gmultiset.v 21.6 KB
Newer Older
1
From stdpp Require Export countable.
Robbert Krebbers's avatar
Robbert Krebbers committed
2
From stdpp Require Import gmap.
3
Set Default Proof Using "Type".
Robbert Krebbers's avatar
Robbert Krebbers committed
4 5

Record gmultiset A `{Countable A} := GMultiSet { gmultiset_car : gmap A nat }.
6 7
Arguments GMultiSet {_ _ _} _ : assert.
Arguments gmultiset_car {_ _ _} _ : assert.
Robbert Krebbers's avatar
Robbert Krebbers committed
8

9
Instance gmultiset_eq_dec `{Countable A} : EqDecision (gmultiset A).
Robbert Krebbers's avatar
Robbert Krebbers committed
10 11
Proof. solve_decision. Defined.

12
Program Instance gmultiset_countable `{Countable A} :
Robbert Krebbers's avatar
Robbert Krebbers committed
13
    Countable (gmultiset A) := {|
14
  encode X := encode (gmultiset_car X); decode p := GMultiSet <$> decode p
Robbert Krebbers's avatar
Robbert Krebbers committed
15 16 17 18 19 20 21 22
|}.
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.
23
  Global Instance gmultiset_elem_of : ElemOf A (gmultiset A) := λ x X,
Robbert Krebbers's avatar
Robbert Krebbers committed
24
    0 < multiplicity x X.
25
  Global Instance gmultiset_subseteq : SubsetEq (gmultiset A) := λ X Y,  x,
Robbert Krebbers's avatar
Robbert Krebbers committed
26
    multiplicity x X  multiplicity x Y.
27 28
  Global Instance gmultiset_equiv : Equiv (gmultiset A) := λ X Y,  x,
    multiplicity x X = multiplicity x Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
29

30
  Global Instance gmultiset_elements : Elements A (gmultiset A) := λ X,
31
    let (X) := X in ''(x,n)  map_to_list X; replicate (S n) x.
32
  Global Instance gmultiset_size : Size (gmultiset A) := length  elements.
Robbert Krebbers's avatar
Robbert Krebbers committed
33

34 35
  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
36
    GMultiSet {[ x := 0 ]}.
37
  Global Instance gmultiset_union : Union (gmultiset A) := λ X Y,
38 39 40 41 42 43 44
    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
45 46
    let (X) := X in let (Y) := Y in
    GMultiSet $ union_with (λ x y, Some (S (x + y))) X Y.
47
  Global Instance gmultiset_difference : Difference (gmultiset A) := λ X Y,
Robbert Krebbers's avatar
Robbert Krebbers committed
48 49 50
    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.
51

52
  Global Instance gmultiset_dom : Dom (gmultiset A) (gset A) := λ X,
53
    let (X) := X in dom _ X.
Michael Sammler's avatar
Michael Sammler committed
54
End definitions.
Robbert Krebbers's avatar
Robbert Krebbers committed
55

56 57 58
Typeclasses Opaque gmultiset_elem_of gmultiset_subseteq.
Typeclasses Opaque gmultiset_elements gmultiset_size gmultiset_empty.
Typeclasses Opaque gmultiset_singleton gmultiset_union gmultiset_difference.
59
Typeclasses Opaque gmultiset_dom.
60

Robbert Krebbers's avatar
Robbert Krebbers committed
61 62 63 64 65 66 67 68 69 70 71 72
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.
73 74
Global Instance gmultiset_leibniz : LeibnizEquiv (gmultiset A).
Proof. intros X Y. by rewrite gmultiset_eq. Qed.
75
Global Instance gmultiset_equiv_equivalence : Equivalence (@{gmultiset A}).
76
Proof. constructor; repeat intro; naive_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
77 78 79 80 81 82 83 84

(* 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.
85 86 87 88 89 90 91
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
92
Lemma multiplicity_union X Y x :
93 94 95 96 97 98 99 100 101 102 103 104 105
  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
106 107
Proof.
  destruct X as [X], Y as [Y]; unfold multiplicity; simpl.
Ralf Jung's avatar
Ralf Jung committed
108
  rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
109 110 111 112 113 114
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
115
  destruct (X !! _), (Y !! _); simplify_option_eq; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
116 117
Qed.

