply set_unfold_forall : typeclass_instances.
Hint Extern 0 (SetUnfold ( _, _) _) =>
  class_apply set_unfold_exist : typeclass_instances.

Section set_unfold_simple.
  Context `{SimpleCollection A C}.
  Implicit Types x y : A.
  Implicit Types X Y : C.

149
  Global Instance set_unfold_empty x : SetUnfold (x  ( : C)) False.
150
  Proof. constructor. split. apply not_elem_of_empty. done. Qed.
151
  Global Instance set_unfold_singleton x y : SetUnfold (x  ({[ y ]} : C)) (x = y).
152 153 154 155 156 157 158 159 160 161 162 163
  Proof. constructor; apply elem_of_singleton. Qed.
  Global Instance set_unfold_union x X Y P Q :
    SetUnfold (x  X) P  SetUnfold (x  Y) Q  SetUnfold (x  X  Y) (P  Q).
  Proof.
    intros ??; constructor.
    by rewrite elem_of_union, (set_unfold (x  X) P), (set_unfold (x  Y) Q).
  Qed.
  Global Instance set_unfold_equiv_same X : SetUnfold (X  X) True | 1.
  Proof. done. Qed.
  Global Instance set_unfold_equiv_empty_l X (P : A  Prop) :
    ( x, SetUnfold (x  X) (P x))  SetUnfold (  X) ( x, ¬P x) | 5.
  Proof.
164
    intros ?; constructor. unfold equiv, collection_equiv.
165
    pose proof (not_elem_of_empty (C:=C)); naive_solver.
166
  Qed.
167
  Global Instance set_unfold_equiv_empty_r (P : A  Prop) X :
168
    ( x, SetUnfold (x  X) (P x))  SetUnfold (X  ) ( x, ¬P x) | 5.
169 170
  Proof.
    intros ?; constructor. unfold equiv, collection_equiv.
171
    pose proof (not_elem_of_empty (C:=C)); naive_solver.
172
  Qed.
173
  Global Instance set_unfold_equiv (P Q : A  Prop) X :
174 175
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
    SetUnfold (X  Y) ( x, P x  Q x) | 10.
176
  Proof. constructor. apply forall_proper; naive_solver. Qed.
177
  Global Instance set_unfold_subseteq (P Q : A  Prop) X Y :
178 179
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
    SetUnfold (X  Y) ( x, P x  Q x).
180
  Proof. constructor. apply forall_proper; naive_solver. Qed.
181
  Global Instance set_unfold_subset (P Q : A  Prop) X :
182
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
183
    SetUnfold (X  Y) (( x, P x  Q x)  ¬∀ x, Q x  P x).
184
  Proof.
185 186
    constructor. unfold strict.
    repeat f_equiv; apply forall_proper; naive_solver.
187
  Qed.
188
  Global Instance set_unfold_disjoint (P Q : A  Prop) X Y :
Robbert Krebbers's avatar
Robbert Krebbers committed
189
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
190
    SetUnfold (X ## Y) ( x, P x  Q x  False).
191
  Proof. constructor. unfold disjoint, collection_disjoint. naive_solver. Qed.
192 193 194 195 196 197

  Context `{!LeibnizEquiv C}.
  Global Instance set_unfold_equiv_same_L X : SetUnfold (X = X) True | 1.
  Proof. done. Qed.
  Global Instance set_unfold_equiv_empty_l_L X (P : A  Prop) :
    ( x, SetUnfold (x  X) (P x))  SetUnfold ( = X) ( x, ¬P x) | 5.
198
  Proof. constructor. unfold_leibniz. by apply set_unfold_equiv_empty_l. Qed.
199
  Global Instance set_unfold_equiv_empty_r_L (P : A  Prop) X :
200
    ( x, SetUnfold (x  X) (P x))  SetUnfold (X = ) ( x, ¬P x) | 5.
201
  Proof. constructor. unfold_leibniz. by apply set_unfold_equiv_empty_r. Qed.
202
  Global Instance set_unfold_equiv_L (P Q : A  Prop) X Y :
203 204
    ( x, SetUnfold (x  X) (P x))  ( x, SetUnfold (x  Y) (Q x)) 
    SetUnfold (X = Y) ( x, P x  Q x) | 10.
205
  Proof. constructor. unfold_leibniz. by apply set_unfold_equiv. Qed.
206 207 208 209 210 211 212 213 214 215
End set_unfold_simple.

Section set_unfold.
  Context `{Collection A C}.
  Implicit Types x y : A.
  Implicit Types X Y : C.

