gmultiset.v 21.7 KB
Newer Older
1
(* Copyright (c) 2012-2019, Coq-std++ developers. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
2
3
(* This file is distributed under the terms of the BSD license. *)
From stdpp Require Import gmap.
4
Set Default Proof Using "Type".
Robbert Krebbers's avatar
Robbert Krebbers committed
5
6

Record gmultiset A `{Countable A} := GMultiSet { gmultiset_car : gmap A nat }.
7
8
Arguments GMultiSet {_ _ _} _ : assert.
Arguments gmultiset_car {_ _ _} _ : assert.
Robbert Krebbers's avatar
Robbert Krebbers committed
9

10
Instance gmultiset_eq_dec `{Countable A} : EqDecision (gmultiset A).
Robbert Krebbers's avatar
Robbert Krebbers committed
11
12
Proof. solve_decision. Defined.

13
Program Instance gmultiset_countable `{Countable A} :
Robbert Krebbers's avatar
Robbert Krebbers committed
14
    Countable (gmultiset A) := {|
15
  encode X := encode (gmultiset_car X); decode p := GMultiSet <$> decode p
Robbert Krebbers's avatar
Robbert Krebbers committed
16
17
18
19
20
21
22
23
|}.
Next Obligation. intros A ?? [X]; simpl. by rewrite decode_encode. Qed.

Section definitions.
  Context `{Countable A}.

  Definition multiplicity (x : A) (X : gmultiset A) : nat :=
    match gmultiset_car X !! x with Some n => S n | None => 0 end.
24
  Global Instance gmultiset_elem_of : ElemOf A (gmultiset A) := λ x X,
Robbert Krebbers's avatar
Robbert Krebbers committed
25
    0 < multiplicity x X.
26
  Global Instance gmultiset_subseteq : SubsetEq (gmultiset A) := λ X Y,  x,
Robbert Krebbers's avatar
Robbert Krebbers committed
27
    multiplicity x X  multiplicity x Y.
28
29
  Global Instance gmultiset_equiv : Equiv (gmultiset A) := λ X Y,  x,
    multiplicity x X = multiplicity x Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
30

31
  Global Instance gmultiset_elements : Elements A (gmultiset A) := λ X,
32
    let (X) := X in ''(x,n)  map_to_list X; replicate (S n) x.
33
  Global Instance gmultiset_size : Size (gmultiset A) := length  elements.
Robbert Krebbers's avatar
Robbert Krebbers committed
34

35
36
  Global Instance gmultiset_empty : Empty (gmultiset A) := GMultiSet .
  Global Instance gmultiset_singleton : Singleton A (gmultiset A) := λ x,
Robbert Krebbers's avatar
Robbert Krebbers committed
37
    GMultiSet {[ x := 0 ]}.
38
  Global Instance gmultiset_union : Union (gmultiset A) := λ X Y,
39
40
41
42
43
44
45
    let (X) := X in let (Y) := Y in
    GMultiSet $ union_with (λ x y, Some (x `max` y)) X Y.
  Global Instance gmultiset_intersection : Intersection (gmultiset A) := λ X Y,
    let (X) := X in let (Y) := Y in
    GMultiSet $ intersection_with (λ x y, Some (x `min` y)) X Y.
  (** Often called the "sum" *)
  Global Instance gmultiset_disj_union : DisjUnion (gmultiset A) := λ X Y,
Robbert Krebbers's avatar
Robbert Krebbers committed
46
47
    let (X) := X in let (Y) := Y in
    GMultiSet $ union_with (λ x y, Some (S (x + y))) X Y.
48
  Global Instance gmultiset_difference : Difference (gmultiset A) := λ X Y,
Robbert Krebbers's avatar
Robbert Krebbers committed
49
50
51
    let (X) := X in let (Y) := Y in
    GMultiSet $ difference_with (λ x y,
      let z := x - y in guard (0 < z); Some (pred z)) X Y.
52

53
  Global Instance gmultiset_dom : Dom (gmultiset A) (gset A) := λ X,
54
    let (X) := X in dom _ X.
55
End definitions. 
Robbert Krebbers's avatar
Robbert Krebbers committed
56

57
58
59
Typeclasses Opaque gmultiset_elem_of gmultiset_subseteq.
Typeclasses Opaque gmultiset_elements gmultiset_size gmultiset_empty.
Typeclasses Opaque gmultiset_singleton gmultiset_union gmultiset_difference.
60
Typeclasses Opaque gmultiset_dom.
61

Robbert Krebbers's avatar
Robbert Krebbers committed
62
63
64
65
66
67
68
69
70
71
72
73
Section lemmas.
Context `{Countable A}.
Implicit Types x y : A.
Implicit Types X Y : gmultiset A.

