gmultiset.v 19.9 KB
 Robbert Krebbers committed Jan 29, 2019 1 ``````(* Copyright (c) 2012-2019, Coq-std++ developers. *) `````` Robbert Krebbers committed Nov 15, 2016 2 3 ``````(* This file is distributed under the terms of the BSD license. *) From stdpp Require Import gmap. `````` Ralf Jung committed Jan 31, 2017 4 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Nov 15, 2016 5 6 `````` Record gmultiset A `{Countable A} := GMultiSet { gmultiset_car : gmap A nat }. `````` Robbert Krebbers committed Sep 08, 2017 7 8 ``````Arguments GMultiSet {_ _ _} _ : assert. Arguments gmultiset_car {_ _ _} _ : assert. `````` Robbert Krebbers committed Nov 15, 2016 9 `````` `````` 10 ``````Instance gmultiset_eq_dec `{Countable A} : EqDecision (gmultiset A). `````` Robbert Krebbers committed Nov 15, 2016 11 12 ``````Proof. solve_decision. Defined. `````` 13 ``````Program Instance gmultiset_countable `{Countable A} : `````` Robbert Krebbers committed Nov 15, 2016 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. `````` Robbert Krebbers committed Sep 17, 2017 24 `````` Global Instance gmultiset_elem_of : ElemOf A (gmultiset A) := λ x X, `````` Robbert Krebbers committed Nov 15, 2016 25 `````` 0 < multiplicity x X. `````` Robbert Krebbers committed Sep 17, 2017 26 `````` Global Instance gmultiset_subseteq : SubsetEq (gmultiset A) := λ X Y, ∀ x, `````` Robbert Krebbers committed Nov 15, 2016 27 `````` multiplicity x X ≤ multiplicity x Y. `````` Robbert Krebbers committed Apr 09, 2018 28 29 `````` Global Instance gmultiset_equiv : Equiv (gmultiset A) := λ X Y, ∀ x, multiplicity x X = multiplicity x Y. `````` Robbert Krebbers committed Nov 15, 2016 30 `````` `````` Robbert Krebbers committed Sep 17, 2017 31 `````` Global Instance gmultiset_elements : Elements A (gmultiset A) := λ X, `````` Robbert Krebbers committed Nov 21, 2017 32 `````` let (X) := X in ''(x,n) ← map_to_list X; replicate (S n) x. `````` Robbert Krebbers committed Sep 17, 2017 33 `````` Global Instance gmultiset_size : Size (gmultiset A) := length ∘ elements. `````` Robbert Krebbers committed Nov 15, 2016 34 `````` `````` Robbert Krebbers committed Sep 17, 2017 35 36 `````` Global Instance gmultiset_empty : Empty (gmultiset A) := GMultiSet ∅. Global Instance gmultiset_singleton : Singleton A (gmultiset A) := λ x, `````` Robbert Krebbers committed Nov 15, 2016 37 `````` GMultiSet {[ x := 0 ]}. `````` Robbert Krebbers committed Sep 17, 2017 38 `````` Global Instance gmultiset_union : Union (gmultiset A) := λ X Y, `````` Robbert Krebbers committed Feb 21, 2019 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 committed Nov 15, 2016 46 47 `````` let (X) := X in let (Y) := Y in GMultiSet \$ union_with (λ x y, Some (S (x + y))) X Y. `````` Robbert Krebbers committed Sep 17, 2017 48 `````` Global Instance gmultiset_difference : Difference (gmultiset A) := λ X Y, `````` Robbert Krebbers committed Nov 15, 2016 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. `````` Robbert Krebbers committed Nov 22, 2016 52 `````` `````` Robbert Krebbers committed Sep 17, 2017 53 `````` Global Instance gmultiset_dom : Dom (gmultiset A) (gset A) := λ X, `````` Robbert Krebbers committed Nov 22, 2016 54 `````` let (X) := X in dom _ X. `````` Robbert Krebbers committed Apr 11, 2018 55 ``````End definitions. `````` Robbert Krebbers committed Nov 15, 2016 56 `````` `````` Robbert Krebbers committed Nov 21, 2016 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. `````` Robbert Krebbers committed Nov 22, 2016 60 ``````Typeclasses Opaque gmultiset_dom. `````` Robbert Krebbers committed Nov 21, 2016 61 `````` `````` Robbert Krebbers committed Nov 15, 2016 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. `````` Robbert Krebbers committed Apr 09, 2018 74 75 ``````Global Instance gmultiset_leibniz : LeibnizEquiv (gmultiset A). Proof. intros