  Global Instance set_unfold_intersection x X Y P Q :
    SetUnfold (x  X) P  SetUnfold (x  Y) Q  SetUnfold (x  X  Y) (P  Q).
  Proof.
216 217
    intros ??; constructor. rewrite elem_of_intersection.
    by rewrite (set_unfold (x  X) P), (set_unfold (x  Y) Q).
218 219 220 221
  Qed.
  Global Instance set_unfold_difference x X Y P Q :
    SetUnfold (x  X) P  SetUnfold (x  Y) Q  SetUnfold (x  X  Y) (P  ¬Q).
  Proof.
222 223
    intros ??; constructor. rewrite elem_of_difference.
    by rewrite (set_unfold (x  X) P), (set_unfold (x  Y) Q).
224 225 226 227
  Qed.
End set_unfold.

Section set_unfold_monad.
228
  Context `{CollectionMonad M}.
229

230 231
  Global Instance set_unfold_ret {A} (x y : A) :
    SetUnfold (x  mret (M:=M) y) (x = y).
232
  Proof. constructor; apply elem_of_ret. Qed.
233
  Global Instance set_unfold_bind {A B} (f : A  M B) X (P Q : A  Prop) :
234 235 236
    ( y, SetUnfold (y  X) (P y))  ( y, SetUnfold (x  f y) (Q y)) 
    SetUnfold (x  X = f) ( y, Q y  P y).
  Proof. constructor. rewrite elem_of_bind; naive_solver. Qed.
237
  Global Instance set_unfold_fmap {A B} (f : A  B) (X : M A) (P : A  Prop) :
238 239 240
    ( y, SetUnfold (y  X) (P y)) 
    SetUnfold (x  f <$> X) ( y, x = f y  P y).
  Proof. constructor. rewrite elem_of_fmap; naive_solver. Qed.
241
  Global Instance set_unfold_join {A} (X : M (M A)) (P : M A  Prop) :
242 243 244 245
    ( Y, SetUnfold (Y  X) (P Y))  SetUnfold (x  mjoin X) ( Y, x  Y  P Y).
  Proof. constructor. rewrite elem_of_join; naive_solver. Qed.
End set_unfold_monad.

Robbert Krebbers's avatar
Robbert Krebbers committed
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
Section set_unfold_list.
  Context {A : Type}.
  Implicit Types x : A.
  Implicit Types l : list A.

  Global Instance set_unfold_nil x : SetUnfold (x  []) False.
  Proof. constructor; apply elem_of_nil. Qed.
  Global Instance set_unfold_cons x y l P :
    SetUnfold (x  l) P  SetUnfold (x  y :: l) (x = y  P).
  Proof. constructor. by rewrite elem_of_cons, (set_unfold (x  l) P). Qed.
  Global Instance set_unfold_app x l k P Q :
    SetUnfold (x  l) P  SetUnfold (x  k) Q  SetUnfold (x  l ++ k) (P  Q).
  Proof.
    intros ??; constructor.
    by rewrite elem_of_app, (set_unfold (x  l) P), (set_unfold (x  k) Q).
  Qed.
  Global Instance set_unfold_included l k (P Q : A  Prop) :
    ( x, SetUnfold (x  l) (P x))  ( x, SetUnfold (x  k) (Q x)) 
    SetUnfold (l  k) ( x, P x  Q x).
  Proof.
    constructor; unfold subseteq, list_subseteq.
    apply forall_proper; naive_solver.
  Qed.
End set_unfold_list.

271 272 273
Ltac set_unfold :=
  let rec unfold_hyps :=
    try match goal with
274 275 276 277 278 279 280
    | H : ?P |- _ =>
       lazymatch type of P with
       | Prop =>
         apply set_unfold_1 in H; revert H;
         first [unfold_hyps; intros H | intros H; fail 1]
       | _ => fail
       end
281 282 283
    end in
  apply set_unfold_2; unfold_hyps; csimpl in *.

284 285
(** Since [firstorder] already fails or loops on very small goals generated by
[set_solver], we use the [naive_solver] tactic as a substitute. *)
286
Tactic Notation "set_solver" "by" tactic3(tac) :=
287
  try fast_done;
288 289 290 291 292 293 294 295 296 297 298 299 300
  intros; setoid_subst;
  set_unfold;
  intros; setoid_subst;
  try match goal with |- _  _ => apply dec_stable end;
  naive_solver tac.
Tactic Notation "set_solver" "-" hyp_list(Hs) "by" tactic3(tac) :=
  clear Hs; set_solver by tac.
Tactic Notation "set_solver" "+" hyp_list(Hs) "by" tactic3(tac) :=
  clear -Hs; set_solver by tac.
Tactic Notation "set_solver" := set_solver by idtac.
Tactic Notation "set_solver" "-" hyp_list(Hs) := clear Hs; set_solver.
Tactic Notation "set_solver" "+" hyp_list(Hs) := clear -Hs; set_solver.

301 302 303 304
Hint Extern 1000 (_  _) => set_solver : set_solver.
Hint Extern 1000 (_  _) => set_solver : set_solver.
Hint Extern 1000 (_  _) => set_solver : set_solver.