118
(* Set_ *)
119 120 121
Lemma elem_of_multiplicity x X : x  X  0 < multiplicity x X.
Proof. done. Qed.

122
Global Instance gmultiset_simple_set : SemiSet A (gmultiset A).
123 124
Proof.
  split.
Ralf Jung's avatar
Ralf Jung committed
125
  - intros x. rewrite elem_of_multiplicity, multiplicity_empty. lia.
126 127 128
  - intros x y.
    rewrite elem_of_multiplicity, multiplicity_singleton'.
    destruct (decide (x = y)); intuition lia.
Ralf Jung's avatar
Ralf Jung committed
129
  - intros X Y x. rewrite !elem_of_multiplicity, multiplicity_union. lia.
130
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
131
Global Instance gmultiset_elem_of_dec : RelDecision (@{gmultiset A}).
132
Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined.
133

134 135 136
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.

137
Global Instance set_unfold_gmultiset_disj_union x X Y P Q :
138 139
  SetUnfoldElemOf x X P  SetUnfoldElemOf x Y Q 
  SetUnfoldElemOf x (X  Y) (P  Q).
140
Proof.
141
  intros ??; constructor. rewrite gmultiset_elem_of_disj_union.
142
  by rewrite <-(set_unfold_elem_of x X P), <-(set_unfold_elem_of x Y Q).
143 144
Qed.

145
(* Algebraic laws *)
146 147
(** For union *)
Global Instance gmultiset_union_comm : Comm (=@{gmultiset A}) ().
Robbert Krebbers's avatar
Robbert Krebbers committed
148
Proof.
Ralf Jung's avatar
Ralf Jung committed
149
  intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
150
Qed.
151
Global Instance gmultiset_union_assoc : Assoc (=@{gmultiset A}) ().
Robbert Krebbers's avatar
Robbert Krebbers committed
152
Proof.
Ralf Jung's avatar
Ralf Jung committed
153
  intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
154
Qed.
155
Global Instance gmultiset_union_left_id : LeftId (=@{gmultiset A})  ().
Robbert Krebbers's avatar
Robbert Krebbers committed
156 157 158 159
Proof.
  intros X. apply gmultiset_eq; intros x.
  by rewrite multiplicity_union, multiplicity_empty.
Qed.
160
Global Instance gmultiset_union_right_id : RightId (=@{gmultiset A})  ().
Robbert Krebbers's avatar
Robbert Krebbers committed
161
Proof. intros X. by rewrite (comm_L ()), (left_id_L _ _). Qed.
162 163 164 165
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
166

167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187
(** 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.

188 189 190 191 192 193 194 195 196 197 198 199 200 201 202
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.

203 204 205 206 207 208 209 210 211 212 213 214 215 216 217
(** 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})  ().
218
Proof. intros X. by rewrite (comm_L ()), (left_id_L _ _). Qed.
219

220
Global Instance gmultiset_disj_union_inj_1 X : Inj (=) (=) (X .).
Robbert Krebbers's avatar
Robbert Krebbers committed
221 222
Proof.
  intros Y1 Y2. rewrite !gmultiset_eq. intros HX x; generalize (HX x).
223
  rewrite !multiplicity_disj_union. lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
224
Qed.
225
Global Instance gmultiset_disj_union_inj_2 X : Inj (=) (=) (. X).
226
Proof. intros Y1 Y2. rewrite <-!(comm_L _ X). apply (inj (X .)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
227

228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245
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.

246
(** Misc *)
247
Lemma gmultiset_non_empty_singleton x : {[ x ]} @{gmultiset A} .
Robbert Krebbers's avatar
Robbert Krebbers committed
248
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
249 250
  rewrite gmultiset_eq. intros Hx; generalize (Hx x).
  by rewrite multiplicity_singleton, multiplicity_empty.
Robbert Krebbers's avatar
Robbert Krebbers committed
251 252
Qed.