Lemma gmultiset_eq X Y : X = Y   x, multiplicity x X = multiplicity x Y.
Proof.
  split; [by intros ->|intros HXY].
  destruct X as [X], Y as [Y]; f_equal; apply map_eq; intros x.
  specialize (HXY x); unfold multiplicity in *; simpl in *.
  repeat case_match; naive_solver.
Qed.
74
75
Global Instance gmultiset_leibniz : LeibnizEquiv (gmultiset A).
Proof. intros X Y. by rewrite gmultiset_eq. Qed.
76
Global Instance gmultiset_equiv_equivalence : Equivalence (@{gmultiset A}).
77
Proof. constructor; repeat intro; naive_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
78
79
80
81
82
83
84
85

(* Multiplicity *)
Lemma multiplicity_empty x : multiplicity x  = 0.
Proof. done. Qed.
Lemma multiplicity_singleton x : multiplicity x {[ x ]} = 1.
Proof. unfold multiplicity; simpl. by rewrite lookup_singleton. Qed.
Lemma multiplicity_singleton_ne x y : x  y  multiplicity x {[ y ]} = 0.
Proof. intros. unfold multiplicity; simpl. by rewrite lookup_singleton_ne. Qed.
86
87
88
89
90
91
92
Lemma multiplicity_singleton' x y :
  multiplicity x {[ y ]} = if decide (x = y) then 1 else 0.
Proof.
  destruct (decide _) as [->|].
  - by rewrite multiplicity_singleton.
  - by rewrite multiplicity_singleton_ne.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
93
Lemma multiplicity_union X Y x :
94
95
96
97
98
99
100
101
102
103
104
105
106
  multiplicity x (X  Y) = multiplicity x X `max` multiplicity x Y.
Proof.
  destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
  rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; lia.
Qed.
Lemma multiplicity_intersection X Y x :
  multiplicity x (X  Y) = multiplicity x X `min` multiplicity x Y.
Proof.
  destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
  rewrite lookup_intersection_with. destruct (X !! _), (Y !! _); simpl; lia.
Qed.
Lemma multiplicity_disj_union X Y x :
  multiplicity x (X  Y) = multiplicity x X + multiplicity x Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
107
108
Proof.
  destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
Ralf Jung's avatar
Ralf Jung committed
109
  rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
110
111
112
113
114
115
Qed.
Lemma multiplicity_difference X Y x :
  multiplicity x (X  Y) = multiplicity x X - multiplicity x Y.
Proof.
  destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
  rewrite lookup_difference_with.
Ralf Jung's avatar
Ralf Jung committed
116
  destruct (X !! _), (Y !! _); simplify_option_eq; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
117
118
Qed.

119
(* Set_ *)
120
121
122
Lemma elem_of_multiplicity x X : x  X  0 < multiplicity x X.
Proof. done. Qed.

123
Global Instance gmultiset_simple_set : SemiSet A (gmultiset A).
124
125
Proof.
  split.
Ralf Jung's avatar
Ralf Jung committed
126
  - intros x. rewrite elem_of_multiplicity, multiplicity_empty. lia.
127
128
129
  - intros x y.
    rewrite elem_of_multiplicity, multiplicity_singleton'.
    destruct (decide (x = y)); intuition lia.
Ralf Jung's avatar
Ralf Jung committed
130
  - intros X Y x. rewrite !elem_of_multiplicity, multiplicity_union. lia.
131
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
132
Global Instance gmultiset_elem_of_dec : RelDecision (@{gmultiset A}).
133
Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined.
134

135
136
137
Lemma gmultiset_elem_of_disj_union X Y x : x  X  Y  x  X  x  Y.
Proof. rewrite !elem_of_multiplicity, multiplicity_disj_union. lia. Qed.