305

306 307
(** * Collections with [∪], [∅] and [{[_]}] *)
Section simple_collection.
308
  Context `{SimpleCollection A C}.
309 310 311 312 313 314 315 316 317 318 319
  Implicit Types x y : A.
  Implicit Types X Y : C.
  Implicit Types Xs Ys : list C.

  (** Equality *)
  Lemma elem_of_equiv X Y : X  Y   x, x  X  x  Y.
  Proof. set_solver. Qed.
  Lemma collection_equiv_spec X Y : X  Y  X  Y  Y  X.
  Proof. set_solver. Qed.

  (** Subset relation *)
320
  Global Instance collection_subseteq_antisymm: AntiSymm () (@{C}).
321 322
  Proof. intros ??. set_solver. Qed.

323
  Global Instance collection_subseteq_preorder: PreOrder (@{C}).
324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345
  Proof. split. by intros ??. intros ???; set_solver. Qed.

  Lemma subseteq_union X Y : X  Y  X  Y  Y.
  Proof. set_solver. Qed.
  Lemma subseteq_union_1 X Y : X  Y  X  Y  Y.
  Proof. by rewrite subseteq_union. Qed.
  Lemma subseteq_union_2 X Y : X  Y  Y  X  Y.
  Proof. by rewrite subseteq_union. Qed.

  Lemma union_subseteq_l X Y : X  X  Y.
  Proof. set_solver. Qed.
  Lemma union_subseteq_r X Y : Y  X  Y.
  Proof. set_solver. Qed.
  Lemma union_least X Y Z : X  Z  Y  Z  X  Y  Z.
  Proof. set_solver. Qed.

  Lemma elem_of_subseteq X Y : X  Y   x, x  X  x  Y.
  Proof. done. Qed.
  Lemma elem_of_subset X Y : X  Y  ( x, x  X  x  Y)  ¬( x, x  Y  x  X).
  Proof. set_solver. Qed.

  (** Union *)
346 347
  Lemma union_subseteq X Y Z : X  Y  Z  X  Z  Y  Z.
  Proof. set_solver. Qed.
348 349 350 351 352 353
  Lemma not_elem_of_union x X Y : x  X  Y  x  X  x  Y.
  Proof. set_solver. Qed.
  Lemma elem_of_union_l x X Y : x  X  x  X  Y.
  Proof. set_solver. Qed.
  Lemma elem_of_union_r x X Y : x  Y  x  X  Y.
  Proof. set_solver. Qed.
354
  Lemma union_mono_l X Y1 Y2 : Y1  Y2  X  Y1  X  Y2.
355
  Proof. set_solver. Qed.
356
  Lemma union_mono_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
357
  Proof. set_solver. Qed.
358
  Lemma union_mono X1 X2 Y1 Y2 : X1  X2  Y1  Y2  X1  Y1  X2  Y2.
359 360
  Proof. set_solver. Qed.

361
  Global Instance union_idemp : IdemP (@{C}) ().
362
  Proof. intros X. set_solver. Qed.
363
  Global Instance union_empty_l : LeftId (@{C})  ().
364
  Proof. intros X. set_solver. Qed.
365
  Global Instance union_empty_r : RightId (@{C})  ().
366
  Proof. intros X. set_solver. Qed.
367
  Global Instance union_comm : Comm (@{C}) ().
368
  Proof. intros X Y. set_solver. Qed.
369
  Global Instance union_assoc : Assoc (@{C}) ().
370 371 372 373 374
  Proof. intros X Y Z. set_solver. Qed.

  Lemma empty_union X Y : X  Y    X    Y  .
  Proof. set_solver. Qed.

375
  Lemma union_cancel_l X Y Z : Z ## X  Z ## Y  Z  X  Z  Y  X  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
376
  Proof. set_solver. Qed.
377
  Lemma union_cancel_r X Y Z : X ## Z  Y ## Z  X  Z  Y  Z  X  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
378 379
  Proof. set_solver. Qed.

380
  (** Empty *)
Robbert Krebbers's avatar
Robbert Krebbers committed
381 382
  Lemma empty_subseteq X :   X.
  Proof. set_solver. Qed.
383 384
  Lemma elem_of_equiv_empty X : X     x, x  X.
  Proof. set_solver. Qed.
385
  Lemma elem_of_empty x : x  ( : C)  False.
386 387 388 389 390 391 392 393 394 395 396
  Proof. set_solver. Qed.
  Lemma equiv_empty X : X    X  .
  Proof. set_solver. Qed.
  Lemma union_positive_l X Y : X  Y    X  .
  Proof. set_solver. Qed.
  Lemma union_positive_l_alt X Y : X    X  Y  .
  Proof. set_solver. Qed.
  Lemma non_empty_inhabited x X : x  X  X  .
  Proof. set_solver. Qed.

