collections.v 24.8 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
2 3 4 5 6 7
(* 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. *)
Require Export base tactics orders.

8 9 10
Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y,
   x, x  X  x  Y.

11
(** * Basic theorems *)
12 13
Section simple_collection.
  Context `{SimpleCollection A C}.
Robbert Krebbers's avatar
Robbert Krebbers committed
14

15
  Lemma elem_of_empty x : x    False.
16
  Proof. split. apply not_elem_of_empty. done. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
17 18 19 20
  Lemma elem_of_union_l x X Y : x  X  x  X  Y.
  Proof. intros. apply elem_of_union. auto. Qed.
  Lemma elem_of_union_r x X Y : x  Y  x  X  Y.
  Proof. intros. apply elem_of_union. auto. Qed.
21 22 23
  Global Instance: EmptySpec C.
  Proof. firstorder auto. Qed.
  Global Instance: JoinSemiLattice C.
24
  Proof. firstorder auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
25
  Lemma elem_of_subseteq X Y : X  Y   x, x  X  x  Y.
26
  Proof. done. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
27 28
  Lemma elem_of_equiv X Y : X  Y   x, x  X  x  Y.
  Proof. firstorder. Qed.
29 30
  Lemma elem_of_equiv_alt X Y :
    X  Y  ( x, x  X  x  Y)  ( x, x  Y  x  X).
Robbert Krebbers's avatar
Robbert Krebbers committed
31
  Proof. firstorder. Qed.
32 33
  Lemma elem_of_equiv_empty X : X     x, x  X.
  Proof. firstorder. Qed.
34 35 36 37 38 39
  Lemma collection_positive_l X Y : X  Y    X  .
  Proof.
    rewrite !elem_of_equiv_empty. setoid_rewrite elem_of_union. naive_solver.
  Qed.
  Lemma collection_positive_l_alt X Y : X    X  Y  .
  Proof. eauto using collection_positive_l. Qed.
40 41 42 43
  Lemma elem_of_singleton_1 x y : x  {[y]}  x = y.
  Proof. by rewrite elem_of_singleton. Qed.
  Lemma elem_of_singleton_2 x y : x = y  x  {[y]}.
  Proof. by rewrite elem_of_singleton. Qed.
44 45 46
  Lemma elem_of_subseteq_singleton x X : x  X  {[ x ]}  X.
  Proof.
    split.
47
    * intros ??. rewrite elem_of_singleton. by intros ->.
48 49
    * intros Ex. by apply (Ex x), elem_of_singleton.
  Qed.
50
  Global Instance singleton_proper : Proper ((=) ==> ()) singleton.
51
  Proof. by repeat intro; subst. Qed.
52
  Global Instance elem_of_proper: Proper ((=) ==> () ==> iff) () | 5.
53
  Proof. intros ???; subst. firstorder. Qed.
54
  Lemma elem_of_union_list Xs x : x   Xs   X, X  Xs  x  X.
55 56
  Proof.
    split.
57 58 59 60
    * 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.
    * intros [X []]. induction 1; simpl; [by apply elem_of_union_l |].
      intros. apply elem_of_union_r; auto.
61 62 63 64 65 66 67 68
  Qed.
  Lemma non_empty_singleton x : {[ x ]}  .
  Proof. intros [E _]. by apply (elem_of_empty x), E, elem_of_singleton. Qed.
  Lemma not_elem_of_singleton x y : x  {[ y ]}  x  y.
  Proof. by rewrite elem_of_singleton. Qed.
  Lemma not_elem_of_union x X Y : x  X  Y  x  X  x  Y.
  Proof. rewrite elem_of_union. tauto. Qed.

69 70 71 72 73 74 75 76 77
  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 elem_of_equiv_alt_L X Y :
      X = Y  ( x, x  X  x  Y)  ( x, x  Y  x  X).
    Proof. unfold_leibniz. apply elem_of_equiv_alt. Qed.
    Lemma elem_of_equiv_empty_L X : X =    x, x  X.
    Proof. unfold_leibniz. apply elem_of_equiv_empty. Qed.
78 79 80 81
    Lemma collection_positive_l_L X Y : X  Y =   X = .
    Proof. unfold_leibniz. apply collection_positive_l. Qed.
    Lemma collection_positive_l_alt_L X Y : X    X  Y  .
    Proof. unfold_leibniz. apply collection_positive_l_alt. Qed.
82 83 84 85 86 87 88 89 90 91 92 93
    Lemma non_empty_singleton_L x : {[ x ]}  .
    Proof. unfold_leibniz. apply non_empty_singleton. Qed.
  End leibniz.