138
Global Instance set_unfold_gmultiset_disj_union x X Y P Q :
139
140
  SetUnfoldElemOf x X P  SetUnfoldElemOf x Y Q 
  SetUnfoldElemOf x (X  Y) (P  Q).
141
Proof.
142
  intros ??; constructor. rewrite gmultiset_elem_of_disj_union.
143
  by rewrite <-(set_unfold_elem_of x X P), <-(set_unfold_elem_of x Y Q).
144
145
Qed.

146
(* Algebraic laws *)
147
148
(** For union *)
Global Instance gmultiset_union_comm : Comm (=@{gmultiset A}) ().
Robbert Krebbers's avatar
Robbert Krebbers committed
149
Proof.
Ralf Jung's avatar
Ralf Jung committed
150
  intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
151
Qed.
152
Global Instance gmultiset_union_assoc : Assoc (=@{gmultiset A}) ().
Robbert Krebbers's avatar
Robbert Krebbers committed
153
Proof.
Ralf Jung's avatar
Ralf Jung committed
154
  intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
155
Qed.
156
Global Instance gmultiset_union_left_id : LeftId (=@{gmultiset A})  ().
Robbert Krebbers's avatar
Robbert Krebbers committed
157
158
159
160
Proof.
  intros X. apply gmultiset_eq; intros x.
  by rewrite multiplicity_union, multiplicity_empty.
Qed.
161
Global Instance gmultiset_union_right_id : RightId (=@{gmultiset A})  ().
Robbert Krebbers's avatar
Robbert Krebbers committed
162
Proof. intros X. by rewrite (comm_L ()), (left_id_L _ _). Qed.
163
164
165
166
Global Instance gmultiset_union_idemp : IdemP (=@{gmultiset A}) ().
Proof.
  intros X. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
167

168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
(** For intersection *)
Global Instance gmultiset_intersection_comm : Comm (=@{gmultiset A}) ().
Proof.
  intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_intersection; lia.
Qed.
Global Instance gmultiset_intersection_assoc : Assoc (=@{gmultiset A}) ().
Proof.
  intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_intersection; lia.
Qed.
Global Instance gmultiset_intersection_left_absorb : LeftAbsorb (=@{gmultiset A})  ().
Proof.
  intros X. apply gmultiset_eq; intros x.
  by rewrite multiplicity_intersection, multiplicity_empty.
Qed.
Global Instance gmultiset_intersection_right_absorb : RightAbsorb (=@{gmultiset A})  ().
Proof. intros X. by rewrite (comm_L ()), (left_absorb_L _ _). Qed.
Global Instance gmultiset_intersection_idemp : IdemP (=@{gmultiset A}) ().
Proof.
  intros X. apply gmultiset_eq; intros x. rewrite !multiplicity_intersection; lia.
Qed.

189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
Lemma gmultiset_union_intersection_l X Y Z : X  (Y  Z) = (X  Y)  (X  Z).
Proof.
  apply gmultiset_eq; intros y.
  rewrite multiplicity_union, !multiplicity_intersection, !multiplicity_union. lia.
Qed.
Lemma gmultiset_union_intersection_r X Y Z : (X  Y)  Z = (X  Z)  (Y  Z).
Proof. by rewrite <-!(comm_L _ Z), gmultiset_union_intersection_l. Qed.
Lemma gmultiset_intersection_union_l X Y Z : X  (Y  Z) = (X  Y)  (X  Z).
Proof.
  apply gmultiset_eq; intros y.
  rewrite multiplicity_union, !multiplicity_intersection, !multiplicity_union. lia.
Qed.
Lemma gmultiset_intersection_union_r X Y Z : (X  Y)  Z = (X  Z)  (Y  Z).
Proof. by rewrite <-!(comm_L _ Z), gmultiset_intersection_union_l. Qed.