  (** Singleton *)
397
  Lemma elem_of_singleton_1 x y : x  ({[y]} : C)  x = y.
398
  Proof. by rewrite elem_of_singleton. Qed.
399
  Lemma elem_of_singleton_2 x y : x = y  x  ({[y]} : C).
400 401 402 403 404
  Proof. by rewrite elem_of_singleton. Qed.
  Lemma elem_of_subseteq_singleton x X : x  X  {[ x ]}  X.
  Proof. set_solver. Qed.
  Lemma non_empty_singleton x : ({[ x ]} : C)  .
  Proof. set_solver. Qed.
405
  Lemma not_elem_of_singleton x y : x  ({[ y ]} : C)  x  y.
406 407 408
  Proof. by rewrite elem_of_singleton. Qed.

  (** Disjointness *)
409
  Lemma elem_of_disjoint X Y : X ## Y   x, x  X  x  Y  False.
410 411
  Proof. done. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
412
  Global Instance disjoint_sym : Symmetric (##@{C}).
413
  Proof. intros X Y. set_solver. Qed.
414
  Lemma disjoint_empty_l Y :  ## Y.
415
  Proof. set_solver. Qed.
416
  Lemma disjoint_empty_r X : X ## .
417
  Proof. set_solver. Qed.
418
  Lemma disjoint_singleton_l x Y : {[ x ]} ## Y  x  Y.
419
  Proof. set_solver. Qed.
420
  Lemma disjoint_singleton_r y X : X ## {[ y ]}  y  X.
421
  Proof. set_solver. Qed.
422
  Lemma disjoint_union_l X1 X2 Y : X1  X2 ## Y  X1 ## Y  X2 ## Y.
423
  Proof. set_solver. Qed.
424
  Lemma disjoint_union_r X Y1 Y2 : X ## Y1  Y2  X ## Y1  X ## Y2.
425 426 427 428
  Proof. set_solver. Qed.

  (** Big unions *)
  Lemma elem_of_union_list Xs x : x   Xs   X, X  Xs  x  X.
429 430
  Proof.
    split.
431 432
    - induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|].
      setoid_rewrite elem_of_cons. apply elem_of_union in HXs. naive_solver.
Ralf Jung's avatar
Ralf Jung committed
433
    - intros [X [Hx]]. induction Hx; simpl; [by apply elem_of_union_l |].
434
      intros. apply elem_of_union_r; auto.
435
  Qed.
436

437 438 439 440 441 442 443
  Lemma union_list_nil :  @nil C = .
  Proof. done. Qed.
  Lemma union_list_cons X Xs :  (X :: Xs) = X   Xs.
  Proof. done. Qed.
  Lemma union_list_singleton X :  [X]  X.
  Proof. simpl. by rewrite (right_id  _). Qed.
  Lemma union_list_app Xs1 Xs2 :  (Xs1 ++ Xs2)   Xs1   Xs2.
444
  Proof.
445 446
    induction Xs1 as [|X Xs1 IH]; simpl; [by rewrite (left_id  _)|].
    by rewrite IH, (assoc _).
447
  Qed.
448
  Lemma union_list_reverse Xs :  (reverse Xs)   Xs.
449
  Proof.
450 451 452
    induction Xs as [|X Xs IH]; simpl; [done |].
    by rewrite reverse_cons, union_list_app,
      union_list_singleton, (comm _), IH.
453
  Qed.
454 455
  Lemma union_list_mono Xs Ys : Xs * Ys   Xs   Ys.
  Proof. induction 1; simpl; auto using union_mono. Qed.
456
  Lemma empty_union_list Xs :  Xs    Forall ( ) Xs.
457
  Proof.
458 459 460
    split.
    - induction Xs; simpl; rewrite ?empty_union; intuition.
    - induction 1 as [|?? E1 ? E2]; simpl. done. by apply empty_union.
461
  Qed.
462

463 464 465 466 467 468 469 470 471
  Section leibniz.
    Context `{!LeibnizEquiv C}.

    Lemma elem_of_equiv_L X Y : X = Y   x, x  X  x  Y.
    Proof. unfold_leibniz. apply elem_of_equiv. Qed.
    Lemma collection_equiv_spec_L X Y : X = Y  X  Y  Y  X.
    Proof. unfold_leibniz. apply collection_equiv_spec. Qed.

    (** Subset relation *)
472
    Global Instance collection_subseteq_partialorder : PartialOrder (@{C}).
473 474 475 476 477 478 479 480 481 482
    Proof. split. apply _. intros ??. unfold_leibniz. apply (anti_symm _). Qed.

    Lemma subseteq_union_L X Y : X  Y  X  Y = Y.
    Proof. unfold_leibniz. apply subseteq_union. Qed.
    Lemma subseteq_union_1_L X Y : X  Y  X  Y = Y.
    Proof. unfold_leibniz. apply subseteq_union_1. Qed.
    Lemma subseteq_union_2_L X Y : X  Y = Y  X  Y.
    Proof. unfold_leibniz. apply subseteq_union_2. Qed.