  Section dec.
    Context `{ X Y : C, Decision (X  Y)}.
    Global Instance elem_of_dec_slow (x : A) (X : C) : Decision (x  X) | 100.
    Proof.
      refine (cast_if (decide_rel () {[ x ]} X));
        by rewrite elem_of_subseteq_singleton.
    Defined.
  End dec.
94 95
End simple_collection.

96
Definition of_option `{Singleton A C, Empty C} (x : option A) : C :=
97
  match x with None =>  | Some a => {[ a ]} end.
98 99
Fixpoint of_list `{Singleton A C, Empty C, Union C} (l : list A) : C :=
  match l with [] =>  | x :: l => {[ x ]}  of_list l end.
100

101 102 103 104 105 106 107 108 109 110 111
Section of_option_list.
  Context `{SimpleCollection A C}.
  Lemma elem_of_of_option (x : A) o : x  of_option o  o = Some x.
  Proof.
    destruct o; simpl;
      rewrite ?elem_of_empty, ?elem_of_singleton; naive_solver.
  Qed.
  Lemma elem_of_of_list (x : A) l : x  of_list l  x  l.
  Proof.
    split.
    * induction l; simpl; [by rewrite elem_of_empty|].
Robbert Krebbers's avatar
Robbert Krebbers committed
112
      rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto.
113 114 115
    * induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto.
  Qed.
End of_option_list.
116 117 118

Global Instance collection_guard `{CollectionMonad M} : MGuard M :=
  λ P dec A x, match dec with left H => x H | _ =>  end.
119 120 121 122 123 124 125 126 127

Section collection_monad_base.
  Context `{CollectionMonad M}.
  Lemma elem_of_guard `{Decision P} {A} (x : A) (X : M A) :
    x  guard P; X  P  x  X.
  Proof.
    unfold mguard, collection_guard; simpl; case_match;
      rewrite ?elem_of_empty; naive_solver.
  Qed.
128 129 130
  Lemma elem_of_guard_2 `{Decision P} {A} (x : A) (X : M A) :
    P  x  X  x  guard P; X.
  Proof. by rewrite elem_of_guard. Qed.
131 132 133 134 135 136 137 138 139 140 141 142
  Lemma guard_empty `{Decision P} {A} (X : M A) : guard P; X    ¬P  X  .
  Proof.
    rewrite !elem_of_equiv_empty; setoid_rewrite elem_of_guard.
    destruct (decide P); naive_solver.
  Qed.
  Lemma bind_empty {A B} (f : A  M B) X :
    X = f    X     x, x  X  f x  .
  Proof.
    setoid_rewrite elem_of_equiv_empty; setoid_rewrite elem_of_bind.
    naive_solver.
  Qed.
End collection_monad_base.
143

144 145 146 147 148 149 150 151 152
(** * Tactics *)
(** Given a hypothesis [H : _ ∈ _], the tactic [destruct_elem_of H] will
recursively split [H] for [(∪)], [(∩)], [(∖)], [map], [∅], [{[_]}]. *)
Tactic Notation "decompose_elem_of" hyp(H) :=
  let rec go H :=
  lazymatch type of H with
  | _   => apply elem_of_empty in H; destruct H
  | ?x  {[ ?y ]} =>
    apply elem_of_singleton in H; try first [subst y | subst x]
153 154
  | ?x  {[ ?y ]} =>
    apply not_elem_of_singleton in H
155
  | _  _  _ =>
156 157 158 159
    apply elem_of_union in H; destruct H as [H|H]; [go H|go H]
  | _  _  _ =>
    let H1 := fresh H in let H2 := fresh H in apply not_elem_of_union in H;
    destruct H as [H1 H2]; go H1; go H2
160
  | _  _  _ =>
161
    let H1 := fresh H in let H2 := fresh H in apply elem_of_intersection in H;
162 163
    destruct H as [H1 H2]; go H1; go H2
  | _  _  _ =>
164
    let H1 := fresh H in let H2 := fresh H in apply elem_of_difference in H;
165 166
    destruct H as [H1 H2]; go H1; go H2
  | ?x  _ <$> _ =>
167
    apply elem_of_fmap in H; destruct H as [? [? H]]; try (subst x); go H
168
  | _  _ = _ =>
169
    let H1 := fresh H in let H2 := fresh H in apply elem_of_bind in H;
170 171 172 173
    destruct H as [? [H1 H2]]; go H1; go H2
  | ?x  mret ?y =>
    apply elem_of_ret in H; try first [subst y | subst x]
  | _  mjoin _ = _ =>
174
    let H1 := fresh H in let H2 := fresh H in apply elem_of_join in H;
175
    destruct H as [? [H1 H2]]; go H1; go H2
176
  | _  guard _; _ =>
177
    let H1 := fresh H in let H2 := fresh H in apply elem_of_guard in H;
178 179
    destruct H as [H1 H2]; go H2
  | _  of_option _ => apply elem_of_of_option in H
Robbert Krebbers's avatar
Robbert Krebbers committed
180
  | _  of_list _ => apply elem_of_of_list in H
181 182 183 184 185
  | _ => idtac
  end in go H.
Tactic Notation "decompose_elem_of" :=
  repeat_on_hyps (fun H => decompose_elem_of H).

