gmultiset.v 14.3 KB
Newer Older
1
(* Copyright (c) 2012-2017, 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 15 16 17 18 19 20 21 22 23
    Countable (gmultiset A) := {|
  encode X := encode (gmultiset_car X);  decode p := GMultiSet <$> decode p
|}.
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,
Robbert Krebbers's avatar
Robbert Krebbers committed
39 40
    let (X) := X in let (Y) := Y in
    GMultiSet $ union_with (λ x y, Some (S (x + y))) X Y.
41
  Global Instance gmultiset_difference : Difference (gmultiset A) := λ X Y,
Robbert Krebbers's avatar
Robbert Krebbers committed
42 43 44
    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.
45

46
  Global Instance gmultiset_dom : Dom (gmultiset A) (gset A) := λ X,
47
    let (X) := X in dom _ X.
Robbert Krebbers's avatar
Robbert Krebbers committed
48 49
End definitions.

50 51 52
Typeclasses Opaque gmultiset_elem_of gmultiset_subseteq.
Typeclasses Opaque gmultiset_elements gmultiset_size gmultiset_empty.
Typeclasses Opaque gmultiset_singleton gmultiset_union gmultiset_difference.
53
Typeclasses Opaque gmultiset_dom.
54

Robbert Krebbers's avatar
Robbert Krebbers committed
55 56 57 58 59 60 61 62 63 64 65 66
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.
67 68
Global Instance gmultiset_leibniz : LeibnizEquiv (gmultiset A).
Proof. intros X Y. by rewrite gmultiset_eq. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

(* 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.
Lemma multiplicity_union 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_union_with. destruct (X !! _), (Y !! _); simpl; omega.
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.
  destruct (X !! _), (Y !! _); simplify_option_eq; omega.
Qed.

91
(* Collection *)
92 93 94 95 96 97 98 99 100 101 102 103 104
Lemma elem_of_multiplicity x X : x  X  0 < multiplicity x X.
Proof. done. Qed.

Global Instance gmultiset_simple_collection : SimpleCollection A (gmultiset A).
Proof.
  split.
  - intros x. rewrite elem_of_multiplicity, multiplicity_empty. omega.
  - intros x y. destruct (decide (x = y)) as [->|].
    + rewrite elem_of_multiplicity, multiplicity_singleton. split; auto with lia.
    + rewrite elem_of_multiplicity, multiplicity_singleton_ne by done.
      by split; auto with lia.
  - intros X Y x. rewrite !elem_of_multiplicity, multiplicity_union. omega.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
105
Global Instance gmultiset_elem_of_dec : RelDecision (@{gmultiset A}).
106
Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined.
107

108
(* Algebraic laws *)
109
Global Instance gmultiset_comm : Comm (=@{gmultiset A}) ().
Robbert Krebbers's avatar
Robbert Krebbers committed
110 111 112
Proof.
  intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; omega.
Qed.
113
Global Instance gmultiset_assoc : Assoc (=@{gmultiset A}) ().
Robbert Krebbers's avatar
Robbert Krebbers committed
114 115 116
Proof.
  intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; omega.
Qed.
117
Global Instance gmultiset_left_id : LeftId (=@{gmultiset A})  ().
Robbert Krebbers's avatar
Robbert Krebbers committed
118 119 120 121
Proof.
  intros X. apply gmultiset_eq; intros x.
  by rewrite multiplicity_union, multiplicity_empty.
Qed.
122
Global Instance gmultiset_right_id : RightId (=@{gmultiset A})  ().
Robbert Krebbers's avatar
Robbert Krebbers committed
123 124 125 126 127 128 129 130 131 132
Proof. intros X. by rewrite (comm_L ()), (left_id_L _ _). Qed.

Global Instance gmultiset_union_inj_1 X : Inj (=) (=) (X ).
Proof.
  intros Y1 Y2. rewrite !gmultiset_eq. intros HX x; generalize (HX x).
  rewrite !multiplicity_union. omega.
Qed.
Global Instance gmultiset_union_inj_2 X : Inj (=) (=) ( X).
Proof. intros Y1 Y2. rewrite <-!(comm_L _ X). apply (inj _). Qed.