204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
(** For disjoint union (aka sum) *)
Global Instance gmultiset_disj_union_comm : Comm (=@{gmultiset A}) ().
Proof.
  intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_disj_union; lia.
Qed.
Global Instance gmultiset_disj_union_assoc : Assoc (=@{gmultiset A}) ().
Proof.
  intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_disj_union; lia.
Qed.
Global Instance gmultiset_disj_union_left_id : LeftId (=@{gmultiset A})  ().
Proof.
  intros X. apply gmultiset_eq; intros x.
  by rewrite multiplicity_disj_union, multiplicity_empty.
Qed.
Global Instance gmultiset_disj_union_right_id : RightId (=@{gmultiset A})  ().
219
Proof. intros X. by rewrite (comm_L ()), (left_id_L _ _). Qed.
220
221

Global Instance gmultiset_disj_union_inj_1 X : Inj (=) (=) (X ).
Robbert Krebbers's avatar
Robbert Krebbers committed
222
223
Proof.
  intros Y1 Y2. rewrite !gmultiset_eq. intros HX x; generalize (HX x).
224
  rewrite !multiplicity_disj_union. lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
225
Qed.
226
Global Instance gmultiset_disj_union_inj_2 X : Inj (=) (=) ( X).
Robbert Krebbers's avatar
Robbert Krebbers committed
227
228
Proof. intros Y1 Y2. rewrite <-!(comm_L _ X). apply (inj _). Qed.

229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
Lemma gmultiset_disj_union_intersection_l X Y Z : X  (Y  Z) = (X  Y)  (X  Z).
Proof.
  apply gmultiset_eq; intros y.
  rewrite multiplicity_disj_union, !multiplicity_intersection,
    !multiplicity_disj_union. lia.
Qed.
Lemma gmultiset_disj_union_intersection_r X Y Z : (X  Y)  Z = (X  Z)  (Y  Z).
Proof. by rewrite <-!(comm_L _ Z), gmultiset_disj_union_intersection_l. Qed.

Lemma gmultiset_disj_union_union_l X Y Z : X  (Y  Z) = (X  Y)  (X  Z).
Proof.
  apply gmultiset_eq; intros y.
  rewrite multiplicity_disj_union, !multiplicity_union,
    !multiplicity_disj_union. lia.
Qed.
Lemma gmultiset_disj_union_union_r X Y Z : (X  Y)  Z = (X  Z)  (Y  Z).
Proof. by rewrite <-!(comm_L _ Z), gmultiset_disj_union_union_l. Qed.

247
(** Misc *)
248
Lemma gmultiset_non_empty_singleton x : {[ x ]} @{gmultiset A} .
Robbert Krebbers's avatar
Robbert Krebbers committed
249
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
250
251
  rewrite gmultiset_eq. intros Hx; generalize (Hx x).
  by rewrite multiplicity_singleton, multiplicity_empty.
Robbert Krebbers's avatar
Robbert Krebbers committed
252
253
Qed.

254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
(** Conversion from lists *)
Lemma list_to_set_disj_nil : list_to_set_disj [] =@{gmultiset A} .
Proof. done. Qed.
Lemma list_to_set_disj_cons x l :
  list_to_set_disj (x :: l) =@{gmultiset A} {[ x ]}  list_to_set_disj l.
Proof. done. Qed.
Lemma list_to_set_disj_app l1 l2 :
  list_to_set_disj (l1 ++ l2) =@{gmultiset A} list_to_set_disj l1  list_to_set_disj l2.
Proof.
  induction l1 as [|x l1 IH]; simpl.
  - by rewrite (left_id_L _ _).
  - by rewrite IH, (assoc_L _).
Qed.
Global Instance list_to_set_disj_perm :
  Proper (() ==> (=)) (list_to_set_disj (C:=gmultiset A)).
Proof.
  induction 1 as [|x l l' _ IH|x y l|l l' l'' _ IH1 _ IH2]; simpl; auto.
  - by rewrite IH.
  - by rewrite !(assoc_L _), (comm_L _ {[ x ]}).
  - by rewrite IH1.
Qed.