186 187
Ltac decompose_empty := repeat
  match goal with
188 189 190 191
  | H :    |- _ => clear H
  | H :  =  |- _ => clear H
  | H :   _ |- _ => symmetry in H
  | H :  = _ |- _ => symmetry in H
192 193 194
  | H : _  _   |- _ => apply empty_union in H; destruct H
  | H : _  _   |- _ => apply non_empty_union in H; destruct H
  | H : {[ _ ]}   |- _ => destruct (non_empty_singleton _ H)
195 196 197
  | H : _  _ =  |- _ => apply empty_union_L in H; destruct H
  | H : _  _   |- _ => apply non_empty_union_L in H; destruct H
  | H : {[ _ ]} =  |- _ => destruct (non_empty_singleton_L _ H)
198
  | H : guard _ ; _   |- _ => apply guard_empty in H; destruct H
199 200
  end.

201 202 203 204
(** The first pass of our collection tactic consists of eliminating all
occurrences of [(∪)], [(∩)], [(∖)], [(<$>)], [∅], [{[_]}], [(≡)], and [(⊆)],
by rewriting these into logically equivalent propositions. For example we
rewrite [A → x ∈ X ∪ ∅] into [A → x ∈ X ∨ False]. *)
205 206 207 208
Ltac unfold_elem_of :=
  repeat_on_hyps (fun H =>
    repeat match type of H with
    | context [ _  _ ] => setoid_rewrite elem_of_subseteq in H
Robbert Krebbers's avatar
Robbert Krebbers committed
209
    | context [ _  _ ] => setoid_rewrite subset_spec in H
210
    | context [ _   ] => setoid_rewrite elem_of_equiv_empty in H
211
    | context [ _  _ ] => setoid_rewrite elem_of_equiv_alt in H
212 213
    | context [ _ =  ] => setoid_rewrite elem_of_equiv_empty_L in H
    | context [ _ = _ ] => setoid_rewrite elem_of_equiv_alt_L in H
214 215 216 217 218
    | context [ _   ] => setoid_rewrite elem_of_empty in H
    | context [ _  {[ _ ]} ] => setoid_rewrite elem_of_singleton in H
    | context [ _  _  _ ] => setoid_rewrite elem_of_union in H
    | context [ _  _  _ ] => setoid_rewrite elem_of_intersection in H
    | context [ _  _  _ ] => setoid_rewrite elem_of_difference in H
219 220 221 222
    | context [ _  _ <$> _ ] => setoid_rewrite elem_of_fmap in H
    | context [ _  mret _ ] => setoid_rewrite elem_of_ret in H
    | context [ _  _ = _ ] => setoid_rewrite elem_of_bind in H
    | context [ _  mjoin _ ] => setoid_rewrite elem_of_join in H
223
    | context [ _  guard _; _ ] => setoid_rewrite elem_of_guard in H
Robbert Krebbers's avatar
Robbert Krebbers committed
224 225
    | context [ _  of_option _ ] => setoid_rewrite elem_of_of_option in H
    | context [ _  of_list _ ] => setoid_rewrite elem_of_of_list in H
226 227
    end);
  repeat match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
228
  | |- context [ _  _ ] => setoid_rewrite elem_of_subseteq
Robbert Krebbers's avatar
Robbert Krebbers committed
229
  | |- context [ _  _ ] => setoid_rewrite subset_spec
230
  | |- context [ _   ] => setoid_rewrite elem_of_equiv_empty
Robbert Krebbers's avatar
Robbert Krebbers committed
231
  | |- context [ _  _ ] => setoid_rewrite elem_of_equiv_alt
232 233
  | |- context [ _ =  ] => setoid_rewrite elem_of_equiv_empty_L
  | |- context [ _ = _ ] => setoid_rewrite elem_of_equiv_alt_L
234
  | |- context [ _   ] => setoid_rewrite elem_of_empty
235
  | |- context [ _  {[ _ ]} ] => setoid_rewrite elem_of_singleton
Robbert Krebbers's avatar
Robbert Krebbers committed
236 237 238
  | |- context [ _  _  _ ] => setoid_rewrite elem_of_union
  | |- context [ _  _  _ ] => setoid_rewrite elem_of_intersection
  | |- context [ _  _  _ ] => setoid_rewrite elem_of_difference
239 240 241 242
  | |- context [ _  _ <$> _ ] => setoid_rewrite elem_of_fmap
  | |- context [ _  mret _ ] => setoid_rewrite elem_of_ret
  | |- context [ _  _ = _ ] => setoid_rewrite elem_of_bind
  | |- context [ _  mjoin _ ] => setoid_rewrite elem_of_join
243
  | |- context [ _  guard _; _ ] => setoid_rewrite elem_of_guard
Robbert Krebbers's avatar
Robbert Krebbers committed
244 245
  | |- context [ _  of_option _ ] => setoid_rewrite elem_of_of_option
  | |- context [ _  of_list _ ] => setoid_rewrite elem_of_of_list
Robbert Krebbers's avatar
Robbert Krebbers committed
246 247
  end.