133
Lemma gmultiset_non_empty_singleton x : {[ x ]} @{gmultiset A} .
Robbert Krebbers's avatar
Robbert Krebbers committed
134
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
135 136
  rewrite gmultiset_eq. intros Hx; generalize (Hx x).
  by rewrite multiplicity_singleton, multiplicity_empty.
Robbert Krebbers's avatar
Robbert Krebbers committed
137 138
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
139 140 141 142 143 144 145 146
(* Properties of the elements operation *)
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.
147 148 149
  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
150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165
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.
Lemma gmultiset_elements_union X Y :
  elements (X  Y)  elements X ++ elements Y.
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.
166
  { by rewrite (left_id_L _ _ Y), map_to_list_empty. }
Robbert Krebbers's avatar
Robbert Krebbers committed
167 168 169 170 171 172
  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.
173 174
    rewrite (assoc_L _). f_equiv.
    rewrite (comm _); simpl. by rewrite replicate_plus, Permutation_middle.
Robbert Krebbers's avatar
Robbert Krebbers committed
175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
  - 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.
  - intros. destruct (X !! x) as [n|] eqn:Hx; [|omega].
    exists (x,n); split; [|by apply elem_of_map_to_list].
    apply elem_of_replicate; auto with omega.
Qed.
190 191 192 193 194 195
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.
  destruct (X !! x); naive_solver omega.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235

(* Properties of the size operation *)
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.
Lemma gmultiset_size_union X Y : size (X  Y) = size X + size Y.
Proof.
  unfold size, gmultiset_size; simpl.
  by rewrite gmultiset_elements_union, app_length.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
236 237

(* Order stuff *)
238
Global Instance gmultiset_po : PartialOrder (@{gmultiset A}).
Robbert Krebbers's avatar
Robbert Krebbers committed
239 240 241 242 243 244 245
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.

246 247 248 249 250 251 252
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.
  destruct (gmultiset_car X !! x), (gmultiset_car Y !! x); naive_solver omega.
Qed.
253
Global Instance gmultiset_subseteq_dec : RelDecision (@{gmultiset A}).
254
Proof.
255
 refine (λ X Y, cast_if (decide (map_relation ()
256 257 258 259
   (λ _, False) (λ _, True) (gmultiset_car X) (gmultiset_car Y))));
   by rewrite gmultiset_subseteq_alt.
Defined.

Robbert Krebbers's avatar
Robbert Krebbers committed
260 261 262 263 264 265 266 267 268 269 270
Lemma gmultiset_subset_subseteq X Y : X  Y  X  Y.
Proof. apply strict_include. Qed.
Hint Resolve gmultiset_subset_subseteq.

Lemma gmultiset_empty_subseteq X :   X.
Proof. intros x. rewrite multiplicity_empty. omega. Qed.

Lemma gmultiset_union_subseteq_l X Y : X  X  Y.
Proof. intros x. rewrite multiplicity_union. omega. Qed.
Lemma gmultiset_union_subseteq_r X Y : Y  X  Y.
Proof. intros x. rewrite multiplicity_union. omega. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
271
Lemma gmultiset_union_mono X1 X2 Y1 Y2 : X1  X2  Y1  Y2  X1  Y1  X2  Y2.
Robbert Krebbers's avatar
Robbert Krebbers committed
272
Proof. intros ?? x. rewrite !multiplicity_union. by apply Nat.add_le_mono. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
273 274 275 276
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
277 278 279 280 281 282 283 284 285 286 287 288