    (** Union *)
483
    Global Instance union_idemp_L : IdemP (=@{C}) ().
484
    Proof. intros ?. unfold_leibniz. apply (idemp _). Qed.
485
    Global Instance union_empty_l_L : LeftId (=@{C})  ().
486
    Proof. intros ?. unfold_leibniz. apply (left_id _ _). Qed.
487
    Global Instance union_empty_r_L : RightId (=@{C})  ().
488
    Proof. intros ?. unfold_leibniz. apply (right_id _ _). Qed.
489
    Global Instance union_comm_L : Comm (=@{C}) ().
490
    Proof. intros ??. unfold_leibniz. apply (comm _). Qed.
491
    Global Instance union_assoc_L : Assoc (=@{C}) ().
492 493 494 495 496
    Proof. intros ???. unfold_leibniz. apply (assoc _). Qed.

    Lemma empty_union_L X Y : X  Y =   X =   Y = .
    Proof. unfold_leibniz. apply empty_union. Qed.

497
    Lemma union_cancel_l_L X Y Z : Z ## X  Z ## Y  Z  X = Z  Y  X = Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
498
    Proof. unfold_leibniz. apply union_cancel_l. Qed.
499
    Lemma union_cancel_r_L X Y Z : X ## Z  Y ## Z  X  Z = Y  Z  X = Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
500 501
    Proof. unfold_leibniz. apply union_cancel_r. Qed.

502 503 504 505 506 507 508 509 510 511 512 513 514
    (** Empty *)
    Lemma elem_of_equiv_empty_L X : X =    x, x  X.
    Proof. unfold_leibniz. apply elem_of_equiv_empty. Qed.
    Lemma equiv_empty_L X : X    X = .
    Proof. unfold_leibniz. apply equiv_empty. Qed.
    Lemma union_positive_l_L X Y : X  Y =   X = .
    Proof. unfold_leibniz. apply union_positive_l. Qed.
    Lemma union_positive_l_alt_L X Y : X    X  Y  .
    Proof. unfold_leibniz. apply union_positive_l_alt. Qed.
    Lemma non_empty_inhabited_L x X : x  X  X  .
    Proof. unfold_leibniz. apply non_empty_inhabited. Qed.

    (** Singleton *)
515
    Lemma non_empty_singleton_L x : {[ x ]}  ( : C).
516 517 518 519 520 521 522 523 524 525 526 527 528 529
    Proof. unfold_leibniz. apply non_empty_singleton. Qed.

    (** Big unions *)
    Lemma union_list_singleton_L X :  [X] = X.
    Proof. unfold_leibniz. apply union_list_singleton. Qed.
    Lemma union_list_app_L Xs1 Xs2 :  (Xs1 ++ Xs2) =  Xs1   Xs2.
    Proof. unfold_leibniz. apply union_list_app. Qed.
    Lemma union_list_reverse_L Xs :  (reverse Xs) =  Xs.
    Proof. unfold_leibniz. apply union_list_reverse. Qed.
    Lemma empty_union_list_L Xs :  Xs =   Forall (= ) Xs.
    Proof. unfold_leibniz. by rewrite empty_union_list. Qed. 
  End leibniz.

  Section dec.
530
    Context `{!RelDecision (@{C})}.
531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554
    Lemma collection_subseteq_inv X Y : X  Y  X  Y  X  Y.
    Proof. destruct (decide (X  Y)); [by right|left;set_solver]. Qed.
    Lemma collection_not_subset_inv X Y : X  Y  X  Y  X  Y.
    Proof. destruct (decide (X  Y)); [by right|left;set_solver]. Qed.

    Lemma non_empty_union X Y : X  Y    X    Y  .
    Proof. rewrite empty_union. destruct (decide (X  )); intuition. Qed.
    Lemma non_empty_union_list Xs :  Xs    Exists ( ) Xs.
    Proof. rewrite empty_union_list. apply (not_Forall_Exists _). Qed.

    Context `{!LeibnizEquiv C}.
    Lemma collection_subseteq_inv_L X Y : X  Y  X  Y  X = Y.
    Proof. unfold_leibniz. apply collection_subseteq_inv. Qed.
    Lemma collection_not_subset_inv_L X Y : X  Y  X  Y  X = Y.
    Proof. unfold_leibniz. apply collection_not_subset_inv. Qed.
    Lemma non_empty_union_L X Y : X  Y    X    Y  .
    Proof. unfold_leibniz. apply non_empty_union. Qed.
    Lemma non_empty_union_list_L Xs :  Xs    Exists ( ) Xs.
    Proof. unfold_leibniz. apply non_empty_union_list. Qed.
  End dec.
End simple_collection.