248 249 250
(** The tactic [solve_elem_of tac] composes the above tactic with [intuition].
For goals that do not involve [≡], [⊆], [map], or quantifiers this tactic is
generally powerful enough. This tactic either fails or proves the goal. *)
251
Tactic Notation "solve_elem_of" tactic3(tac) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
252
  simpl in *;
253
  decompose_empty;
254 255 256 257 258 259 260 261 262
  unfold_elem_of;
  solve [intuition (simplify_equality; tac)].
Tactic Notation "solve_elem_of" := solve_elem_of auto.

(** For goals with quantifiers we could use the above tactic but with
[firstorder] instead of [intuition] as finishing tactic. However, [firstorder]
fails or loops on very small goals generated by [solve_elem_of] already. We
use the [naive_solver] tactic as a substitute. This tactic either fails or
proves the goal. *)
263
Tactic Notation "esolve_elem_of" tactic3(tac) :=
264
  simpl in *;
265
  decompose_empty;
266 267 268
  unfold_elem_of;
  naive_solver tac.
Tactic Notation "esolve_elem_of" := esolve_elem_of eauto.
269 270
 
(** * More theorems *)
Robbert Krebbers's avatar
Robbert Krebbers committed
271 272 273
Section collection.
  Context `{Collection A C}.

274
  Global Instance: Lattice C.
275
  Proof. split. apply _. firstorder auto. solve_elem_of. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
276 277 278 279
  Lemma intersection_singletons x : {[x]}  {[x]}  {[x]}.
  Proof. esolve_elem_of. Qed.
  Lemma difference_twice X Y : (X  Y)  Y  X  Y.
  Proof. esolve_elem_of. Qed.
280
  Lemma subseteq_empty_difference X Y : X  Y  X  Y  .
Robbert Krebbers's avatar
Robbert Krebbers committed
281 282 283 284 285 286 287 288
  Proof. esolve_elem_of. Qed.
  Lemma difference_diag X : X  X  .
  Proof. esolve_elem_of. Qed.
  Lemma difference_union_distr_l X Y Z : (X  Y)  Z  X  Z  Y  Z.
  Proof. esolve_elem_of. Qed.
  Lemma difference_intersection_distr_l X Y Z : (X  Y)  Z  X  Z  Y  Z.
  Proof. esolve_elem_of. Qed.