Lemma gmultiset_subset X Y : X  Y  size X < size Y  X  Y.
Proof. intros. apply strict_spec_alt; split; naive_solver auto with omega. Qed.
Lemma gmultiset_union_subset_l X Y : Y    X  X  Y.
Proof.
  intros HY%gmultiset_size_non_empty_iff.
  apply gmultiset_subset; auto using gmultiset_union_subseteq_l.
  rewrite gmultiset_size_union; omega.
Qed.
Lemma gmultiset_union_subset_r X Y : X    Y  X  Y.
Proof. rewrite (comm_L ()). apply gmultiset_union_subset_l. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
289
Lemma gmultiset_elem_of_singleton_subseteq x X : x  X  {[ x ]}  X.
Robbert Krebbers's avatar
Robbert Krebbers committed
290
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
291 292 293 294 295
  rewrite elem_of_multiplicity. split.
  - intros Hx y; destruct (decide (x = y)) as [->|].
    + rewrite multiplicity_singleton; omega.
    + rewrite multiplicity_singleton_ne by done; omega.
  - intros Hx. generalize (Hx x). rewrite multiplicity_singleton. omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
296 297
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
298 299 300
Lemma gmultiset_elem_of_subseteq X1 X2 x : x  X1  X1  X2  x  X2.
Proof. rewrite !gmultiset_elem_of_singleton_subseteq. by intros ->. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
301 302 303 304 305 306
Lemma gmultiset_union_difference X Y : X  Y  Y = X  Y  X.
Proof.
  intros HXY. apply gmultiset_eq; intros x; specialize (HXY x).
  rewrite multiplicity_union, multiplicity_difference; omega.
Qed.
Lemma gmultiset_union_difference' x Y : x  Y  Y = {[ x ]}  Y  {[ x ]}.
Robbert Krebbers's avatar
Robbert Krebbers committed
307 308 309 310
Proof.
  intros. by apply gmultiset_union_difference,
    gmultiset_elem_of_singleton_subseteq.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
311

Robbert Krebbers's avatar
Robbert Krebbers committed
312 313 314 315 316 317
Lemma gmultiset_size_difference X Y : Y  X  size (X  Y) = size X - size Y.
Proof.
  intros HX%gmultiset_union_difference.
  rewrite HX at 2; rewrite gmultiset_size_union. omega.
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
318 319 320 321 322 323 324 325 326 327 328 329 330
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).
  rewrite multiplicity_difference, multiplicity_empty; omega.
Qed.

Lemma gmultiset_difference_subset X Y : X    X  Y  Y  X  Y.
Proof.
  intros. eapply strict_transitive_l; [by apply gmultiset_union_subset_r|].
  by rewrite <-(gmultiset_union_difference X Y).
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
331
(* Mononicity *)
Robbert Krebbers's avatar
Robbert Krebbers committed
332
Lemma gmultiset_elements_submseteq X Y : X  Y  elements X + elements Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
333 334
Proof.
  intros ->%gmultiset_union_difference. rewrite gmultiset_elements_union.
Robbert Krebbers's avatar
Robbert Krebbers committed
335
  by apply submseteq_inserts_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
336 337
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
338
Lemma gmultiset_subseteq_size X Y : X  Y  size X  size Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
339
Proof. intros. by apply submseteq_length, gmultiset_elements_submseteq. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
340 341 342 343

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
344
  { by apply gmultiset_size_non_empty_iff, gmultiset_non_empty_difference. }
Robbert Krebbers's avatar
Robbert Krebbers committed
345 346 347 348
  rewrite (gmultiset_union_difference X Y), gmultiset_size_union by auto. lia.
Qed.

(* Well-foundedness *)
349
Lemma gmultiset_wf : wf (@{gmultiset A}).
Robbert Krebbers's avatar
Robbert Krebbers committed
350 351 352
Proof.
  apply (wf_projected (<) size); auto using gmultiset_subset_size, lt_wf.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
353 354 355 356 357 358 359

Lemma gmultiset_ind (P : gmultiset A  Prop) :
  P   ( x X, P X  P ({[ x ]}  X))   X, P X.
Proof.
  intros Hemp Hinsert X. induction (gmultiset_wf X) as [X _ IH].
  destruct (gmultiset_choose_or_empty X) as [[x Hx]| ->]; auto.
  rewrite (gmultiset_union_difference' x X) by done.
Robbert Krebbers's avatar
Robbert Krebbers committed
360 361
  apply Hinsert, IH, gmultiset_difference_subset,
    gmultiset_elem_of_singleton_subseteq; auto using gmultiset_non_empty_singleton.
Robbert Krebbers's avatar
Robbert Krebbers committed
362
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
363
End lemmas.