253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275
(** 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
276 277 278 279 280 281 282
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.
283 284 285
  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
286 287 288 289 290 291 292 293 294 295
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.
296 297
Lemma gmultiset_elements_disj_union X Y :
  elements (X  Y)  elements X ++ elements Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
298 299 300 301
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.
302
  { by rewrite (left_id_L _ _ Y), map_to_list_empty. }
Robbert Krebbers's avatar
Robbert Krebbers committed
303 304 305 306 307 308
  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.
309 310
    rewrite (assoc_L _). f_equiv.
    rewrite (comm _); simpl. by rewrite replicate_plus, Permutation_middle.
Robbert Krebbers's avatar
Robbert Krebbers committed
311 312 313 314 315 316 317 318 319 320 321
  - 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
322
  - intros. destruct (X !! x) as [n|] eqn:Hx; [|lia].
Robbert Krebbers's avatar
Robbert Krebbers committed
323
    exists (x,n); split; [|by apply elem_of_map_to_list].
Ralf Jung's avatar
Ralf Jung committed
324
    apply elem_of_replicate; auto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
325
Qed.
326 327 328 329
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
330
  destruct (X !! x); naive_solver lia.
331
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
332

Dan Frumin's avatar
Dan Frumin committed
333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349
(** 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
350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382
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.
383
Lemma gmultiset_size_disj_union X Y : size (X  Y) = size X + size Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
384 385
Proof.
  unfold size, gmultiset_size; simpl.
386
  by rewrite gmultiset_elements_disj_union, app_length.
Robbert Krebbers's avatar
Robbert Krebbers committed
387
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
388

Dan Frumin's avatar
Dan Frumin committed
389
(** Order stuff *)
390
Global Instance gmultiset_po : PartialOrder (@{gmultiset A}).
Robbert Krebbers's avatar
Robbert Krebbers committed
391 392 393 394 395 396 397
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.

398 399 400 401 402
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
403
  destruct (gmultiset_car X !! x), (gmultiset_car Y !! x); naive_solver lia.
404
Qed.
405
Global Instance gmultiset_subseteq_dec : RelDecision (@{gmultiset A}).
406
Proof.
407
 refine (λ X Y, cast_if (decide (map_relation ()
408 409 410 411
   (λ _, False) (λ _, True) (gmultiset_car X) (gmultiset_car Y))));
   by rewrite gmultiset_subseteq_alt.
Defined.

Robbert Krebbers's avatar
Robbert Krebbers committed
412 413
Lemma gmultiset_subset_subseteq X Y : X  Y  X  Y.
Proof. apply strict_include. Qed.
Tej Chajed's avatar
Tej Chajed committed
414
Hint Resolve gmultiset_subset_subseteq : core.
Robbert Krebbers's avatar
Robbert Krebbers committed
415 416

Lemma gmultiset_empty_subseteq X :   X.
Ralf Jung's avatar
Ralf Jung committed
417
Proof. intros x. rewrite multiplicity_empty. lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
418 419

Lemma gmultiset_union_subseteq_l X Y : X  X  Y.
Ralf Jung's avatar
Ralf Jung committed
420
Proof. intros x. rewrite multiplicity_union. lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
421
Lemma gmultiset_union_subseteq_r X Y : Y  X  Y.
Ralf Jung's avatar
Ralf Jung committed
422
Proof. intros x. rewrite multiplicity_union. lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
423
Lemma gmultiset_union_mono X1 X2 Y1 Y2 : X1  X2  Y1  Y2  X1  Y1  X2  Y2.
424 425 426 427
Proof.
  intros HX HY x. rewrite !multiplicity_union.
  specialize (HX x); specialize (HY x); lia.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
428 429 430 431
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
432

433 434 435 436 437 438 439 440 441 442 443
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
444
Lemma gmultiset_subset X Y : X  Y  size X < size Y  X  Y.
Ralf Jung's avatar
Ralf Jung committed
445
Proof. intros. apply strict_spec_alt; split; naive_solver auto with lia. Qed.
446
Lemma gmultiset_disj_union_subset_l X Y : Y    X  X  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
447 448
Proof.
  intros HY%gmultiset_size_non_empty_iff.
449 450
  apply gmultiset_subset; auto using gmultiset_disj_union_subseteq_l.
  rewrite gmultiset_size_disj_union; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
451
Qed.
452 453
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
454

Robbert Krebbers's avatar
Robbert Krebbers committed
455
Lemma gmultiset_elem_of_singleton_subseteq x X : x  X  {[ x ]}  X.
Robbert Krebbers's avatar
Robbert Krebbers committed
456
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
457
  rewrite elem_of_multiplicity. split.
458 459
  - intros Hx y. rewrite multiplicity_singleton'.
    destruct (decide (y = x)); naive_solver lia.
Ralf Jung's avatar
Ralf Jung committed
460
  - intros Hx. generalize (Hx x). rewrite multiplicity_singleton. lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
461 462
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
463 464 465
Lemma gmultiset_elem_of_subseteq X1 X2 x : x  X1  X1  X2  x  X2.
Proof. rewrite !gmultiset_elem_of_singleton_subseteq. by intros ->. Qed.