289 290 291 292 293 294
  Section leibniz.
    Context `{!LeibnizEquiv C}.
    Lemma intersection_singletons_L x : {[x]}  {[x]} = {[x]}.
    Proof. unfold_leibniz. apply intersection_singletons. Qed.
    Lemma difference_twice_L X Y : (X  Y)  Y = X  Y.
    Proof. unfold_leibniz. apply difference_twice. Qed.
295 296
    Lemma subseteq_empty_difference_L X Y : X  Y  X  Y = .
    Proof. unfold_leibniz. apply subseteq_empty_difference. Qed.
297 298 299 300 301 302 303 304 305 306 307 308
    Lemma difference_diag_L X : X  X = .
    Proof. unfold_leibniz. apply difference_diag. Qed.
    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.
    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.
  End leibniz.

  Section dec.
    Context `{ X Y : C, Decision (X  Y)}.
    Lemma not_elem_of_intersection x X Y : x  X  Y  x  X  x  Y.
309
    Proof. rewrite elem_of_intersection. destruct (decide (x  X)); tauto. Qed.
310
    Lemma not_elem_of_difference x X Y : x  X  Y  x  X  x  Y.
311
    Proof. rewrite elem_of_difference. destruct (decide (x  Y)); tauto. Qed.
312 313
    Lemma union_difference X Y : X  Y  Y  X  Y  X.
    Proof.
314 315
      split; intros x; rewrite !elem_of_union, elem_of_difference; [|intuition].
      destruct (decide (x  X)); intuition.
316 317 318 319 320 321
    Qed.
    Lemma non_empty_difference X Y : X  Y  Y  X  .
    Proof.
      intros [HXY1 HXY2] Hdiff. destruct HXY2. intros x.
      destruct (decide (x  X)); esolve_elem_of.
    Qed.
322 323
    Lemma empty_difference_subseteq X Y : X  Y    X  Y.
    Proof. intros ? x ?; apply dec_stable; esolve_elem_of. Qed.
324 325 326 327 328
    Context `{!LeibnizEquiv C}.
    Lemma union_difference_L X Y : X  Y  Y = X  Y  X.
    Proof. unfold_leibniz. apply union_difference. Qed.
    Lemma non_empty_difference_L X Y : X  Y  Y  X  .
    Proof. unfold_leibniz. apply non_empty_difference. Qed.
329 330
    Lemma empty_difference_subseteq_L X Y : X  Y =   X  Y.
    Proof. unfold_leibniz. apply empty_difference_subseteq. Qed.
331 332 333 334 335 336
  End dec.
End collection.

Section collection_ops.
  Context `{CollectionOps A C}.

Robbert Krebbers's avatar
Robbert Krebbers committed
337 338 339 340 341
  Lemma elem_of_intersection_with_list (f : A  A  option A) Xs Y x :
    x  intersection_with_list f Y Xs   xs y,
      Forall2 () xs Xs  y  Y  foldr (λ x, (= f x)) (Some y) xs = Some x.
  Proof.
    split.
342 343 344 345
    * revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|].
      rewrite elem_of_intersection_with in HXs; destruct HXs as (x1&x2&?&?&?).
      destruct (IHXs x2) as (xs & y & hy & ? & ?); trivial.
      eexists (x1 :: xs), y. intuition (simplify_option_equality; auto).
Robbert Krebbers's avatar
Robbert Krebbers committed
346 347 348 349 350 351 352 353 354 355 356
    * intros (xs & y & Hxs & ? & Hx). revert x Hx.
      induction Hxs; intros; simplify_option_equality; [done |].
      rewrite elem_of_intersection_with. naive_solver.
  Qed.

  Lemma intersection_with_list_ind (P Q : A  Prop) f Xs Y :
    ( y, y  Y  P y) 
    Forall (λ X,  x, x  X  Q x) Xs 
    ( x y z, Q x  P y  f x y = Some z  P z) 
     x, x  intersection_with_list f Y Xs  P x.
  Proof.
357
    intros HY HXs Hf. induction Xs; simplify_option_equality; [done |].
Robbert Krebbers's avatar
Robbert Krebbers committed
358 359 360
    intros x Hx. rewrite elem_of_intersection_with in Hx.
    decompose_Forall. destruct Hx as (? & ? & ? & ? & ?). eauto.
  Qed.
361
End collection_ops.
Robbert Krebbers's avatar
Robbert Krebbers committed
362