(** Properties of the elements operation *)
Robbert Krebbers's avatar
Robbert Krebbers committed
277
278
279
280
281
282
283
Lemma gmultiset_elements_empty : elements ( : gmultiset A) = [].
Proof.
  unfold elements, gmultiset_elements; simpl. by rewrite map_to_list_empty.
Qed.
Lemma gmultiset_elements_empty_inv X : elements X = []  X = .
Proof.
  destruct X as [X]; unfold elements, gmultiset_elements; simpl.
284
285
286
  intros; apply (f_equal GMultiSet). destruct (map_to_list X) as [|[]] eqn:?.
  - by apply map_to_list_empty_inv.
  - naive_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
287
288
289
290
291
292
293
294
295
296
Qed.
Lemma gmultiset_elements_empty' X : elements X = []  X = .
Proof.
  split; intros HX; [by apply gmultiset_elements_empty_inv|].
  by rewrite HX, gmultiset_elements_empty.
Qed.
Lemma gmultiset_elements_singleton x : elements ({[ x ]} : gmultiset A) = [ x ].
Proof.
  unfold elements, gmultiset_elements; simpl. by rewrite map_to_list_singleton.
Qed.
297
298
Lemma gmultiset_elements_disj_union X Y :
  elements (X  Y)  elements X ++ elements Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
299
300
301
302
Proof.
  destruct X as [X], Y as [Y]; unfold elements, gmultiset_elements.
  set (f xn := let '(x, n) := xn in replicate (S n) x); simpl.
  revert Y; induction X as [|x n X HX IH] using map_ind; intros Y.
303
  { by rewrite (left_id_L _ _ Y), map_to_list_empty. }
Robbert Krebbers's avatar
Robbert Krebbers committed
304
305
306
307
308
309
  destruct (Y !! x) as [n'|] eqn:HY.
  - rewrite <-(insert_id Y x n'), <-(insert_delete Y) by done.
    erewrite <-insert_union_with by done.
    rewrite !map_to_list_insert, !bind_cons
      by (by rewrite ?lookup_union_with, ?lookup_delete, ?HX).
    rewrite (assoc_L _), <-(comm (++) (f (_,n'))), <-!(assoc_L _), <-IH.
310
311
    rewrite (assoc_L _). f_equiv.
    rewrite (comm _); simpl. by rewrite replicate_plus, Permutation_middle.
Robbert Krebbers's avatar
Robbert Krebbers committed
312
313
314
315
316
317
318
319
320
321
322
  - rewrite <-insert_union_with_l, !map_to_list_insert, !bind_cons
      by (by rewrite ?lookup_union_with, ?HX, ?HY).
    by rewrite <-(assoc_L (++)), <-IH.
Qed.
Lemma gmultiset_elem_of_elements x X : x  elements X  x  X.
Proof.
  destruct X as [X]. unfold elements, gmultiset_elements.
  set (f xn := let '(x, n) := xn in replicate (S n) x); simpl.
  unfold elem_of at 2, gmultiset_elem_of, multiplicity; simpl.
  rewrite elem_of_list_bind. split.
  - intros [[??] [[<- ?]%elem_of_replicate ->%elem_of_map_to_list]]; lia.
Ralf Jung's avatar
Ralf Jung committed
323
  - intros. destruct (X !! x) as [n|] eqn:Hx; [|lia].
Robbert Krebbers's avatar
Robbert Krebbers committed
324
    exists (x,n); split; [|by apply elem_of_map_to_list].
Ralf Jung's avatar
Ralf Jung committed
325
    apply elem_of_replicate; auto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
326
Qed.
327
328
329
330
Lemma gmultiset_elem_of_dom x X : x  dom (gset A) X  x  X.
Proof.
  unfold dom, gmultiset_dom, elem_of at 2, gmultiset_elem_of, multiplicity.
  destruct X as [X]; simpl; rewrite elem_of_dom, <-not_eq_None_Some.
Ralf Jung's avatar
Ralf Jung committed
331
  destruct (X !! x); naive_solver lia.
332
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
333

Dan Frumin's avatar
Dan Frumin committed
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
(** Properties of the set_fold operation *)
Lemma gmultiset_set_fold_empty {B} (f : A  B  B) (b : B) :
  set_fold f b ( : gmultiset A) = b.
Proof. by unfold set_fold; simpl; rewrite gmultiset_elements_empty. Qed.
Lemma gmultiset_set_fold_singleton {B} (f : A  B  B) (b : B) (a : A) :
  set_fold f b ({[a]} : gmultiset A) = f a b.
Proof. by unfold set_fold; simpl; rewrite gmultiset_elements_singleton. Qed.
Lemma gmultiset_set_fold_disj_union (f : A  A  A) (b : A) X Y :
  Comm (=) f 
  Assoc (=) f 
  set_fold f b (X  Y) = set_fold f (set_fold f b X) Y.
Proof.
  intros Hcomm Hassoc. unfold set_fold; simpl.
  by rewrite gmultiset_elements_disj_union, <- foldr_app, (comm (++)).
Qed.