(** * Collections with [∪], [∩], [∖], [∅] and [{[_]}] *)
Robbert Krebbers's avatar
Robbert Krebbers committed
555 556
Section collection.
  Context `{Collection A C}.
557
  Implicit Types x y : A.
558
  Implicit Types X Y : C.
Robbert Krebbers's avatar
Robbert Krebbers committed
559

560 561 562 563 564 565 566 567 568 569 570 571 572 573 574
  (** Intersection *)
  Lemma subseteq_intersection X Y : X  Y  X  Y  X.
  Proof. set_solver. Qed. 
  Lemma subseteq_intersection_1 X Y : X  Y  X  Y  X.
  Proof. apply subseteq_intersection. Qed.
  Lemma subseteq_intersection_2 X Y : X  Y  X  X  Y.
  Proof. apply subseteq_intersection. Qed.

  Lemma intersection_subseteq_l X Y : X  Y  X.
  Proof. set_solver. Qed.
  Lemma intersection_subseteq_r X Y : X  Y  Y.
  Proof. set_solver. Qed.
  Lemma intersection_greatest X Y Z : Z  X  Z  Y  Z  X  Y.
  Proof. set_solver. Qed.

575
  Lemma intersection_mono_l X Y1 Y2 : Y1  Y2  X  Y1  X  Y2.
576
  Proof. set_solver. Qed.
577
  Lemma intersection_mono_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
578
  Proof. set_solver. Qed.
579
  Lemma intersection_mono X1 X2 Y1 Y2 :
580
    X1  X2  Y1  Y2  X1  Y1  X2  Y2.
581
  Proof. set_solver. Qed.
582

583
  Global Instance intersection_idemp : IdemP (@{C}) ().
584
  Proof. intros X; set_solver. Qed.
585
  Global Instance intersection_comm : Comm (@{C}) ().
586
  Proof. intros X Y; set_solver. Qed.
587
  Global Instance intersection_assoc : Assoc (@{C}) ().
588
  Proof. intros X Y Z; set_solver. Qed.
589
  Global Instance intersection_empty_l : LeftAbsorb (@{C})  ().
590
  Proof. intros X; set_solver. Qed.
591
  Global Instance intersection_empty_r: RightAbsorb (@{C})  ().
592 593
  Proof. intros X; set_solver. Qed.

594
  Lemma intersection_singletons x : ({[x]} : C)  {[x]}  {[x]}.
595
  Proof. set_solver. Qed.
596 597 598 599 600 601 602 603 604 605 606

  Lemma union_intersection_l X Y Z : X  (Y  Z)  (X  Y)  (X  Z).
  Proof. set_solver. Qed.
  Lemma union_intersection_r X Y Z : (X  Y)  Z  (X  Z)  (Y  Z).
  Proof. set_solver. Qed.
  Lemma intersection_union_l X Y Z : X  (Y  Z)  (X  Y)  (X  Z).
  Proof. set_solver. Qed.
  Lemma intersection_union_r X Y Z : (X  Y)  Z  (X  Z)  (Y  Z).
  Proof. set_solver. Qed.

  (** Difference *)
Robbert Krebbers's avatar
Robbert Krebbers committed
607
  Lemma difference_twice X Y : (X  Y)  Y  X  Y.
608
  Proof. set_solver. Qed.
609
  Lemma subseteq_empty_difference X Y : X  Y  X  Y  .
610
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
611
  Lemma difference_diag X : X  X  .
612
  Proof. set_solver. Qed.
613 614
  Lemma difference_empty X : X    X.
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
615
  Lemma difference_union_distr_l X Y Z : (X  Y)  Z  X  Z  Y  Z.
616
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
617
  Lemma difference_union_distr_r X Y Z : Z  (X  Y)  (Z  X)  (Z  Y).
618
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
619
  Lemma difference_intersection_distr_l X Y Z : (X  Y)  Z  X  Z  Y  Z.
620
  Proof. set_solver. Qed.
621
  Lemma difference_disjoint X Y : X ## Y  X  Y  X.
622
  Proof. set_solver. Qed.
623 624
  Lemma subset_difference_elem_of {x: A} {s: C} (inx: x  s): s  {[ x ]}  s.
  Proof. set_solver. Qed.
625 626
  Lemma difference_difference X Y Z : (X  Y)  Z  X  (Y  Z).
  Proof. set_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
627

628
  Lemma difference_mono X1 X2 Y1 Y2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
629 630
    X1  X2  Y2  Y1  X1  Y1  X2  Y2.
  Proof. set_solver. Qed.
631
  Lemma difference_mono_l X Y1 Y2 : Y2  Y1  X  Y1  X  Y2.
Robbert Krebbers's avatar
Robbert Krebbers committed
632
  Proof. set_solver. Qed.
633
  Lemma difference_mono_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
634 635
  Proof. set_solver. Qed.

636
  (** Disjointness *)
637
  Lemma disjoint_intersection X Y : X ## Y  X  Y  .
638 639
  Proof. set_solver. Qed.