363
(** * Sets without duplicates up to an equivalence *)
364
Section NoDup.
365
  Context `{SimpleCollection A B} (R : relation A) `{!Equivalence R}.
Robbert Krebbers's avatar
Robbert Krebbers committed
366 367

  Definition elem_of_upto (x : A) (X : B) :=  y, y  X  R x y.
368
  Definition set_NoDup (X : B) :=  x y, x  X  y  X  R x y  x = y.
Robbert Krebbers's avatar
Robbert Krebbers committed
369 370

  Global Instance: Proper (() ==> iff) (elem_of_upto x).
Robbert Krebbers's avatar
Robbert Krebbers committed
371
  Proof. intros ??? E. unfold elem_of_upto. by setoid_rewrite E. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
372 373 374
  Global Instance: Proper (R ==> () ==> iff) elem_of_upto.
  Proof.
    intros ?? E1 ?? E2. split; intros [z [??]]; exists z.
375 376
    * rewrite <-E1, <-E2; intuition.
    * rewrite E1, E2; intuition.
Robbert Krebbers's avatar
Robbert Krebbers committed
377
  Qed.
378
  Global Instance: Proper (() ==> iff) set_NoDup.
Robbert Krebbers's avatar
Robbert Krebbers committed
379 380 381
  Proof. firstorder. Qed.

  Lemma elem_of_upto_elem_of x X : x  X  elem_of_upto x X.
382
  Proof. unfold elem_of_upto. esolve_elem_of. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
383
  Lemma elem_of_upto_empty x : ¬elem_of_upto x .
384
  Proof. unfold elem_of_upto. esolve_elem_of. Qed.
385
  Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]}  R x y.
386
  Proof. unfold elem_of_upto. esolve_elem_of. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
387

388 389
  Lemma elem_of_upto_union X Y x :
    elem_of_upto x (X  Y)  elem_of_upto x X  elem_of_upto x Y.
390
  Proof. unfold elem_of_upto. esolve_elem_of. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
391
  Lemma not_elem_of_upto x X : ¬elem_of_upto x X   y, y  X  ¬R x y.
392
  Proof. unfold elem_of_upto. esolve_elem_of. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
393

394 395 396 397 398 399 400
  Lemma set_NoDup_empty: set_NoDup .
  Proof. unfold set_NoDup. solve_elem_of. Qed.
  Lemma set_NoDup_add x X :
    ¬elem_of_upto x X  set_NoDup X  set_NoDup ({[ x ]}  X).
  Proof. unfold set_NoDup, elem_of_upto. esolve_elem_of. Qed.
  Lemma set_NoDup_inv_add x X :
    x  X  set_NoDup ({[ x ]}  X)  ¬elem_of_upto x X.
401 402
  Proof.
    intros Hin Hnodup [y [??]].
403
    rewrite (Hnodup x y) in Hin; solve_elem_of.
404
  Qed.
405 406 407 408 409
  Lemma set_NoDup_inv_union_l X Y : set_NoDup (X  Y)  set_NoDup X.
  Proof. unfold set_NoDup. solve_elem_of. Qed.
  Lemma set_NoDup_inv_union_r X Y : set_NoDup (X  Y)  set_NoDup Y.
  Proof. unfold set_NoDup. solve_elem_of. Qed.
End NoDup.
Robbert Krebbers's avatar
Robbert Krebbers committed
410

411
(** * Quantifiers *)
Robbert Krebbers's avatar
Robbert Krebbers committed
412
Section quantifiers.
413
  Context `{SimpleCollection A B} (P : A  Prop).
Robbert Krebbers's avatar
Robbert Krebbers committed
414

415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439
  Definition set_Forall X :=  x, x  X  P x.
  Definition set_Exists X :=  x, x  X  P x.

  Lemma set_Forall_empty : set_Forall .
  Proof. unfold set_Forall. solve_elem_of. Qed.
  Lemma set_Forall_singleton x : set_Forall {[ x ]}  P x.
  Proof. unfold set_Forall. solve_elem_of. Qed.
  Lemma set_Forall_union X Y : set_Forall X  set_Forall Y  set_Forall (X  Y).
  Proof. unfold set_Forall. solve_elem_of. Qed.
  Lemma set_Forall_union_inv_1 X Y : set_Forall (X  Y)  set_Forall X.
  Proof. unfold set_Forall. solve_elem_of. Qed.
  Lemma set_Forall_union_inv_2 X Y : set_Forall (X  Y)  set_Forall Y.
  Proof. unfold set_Forall. solve_elem_of. Qed.