(** Properties of the size operation *)
Robbert Krebbers's avatar
Robbert Krebbers committed
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
Lemma gmultiset_size_empty : size ( : gmultiset A) = 0.
Proof. done. Qed.
Lemma gmultiset_size_empty_inv X : size X = 0  X = .
Proof.
  unfold size, gmultiset_size; simpl. rewrite length_zero_iff_nil.
  apply gmultiset_elements_empty_inv.
Qed.
Lemma gmultiset_size_empty_iff X : size X = 0  X = .
Proof.
  split; [apply gmultiset_size_empty_inv|].
  by intros ->; rewrite gmultiset_size_empty.
Qed.
Lemma gmultiset_size_non_empty_iff X : size X  0  X  .
Proof. by rewrite gmultiset_size_empty_iff. Qed.

Lemma gmultiset_choose_or_empty X : ( x, x  X)  X = .
Proof.
  destruct (elements X) as [|x l] eqn:HX; [right|left].
  - by apply gmultiset_elements_empty_inv.
  - exists x. rewrite <-gmultiset_elem_of_elements, HX. by left.
Qed.
Lemma gmultiset_choose X : X     x, x  X.
Proof. intros. by destruct (gmultiset_choose_or_empty X). Qed.
Lemma gmultiset_size_pos_elem_of X : 0 < size X   x, x  X.
Proof.
  intros Hsz. destruct (gmultiset_choose_or_empty X) as [|HX]; [done|].
  contradict Hsz. rewrite HX, gmultiset_size_empty; lia.
Qed.

Lemma gmultiset_size_singleton x : size ({[ x ]} : gmultiset A) = 1.
Proof.
  unfold size, gmultiset_size; simpl. by rewrite gmultiset_elements_singleton.
Qed.
384
Lemma gmultiset_size_disj_union X Y : size (X  Y) = size X + size Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
385
386
Proof.
  unfold size, gmultiset_size; simpl.
387
  by rewrite gmultiset_elements_disj_union, app_length.
Robbert Krebbers's avatar
Robbert Krebbers committed
388
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
389

Dan Frumin's avatar
Dan Frumin committed
390
(** Order stuff *)
391
Global Instance gmultiset_po : PartialOrder (@{gmultiset A}).
Robbert Krebbers's avatar
Robbert Krebbers committed
392
393
394
395
396
397
398
Proof.
  split; [split|].
  - by intros X x.
  - intros X Y Z HXY HYZ x. by trans (multiplicity x Y).
  - intros X Y HXY HYX; apply gmultiset_eq; intros x. by apply (anti_symm ()).
Qed.

399
400
401
402
403
Lemma gmultiset_subseteq_alt X Y :
  X  Y 
  map_relation () (λ _, False) (λ _, True) (gmultiset_car X) (gmultiset_car Y).
Proof.
  apply forall_proper; intros x. unfold multiplicity.
Ralf Jung's avatar
Ralf Jung committed
404
  destruct (gmultiset_car X !! x), (gmultiset_car Y !! x); naive_solver lia.
405
Qed.
406
Global Instance gmultiset_subseteq_dec : RelDecision (@{gmultiset A}).
407
Proof.
408
 refine (λ X Y, cast_if (decide (map_relation ()
409
410
411
412
   (λ _, False) (λ _, True) (gmultiset_car X) (gmultiset_car Y))));
   by rewrite gmultiset_subseteq_alt.
Defined.