466
Lemma gmultiset_disj_union_difference X Y : X  Y  Y = X  Y  X.
Robbert Krebbers's avatar
Robbert Krebbers committed
467 468
Proof.
  intros HXY. apply gmultiset_eq; intros x; specialize (HXY x).
469
  rewrite multiplicity_disj_union, multiplicity_difference; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
470
Qed.
471
Lemma gmultiset_disj_union_difference' x Y : x  Y  Y = {[ x ]}  Y  {[ x ]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
472
Proof.
473
  intros. by apply gmultiset_disj_union_difference,
Robbert Krebbers's avatar
Robbert Krebbers committed
474 475
    gmultiset_elem_of_singleton_subseteq.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
476

Robbert Krebbers's avatar
Robbert Krebbers committed
477 478
Lemma gmultiset_size_difference X Y : Y  X  size (X  Y) = size X - size Y.
Proof.
479 480
  intros HX%gmultiset_disj_union_difference.
  rewrite HX at 2; rewrite gmultiset_size_disj_union. lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
481 482
Qed.

Dan Frumin's avatar
Dan Frumin committed
483 484 485 486 487 488 489
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
490 491 492 493
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
494
  rewrite multiplicity_difference, multiplicity_empty; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
495 496
Qed.

Dan Frumin's avatar
Dan Frumin committed
497 498 499
Lemma gmultiset_difference_diag X : X  X = .
Proof. by apply gmultiset_empty_difference. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
500 501 502
Lemma gmultiset_difference_subset X Y : X    X  Y  Y  X  Y.
Proof.
  intros. eapply strict_transitive_l; [by apply gmultiset_union_subset_r|].
503
  by rewrite <-(gmultiset_disj_union_difference X Y).
Robbert Krebbers's avatar
Robbert Krebbers committed
504 505
Qed.

Dan Frumin's avatar
Dan Frumin committed
506
(** Mononicity *)
Robbert Krebbers's avatar
Robbert Krebbers committed
507
Lemma gmultiset_elements_submseteq X Y : X  Y  elements X + elements Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
508
Proof.
509
  intros ->%gmultiset_disj_union_difference. rewrite gmultiset_elements_disj_union.
Robbert Krebbers's avatar
Robbert Krebbers committed
510
  by apply submseteq_inserts_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
511 512
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
513
Lemma gmultiset_subseteq_size X Y : X  Y  size X  size Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
514
Proof. intros. by apply submseteq_length, gmultiset_elements_submseteq. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
515 516 517 518

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
519
  { by apply gmultiset_size_non_empty_iff, gmultiset_non_empty_difference. }
520 521
  rewrite (gmultiset_disj_union_difference X Y),
    gmultiset_size_disj_union by auto. lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
522 523
Qed.

Dan Frumin's avatar
Dan Frumin committed
524
(** Well-foundedness *)
525
Lemma gmultiset_wf : wf (@{gmultiset A}).
Robbert Krebbers's avatar
Robbert Krebbers committed
526 527 528
Proof.
  apply (wf_projected (<) size); auto using gmultiset_subset_size, lt_wf.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
529 530

Lemma gmultiset_ind (P : gmultiset A  Prop) :
531
  P   ( x X, P X  P ({[ x ]}  X))   X, P X.
Robbert Krebbers's avatar
Robbert Krebbers committed
532 533 534
Proof.
  intros Hemp Hinsert X. induction (gmultiset_wf X) as [X _ IH].
  destruct (gmultiset_choose_or_empty X) as [[x Hx]| ->]; auto.
535
  rewrite (gmultiset_disj_union_difference' x X) by done.
Robbert Krebbers's avatar
Robbert Krebbers committed
536 537
  apply Hinsert, IH, gmultiset_difference_subset,
    gmultiset_elem_of_singleton_subseteq; auto using gmultiset_non_empty_singleton.
Robbert Krebbers's avatar
Robbert Krebbers committed
538
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
539
End lemmas.