  Lemma set_Exists_empty : ¬set_Exists .
  Proof. unfold set_Exists. esolve_elem_of. Qed.
  Lemma set_Exists_singleton x : set_Exists {[ x ]}  P x.
  Proof. unfold set_Exists. esolve_elem_of. Qed.
  Lemma set_Exists_union_1 X Y : set_Exists X  set_Exists (X  Y).
  Proof. unfold set_Exists. esolve_elem_of. Qed.
  Lemma set_Exists_union_2 X Y : set_Exists Y  set_Exists (X  Y).
  Proof. unfold set_Exists. esolve_elem_of. Qed.
  Lemma set_Exists_union_inv X Y :
    set_Exists (X  Y)  set_Exists X  set_Exists Y.
  Proof. unfold set_Exists. esolve_elem_of. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
440 441
End quantifiers.

442
Section more_quantifiers.
443
  Context `{SimpleCollection A B}.
444

445 446 447 448 449 450
  Lemma set_Forall_weaken (P Q : A  Prop) (Hweaken :  x, P x  Q x) X :
    set_Forall P X  set_Forall Q X.
  Proof. unfold set_Forall. naive_solver. Qed.
  Lemma set_Exists_weaken (P Q : A  Prop) (Hweaken :  x, P x  Q x) X :
    set_Exists P X  set_Exists Q X.
  Proof. unfold set_Exists. naive_solver. Qed.
451 452
End more_quantifiers.

453 454 455
(** * Fresh elements *)
(** We collect some properties on the [fresh] operation. In particular we
generalize [fresh] to generate lists of fresh elements. *)
456 457 458 459 460 461 462 463 464 465
Fixpoint fresh_list `{Fresh A C, Union C, Singleton A C}
    (n : nat) (X : C) : list A :=
  match n with
  | 0 => []
  | S n => let x := fresh X in x :: fresh_list n ({[ x ]}  X)
  end.
Inductive Forall_fresh `{ElemOf A C} (X : C) : list A  Prop :=
  | Forall_fresh_nil : Forall_fresh X []
  | Forall_fresh_cons x xs :
     x  xs  x  X  Forall_fresh X xs  Forall_fresh X (x :: xs).
466

467 468
Section fresh.
  Context `{FreshSpec A C}.
469

470 471
  Global Instance fresh_proper: Proper (() ==> (=)) fresh.
  Proof. intros ???. by apply fresh_proper_alt, elem_of_equiv. Qed.
472 473
  Global Instance fresh_list_proper: Proper ((=) ==> () ==> (=)) fresh_list.
  Proof.
474 475
    intros ? n ->. induction n as [|n IH]; intros ?? E; f_equal'; [by rewrite E|].
    apply IH. by rewrite E.
476
  Qed.
477 478 479 480 481 482 483 484 485 486 487 488 489 490 491

  Lemma Forall_fresh_NoDup X xs : Forall_fresh X xs  NoDup xs.
  Proof. induction 1; by constructor. Qed.
  Lemma Forall_fresh_elem_of X xs x : Forall_fresh X xs  x  xs  x  X.
  Proof.
    intros HX; revert x; rewrite <-Forall_forall.
    by induction HX; constructor.
  Qed.
  Lemma Forall_fresh_alt X xs :
    Forall_fresh X xs  NoDup xs   x, x  xs  x  X.
  Proof.
    split; eauto using Forall_fresh_NoDup, Forall_fresh_elem_of.
    rewrite <-Forall_forall.
    intros [Hxs Hxs']. induction Hxs; decompose_Forall_hyps; constructor; auto.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
492 493 494
  Lemma Forall_fresh_subseteq X Y xs :
    Forall_fresh X xs  Y  X  Forall_fresh Y xs.
  Proof. rewrite !Forall_fresh_alt; esolve_elem_of. Qed.
495