Robbert Krebbers's avatar
Robbert Krebbers committed
413
414
Lemma gmultiset_subset_subseteq X Y : X  Y  X  Y.
Proof. apply strict_include. Qed.
Tej Chajed's avatar
Tej Chajed committed
415
Hint Resolve gmultiset_subset_subseteq : core.
Robbert Krebbers's avatar
Robbert Krebbers committed
416
417

Lemma gmultiset_empty_subseteq X :   X.
Ralf Jung's avatar
Ralf Jung committed
418
Proof. intros x. rewrite multiplicity_empty. lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
419
420

Lemma gmultiset_union_subseteq_l X Y : X  X  Y.
Ralf Jung's avatar
Ralf Jung committed
421
Proof. intros x. rewrite multiplicity_union. lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
422
Lemma gmultiset_union_subseteq_r X Y : Y  X  Y.
Ralf Jung's avatar
Ralf Jung committed
423
Proof. intros x. rewrite multiplicity_union. lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
424
Lemma gmultiset_union_mono X1 X2 Y1 Y2 : X1  X2  Y1  Y2  X1  Y1  X2  Y2.
425
426
427
428
Proof.
  intros HX HY x. rewrite !multiplicity_union.
  specialize (HX x); specialize (HY x); lia.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
429
430
431
432
Lemma gmultiset_union_mono_l X Y1 Y2 : Y1  Y2  X  Y1  X  Y2.
Proof. intros. by apply gmultiset_union_mono. Qed.
Lemma gmultiset_union_mono_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
Proof. intros. by apply gmultiset_union_mono. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
433

434
435
436
437
438
439
440
441
442
443
444
Lemma gmultiset_disj_union_subseteq_l X Y : X  X  Y.
Proof. intros x. rewrite multiplicity_disj_union. lia. Qed.
Lemma gmultiset_disj_union_subseteq_r X Y : Y  X  Y.
Proof. intros x. rewrite multiplicity_disj_union. lia. Qed.
Lemma gmultiset_disj_union_mono X1 X2 Y1 Y2 : X1  X2  Y1  Y2  X1  Y1  X2  Y2.
Proof. intros ?? x. rewrite !multiplicity_disj_union. by apply Nat.add_le_mono. Qed.
Lemma gmultiset_disj_union_mono_l X Y1 Y2 : Y1  Y2  X  Y1  X  Y2.
Proof. intros. by apply gmultiset_disj_union_mono. Qed.
Lemma gmultiset_disj_union_mono_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
Proof. intros. by apply gmultiset_disj_union_mono. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
445
Lemma gmultiset_subset X Y : X  Y  size X < size Y  X  Y.
Ralf Jung's avatar
Ralf Jung committed
446
Proof. intros. apply strict_spec_alt; split; naive_solver auto with lia. Qed.
447
Lemma gmultiset_disj_union_subset_l X Y : Y    X  X  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
448
449
Proof.
  intros HY%gmultiset_size_non_empty_iff.
450
451
  apply gmultiset_subset; auto using gmultiset_disj_union_subseteq_l.
  rewrite gmultiset_size_disj_union; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
452
Qed.
453
454
Lemma gmultiset_union_subset_r X Y : X    Y  X  Y.
Proof. rewrite (comm_L ()). apply gmultiset_disj_union_subset_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
455

Robbert Krebbers's avatar
Robbert Krebbers committed
456
Lemma gmultiset_elem_of_singleton_subseteq x X : x  X  {[ x ]}  X.
Robbert Krebbers's avatar
Robbert Krebbers committed
457
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
458
  rewrite elem_of_multiplicity. split.
459
460
  - intros Hx y. rewrite multiplicity_singleton'.
    destruct (decide (y = x)); naive_solver lia.
Ralf Jung's avatar
Ralf Jung committed
461
  - intros Hx. generalize (Hx x). rewrite multiplicity_singleton. lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
462
463
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
464
465
466
Lemma gmultiset_elem_of_subseteq X1 X2 x : x  X1  X1  X2  x  X2.
Proof. rewrite !gmultiset_elem_of_singleton_subseteq. by intros ->. Qed.