640 641
  Section leibniz.
    Context `{!LeibnizEquiv C}.
642 643 644 645 646 647 648 649 650

    (** Intersection *)
    Lemma subseteq_intersection_L X Y : X  Y  X  Y = X.
    Proof. unfold_leibniz. apply subseteq_intersection. Qed.
    Lemma subseteq_intersection_1_L X Y : X  Y  X  Y = X.
    Proof. unfold_leibniz. apply subseteq_intersection_1. Qed.
    Lemma subseteq_intersection_2_L X Y : X  Y = X  X  Y.
    Proof. unfold_leibniz. apply subseteq_intersection_2. Qed.

651
    Global Instance intersection_idemp_L : IdemP (=@{C}) ().
652
    Proof. intros ?. unfold_leibniz. apply (idemp _). Qed.
653
    Global Instance intersection_comm_L : Comm (=@{C}) ().
654
    Proof. intros ??. unfold_leibniz. apply (comm _). Qed.
655
    Global Instance intersection_assoc_L : Assoc (=@{C}) ().
656
    Proof. intros ???. unfold_leibniz. apply (assoc _). Qed.
657
    Global Instance intersection_empty_l_L: LeftAbsorb (=@{C})  ().
658
    Proof. intros ?. unfold_leibniz. apply (left_absorb _ _). Qed.
659
    Global Instance intersection_empty_r_L: RightAbsorb (=@{C})  ().
660 661
    Proof. intros ?. unfold_leibniz. apply (right_absorb _ _). Qed.

662
    Lemma intersection_singletons_L x : {[x]}  {[x]} = ({[x]} : C).
663
    Proof. unfold_leibniz. apply intersection_singletons. Qed.
664 665 666 667 668

    Lemma union_intersection_l_L X Y Z : X  (Y  Z) = (X  Y)  (X  Z).
    Proof. unfold_leibniz; apply union_intersection_l. Qed.
    Lemma union_intersection_r_L X Y Z : (X  Y)  Z = (X  Z)  (Y  Z).
    Proof. unfold_leibniz; apply union_intersection_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
669
    Lemma intersection_union_l_L X Y Z : X  (Y  Z) = (X  Y)  (X  Z).
670
    Proof. unfold_leibniz; apply intersection_union_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
671
    Lemma intersection_union_r_L X Y Z : (X  Y)  Z = (X  Z)  (Y  Z).
672 673 674
    Proof. unfold_leibniz; apply intersection_union_r. Qed.

    (** Difference *)
675 676
    Lemma difference_twice_L X Y : (X  Y)  Y = X  Y.
    Proof. unfold_leibniz. apply difference_twice. Qed.
677 678
    Lemma subseteq_empty_difference_L X Y : X  Y  X  Y = .
    Proof. unfold_leibniz. apply subseteq_empty_difference. Qed.
679 680
    Lemma difference_diag_L X : X  X = .
    Proof. unfold_leibniz. apply difference_diag. Qed.
681 682
    Lemma difference_empty_L X : X   = X.
    Proof. unfold_leibniz. apply difference_empty. Qed.
683 684
    Lemma difference_union_distr_l_L X Y Z : (X  Y)  Z = X  Z  Y  Z.
    Proof. unfold_leibniz. apply difference_union_distr_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
685 686
    Lemma difference_union_distr_r_L X Y Z : Z  (X  Y) = (Z  X)  (Z  Y).
    Proof. unfold_leibniz. apply difference_union_distr_r. Qed.
687 688 689
    Lemma difference_intersection_distr_l_L X Y Z :
      (X  Y)  Z = X  Z  Y  Z.
    Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed.
690
    Lemma difference_disjoint_L X Y : X ## Y  X  Y = X.
691
    Proof. unfold_leibniz. apply difference_disjoint. Qed.
692 693
    Lemma difference_difference_L X Y Z : (X  Y)  Z = X  (Y  Z).
    Proof. unfold_leibniz. apply difference_difference. Qed.
694 695

    (** Disjointness *)
696
    Lemma disjoint_intersection_L X Y : X ## Y  X  Y = .
697
    Proof. unfold_leibniz. apply disjoint_intersection. Qed.
698 699 700
  End leibniz.