496 497
  Lemma fresh_list_length n X : length (fresh_list n X) = n.
  Proof. revert X. induction n; simpl; auto. Qed.
498
  Lemma fresh_list_is_fresh n X x : x  fresh_list n X  x  X.
499
  Proof.
500
    revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|].
501 502
    rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|].
    apply IH in Hin; solve_elem_of.
503
  Qed.
504
  Lemma NoDup_fresh_list n X : NoDup (fresh_list n X).
505
  Proof.
506
    revert X. induction n; simpl; constructor; auto.
507 508 509 510 511
    intros Hin; apply fresh_list_is_fresh in Hin; solve_elem_of.
  Qed.
  Lemma Forall_fresh_list X n : Forall_fresh X (fresh_list n X).
  Proof.
    rewrite Forall_fresh_alt; eauto using NoDup_fresh_list, fresh_list_is_fresh.
512 513
  Qed.
End fresh.
514

515
(** * Properties of implementations of collections that form a monad *)
516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531
Section collection_monad.
  Context `{CollectionMonad M}.

  Global Instance collection_fmap_proper {A B} (f : A  B) :
    Proper (() ==> ()) (fmap f).
  Proof. intros X Y E. esolve_elem_of. Qed.
  Global Instance collection_ret_proper {A} :
    Proper ((=) ==> ()) (@mret M _ A).
  Proof. intros X Y E. esolve_elem_of. Qed.
  Global Instance collection_bind_proper {A B} (f : A  M B) :
    Proper (() ==> ()) (mbind f).
  Proof. intros X Y E. esolve_elem_of. Qed.
  Global Instance collection_join_proper {A} :
    Proper (() ==> ()) (@mjoin M _ A).
  Proof. intros X Y E. esolve_elem_of. Qed.

532 533 534 535
  Lemma collection_bind_singleton {A B} (f : A  M B) x : {[ x ]} = f  f x.
  Proof. esolve_elem_of. Qed.
  Lemma collection_guard_True {A} `{Decision P} (X : M A) : P  guard P; X  X.
  Proof. esolve_elem_of. Qed.
536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555
  Lemma collection_fmap_compose {A B C} (f : A  B) (g : B  C) X :
    g  f <$> X  g <$> (f <$> X).
  Proof. esolve_elem_of. Qed.
  Lemma elem_of_fmap_1 {A B} (f : A  B) (X : M A) (y : B) :
    y  f <$> X   x, y = f x  x  X.
  Proof. esolve_elem_of. Qed.
  Lemma elem_of_fmap_2 {A B} (f : A  B) (X : M A) (x : A) :
    x  X  f x  f <$> X.
  Proof. esolve_elem_of. Qed.
  Lemma elem_of_fmap_2_alt {A B} (f : A  B) (X : M A) (x : A) (y : B) :
    x  X  y = f x  y  f <$> X.
  Proof. esolve_elem_of. Qed.

  Lemma elem_of_mapM {A B} (f : A  M B) l k :
    l  mapM f k  Forall2 (λ x y, x  f y) l k.
  Proof.
    split.
    * revert l. induction k; esolve_elem_of.
    * induction 1; esolve_elem_of.
  Qed.
556
  Lemma collection_mapM_length {A B} (f : A  M B) l k :
557 558 559
    l  mapM f k  length l = length k.
  Proof. revert l; induction k; esolve_elem_of. Qed.
  Lemma elem_of_mapM_fmap {A B} (f : A  B) (g : B  M A) l k :
560
    Forall (λ x,  y, y  g x  f y = x) l  k  mapM g l  fmap f k = l.
561
  Proof.
562 563
    intros Hl. revert k. induction Hl; simpl; intros;
      decompose_elem_of; f_equal'; auto.
564 565
  Qed.
  Lemma elem_of_mapM_Forall {A B} (f : A  M B) (P : B  Prop) l k :
566
    l  mapM f k  Forall (λ x,  y, y  f x  P y) k  Forall P l.
Robbert Krebbers's avatar
Robbert Krebbers committed
567
  Proof. rewrite elem_of_mapM. apply Forall2_Forall_l. Qed.
568 569
  Lemma elem_of_mapM_Forall2_l {A B C} (f : A  M B) (P: B  C  Prop) l1 l2 k :
    l1  mapM f k  Forall2 (λ x y,  z, z  f x  P z y) k l2 
Robbert Krebbers's avatar
Robbert Krebbers committed
570 571 572 573 574
    Forall2 P l1 l2.
  Proof.
    rewrite elem_of_mapM. intros Hl1. revert l2.
    induction Hl1; inversion_clear 1; constructor; auto.
  Qed.
575
End collection_monad.