467
Lemma gmultiset_disj_union_difference X Y : X  Y  Y = X  Y  X.
Robbert Krebbers's avatar
Robbert Krebbers committed
468
469
Proof.
  intros HXY. apply gmultiset_eq; intros x; specialize (HXY x).
470
  rewrite multiplicity_disj_union, multiplicity_difference; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
471
Qed.
472
Lemma gmultiset_disj_union_difference' x Y : x  Y  Y = {[ x ]}  Y  {[ x ]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
473
Proof.
474
  intros. by apply gmultiset_disj_union_difference,
Robbert Krebbers's avatar
Robbert Krebbers committed
475
476
    gmultiset_elem_of_singleton_subseteq.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
477

Robbert Krebbers's avatar
Robbert Krebbers committed
478
479
Lemma gmultiset_size_difference X Y : Y  X  size (X  Y) = size X - size Y.
Proof.
480
481
  intros HX%gmultiset_disj_union_difference.
  rewrite HX at 2; rewrite gmultiset_size_disj_union. lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
482
483
Qed.

Dan Frumin's avatar
Dan Frumin committed
484
485
486
487
488
489
490
Lemma gmultiset_empty_difference X Y : Y  X  Y  X = .
Proof.
  intros HYX. unfold_leibniz. intros x.
  rewrite multiplicity_difference, multiplicity_empty.
  specialize (HYX x). lia.
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
491
492
493
494
Lemma gmultiset_non_empty_difference X Y : X  Y  Y  X  .
Proof.
  intros [_ HXY2] Hdiff; destruct HXY2; intros x.
  generalize (f_equal (multiplicity x) Hdiff).
Ralf Jung's avatar
Ralf Jung committed
495
  rewrite multiplicity_difference, multiplicity_empty; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
496
497
Qed.

Dan Frumin's avatar
Dan Frumin committed
498
499
500
Lemma gmultiset_difference_diag X : X  X = .
Proof. by apply gmultiset_empty_difference. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
501
502
503
Lemma gmultiset_difference_subset X Y : X    X  Y  Y  X  Y.
Proof.
  intros. eapply strict_transitive_l; [by apply gmultiset_union_subset_r|].
504
  by rewrite <-(gmultiset_disj_union_difference X Y).
Robbert Krebbers's avatar
Robbert Krebbers committed
505
506
Qed.

Dan Frumin's avatar
Dan Frumin committed
507
(** Mononicity *)
Robbert Krebbers's avatar
Robbert Krebbers committed
508
Lemma gmultiset_elements_submseteq X Y : X  Y  elements X + elements Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
509
Proof.
510
  intros ->%gmultiset_disj_union_difference. rewrite gmultiset_elements_disj_union.
Robbert Krebbers's avatar
Robbert Krebbers committed
511
  by apply submseteq_inserts_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
512
513
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
514
Lemma gmultiset_subseteq_size X Y : X  Y  size X  size Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
515
Proof. intros. by apply submseteq_length, gmultiset_elements_submseteq. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
516
517
518
519

Lemma gmultiset_subset_size X Y : X  Y  size X < size Y.
Proof.
  intros HXY. assert (size (Y  X)  0).
Robbert Krebbers's avatar
Robbert Krebbers committed
520
  { by apply gmultiset_size_non_empty_iff, gmultiset_non_empty_difference. }
521
522
  rewrite (gmultiset_disj_union_difference X Y),
    gmultiset_size_disj_union by auto. lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
523
524
Qed.

Dan Frumin's avatar
Dan Frumin committed
525
(** Well-foundedness *)
526
Lemma gmultiset_wf : wf (@{gmultiset A}).
Robbert Krebbers's avatar
Robbert Krebbers committed
527
528
529
Proof.
  apply (wf_projected (<) size); auto using gmultiset_subset_size, lt_wf.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
530
531

Lemma gmultiset_ind (P : gmultiset A  Prop) :
532
  P   ( x X, P X  P ({[ x ]}  X))   X, P X.
Robbert Krebbers's avatar
Robbert Krebbers committed
533
534
535
Proof.
  intros Hemp Hinsert X. induction (gmultiset_wf X) as [X _ IH].
  destruct (gmultiset_choose_or_empty X) as [[x Hx]| ->]; auto.
536
  rewrite (gmultiset_disj_union_difference' x X) by done.
Robbert Krebbers's avatar
Robbert Krebbers committed
537
538
  apply Hinsert, IH, gmultiset_difference_subset,
    gmultiset_elem_of_singleton_subseteq; auto using gmultiset_non_empty_singleton.
Robbert Krebbers's avatar
Robbert Krebbers committed
539
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
540
End lemmas.