  Section dec.
Robbert Krebbers's avatar
Robbert Krebbers committed
701
    Context `{!RelDecision (@{C})}.
702
    Lemma not_elem_of_intersection x X Y : x  X  Y  x  X  x  Y.
703
    Proof. rewrite elem_of_intersection. destruct (decide (x  X)); tauto. Qed.
704
    Lemma not_elem_of_difference x X Y : x  X  Y  x  X  x  Y.
705
    Proof. rewrite elem_of_difference. destruct (decide (x  Y)); tauto. Qed.
706 707
    Lemma union_difference X Y : X  Y  Y  X  Y  X.
    Proof.
708
      intros ? x; split; rewrite !elem_of_union, elem_of_difference; [|intuition].
709
      destruct (decide (x  X)); intuition.
710
    Qed.
711 712 713 714 715
    Lemma difference_union X Y : X  Y  Y  X  Y.
    Proof.
      intros x. rewrite !elem_of_union; rewrite elem_of_difference.
      split; [ | destruct (decide (x  Y)) ]; intuition.
    Qed.
716
    Lemma subseteq_disjoint_union X Y : X  Y   Z, Y  X  Z  X ## Z.
717 718 719 720
    Proof.
      split; [|set_solver].
      exists (Y  X); split; [auto using union_difference|set_solver].
    Qed.
721
    Lemma non_empty_difference X Y : X  Y  Y  X  .
722
    Proof. intros [HXY1 HXY2] Hdiff. destruct HXY2. set_solver. Qed.
723
    Lemma empty_difference_subseteq X Y : X  Y    X  Y.
724
    Proof. set_solver. Qed.
725 726 727 728
    Lemma singleton_union_difference X Y x :
      {[x]}  (X  Y)  ({[x]}  X)  (Y  {[x]}).
    Proof.
      intro y; split; intros Hy; [ set_solver | ].
729
      destruct (decide (y  ({[x]} : C))); set_solver.
730
    Qed.
731

732 733 734
    Context `{!LeibnizEquiv C}.
    Lemma union_difference_L X Y : X  Y  Y = X  Y  X.
    Proof. unfold_leibniz. apply union_difference. Qed.
735 736
    Lemma difference_union_L X Y : X  Y  Y = X  Y.
    Proof. unfold_leibniz. apply difference_union. Qed.
737 738
    Lemma non_empty_difference_L X Y : X  Y  Y  X  .
    Proof. unfold_leibniz. apply non_empty_difference. Qed.
739 740
    Lemma empty_difference_subseteq_L X Y : X  Y =   X  Y.
    Proof. unfold_leibniz. apply empty_difference_subseteq. Qed.
741
    Lemma subseteq_disjoint_union_L X Y : X  Y   Z, Y = X  Z  X ## Z.
742
    Proof. unfold_leibniz. apply subseteq_disjoint_union. Qed.
743 744 745
    Lemma singleton_union_difference_L X Y x :
      {[x]}  (X  Y) = ({[x]}  X)  (Y  {[x]}).
    Proof. unfold_leibniz. apply singleton_union_difference. Qed.
746 747 748
  End dec.
End collection.

749 750 751 752 753 754 755 756 757

(** * Conversion of option and list *)
Definition of_option `{Singleton A C, Empty C} (mx : option A) : C :=
  match mx with None =>  | Some x => {[ x ]} end.
Fixpoint of_list `{Singleton A C, Empty C, Union C} (l : list A) : C :=
  match l with [] =>  | x :: l => {[ x ]}  of_list l end.

Section of_option_list.
  Context `{SimpleCollection A C}.
Robbert Krebbers's avatar
Robbert Krebbers committed
758 759
  Implicit Types l : list A.

760
  Lemma elem_of_of_option (x : A) mx: x  of_option (C:=C) mx  mx = Some x.
761
  Proof. destruct mx; set_solver. Qed.
762
  Lemma not_elem_of_of_option (x : A) mx: x  of_option (C:=C) mx  mx  Some x.
Robbert Krebbers's avatar
Robbert Krebbers committed
763 764
  Proof. by rewrite elem_of_of_option. Qed.

765
  Lemma elem_of_of_list (x : A) l : x  of_list (C:=C) l  x  l.
766 767 768 769 770 771
  Proof.
    split.
    - induction l; simpl; [by rewrite elem_of_empty|].
      rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto.
    - induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto.
  Qed.
772
  Lemma not_elem_of_of_list (x : A) l : x  of_list (C:=C) l  x  l.
Robbert Krebbers's avatar
Robbert Krebbers committed
773 774
  Proof. by rewrite elem_of_of_list. Qed.

775
  Global Instance set_unfold_of_option (mx : option A) x :
776
    SetUnfold (x  of_option (C:=C) mx) (mx = Some x).
777 778
  Proof. constructor; apply elem_of_of_option. Qed.
  Global Instance set_unfold_of_list (l : list A) x P :
779
    SetUnfold (x  l) P  SetUnfold (x  of_list (C:=C) l) P.
780 781
  Proof. constructor. by rewrite elem_of_of_list, (set_unfold (x  l) P). Qed.

782
  Lemma of_list_nil : of_list [] =@{C} .
Robbert Krebbers's avatar
Robbert Krebbers committed
783
  Proof. done. Qed.
784
  Lemma of_list_cons x l : of_list (x :: l) =@{C} {[ x ]}  of_list l.
Robbert Krebbers's avatar
Robbert Krebbers committed
785
  Proof. done. Qed.