X Y. by rewrite gmultiset_eq. Qed. `````` Robbert Krebbers committed Feb 20, 2019 76 ``````Global Instance gmultiset_equiv_equivalence : Equivalence (≡@{gmultiset A}). `````` Robbert Krebbers committed Apr 11, 2018 77 ``````Proof. constructor; repeat intro; naive_solver. Qed. `````` Robbert Krebbers committed Nov 15, 2016 78 79 80 81 82 83 84 85 86 `````` (* 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 : `````` Robbert Krebbers committed Feb 21, 2019 87 88 89 90 91 92 93 94 95 96 97 98 99 `````` 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 committed Nov 15, 2016 100 101 ``````Proof. destruct X as [X], Y as [Y]; unfold multiplicity; simpl. `````` Ralf Jung committed Jun 20, 2018 102 `````` rewrite lookup_union_with. destruct (X !! _), (Y !! _); simpl; lia. `````` Robbert Krebbers committed Nov 15, 2016 103 104 105 106 107 108 ``````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 committed Jun 20, 2018 109 `````` destruct (X !! _), (Y !! _); simplify_option_eq; lia. `````` Robbert Krebbers committed Nov 15, 2016 110 111 ``````Qed. `````` Robbert Krebbers committed Feb 20, 2019 112 ``````(* Set_ *) `````` Robbert Krebbers committed Nov 17, 2016 113 114 115 ``````Lemma elem_of_multiplicity x X : x ∈ X ↔ 0 < multiplicity x X. Proof. done. Qed. `````` Robbert Krebbers committed Feb 20, 2019 116 ``````Global Instance gmultiset_simple_set : SemiSet A (gmultiset A). `````` Robbert Krebbers committed Nov 17, 2016 117 118 ``````Proof. split. `````` Ralf Jung committed Jun 20, 2018 119 `````` - intros x. rewrite elem_of_multiplicity, multiplicity_empty. lia. `````` Robbert Krebbers committed Nov 17, 2016 120 121 122 123 `````` - 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. `````` Ralf Jung committed Jun 20, 2018 124 `````` - intros X Y x. rewrite !elem_of_multiplicity, multiplicity_union. lia. `````` Robbert Krebbers committed Nov 17, 2016 125 ``````Qed. `````` Robbert Krebbers committed Apr 05, 2018 126 ``````Global Instance gmultiset_elem_of_dec : RelDecision (∈@{gmultiset A}). `````` 127 ``````Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined. `````` Robbert Krebbers committed Nov 17, 2016 128 `````` `````` Robbert Krebbers committed Feb 21, 2019 129 130 131 ``````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. `````` Robbert Krebbers committed Feb 21, 2019 132 133 134 ``````Global Instance set_unfold_gmultiset_disj_union x X Y P Q : SetUnfold (x ∈ X) P → SetUnfold (x ∈ Y) Q → SetUnfold (x ∈ X ⊎ Y) (P ∨ Q). Proof. `````` Robbert Krebbers committed Feb 21, 2019 135 136 `````` intros ??; constructor. rewrite gmultiset_elem_of_disj_union. by rewrite <-(set_unfold (x ∈ X) P), <-(set_unfold (x ∈ Y) Q). `````` Robbert Krebbers committed Feb 21, 2019 137 138 ``````Qed. `````` Robbert Krebbers committed Nov 21, 2016 139 ``````(* Algebraic laws *) `````` Robbert Krebbers committed Feb 21, 2019 140 141 ``````(** For union *) Global Instance gmultiset_union_comm : Comm (=@{gmultiset A}) (∪). `````` Robbert Krebbers committed Nov 15, 2016 142 ``````Proof. `````` Ralf Jung committed Jun 20, 2018 143 `````` intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia. `````` Robbert Krebbers committed Nov 15, 2016 144 ``````Qed. `````` Robbert Krebbers committed Feb 21, 2019 145 ``````Global Instance gmultiset_union_assoc : Assoc (=@{gmultiset A}) (∪). `````` Robbert Krebbers committed Nov 15, 2016 146 ``````Proof. `````` Ralf Jung committed Jun 20, 2018 147 `````` intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia. `````` Robbert Krebbers committed Nov 15, 2016 148 ``````Qed. `````` Robbert Krebbers committed Feb 21, 2019 149 ``````Global Instance gmultiset_union_left_id : LeftId (=@{gmultiset A}) ∅ (∪). `````` Robbert Krebbers committed Nov 15, 2016 150 151 152 153 ``````Proof. intros X. apply gmultiset_eq; intros x. by rewrite multiplicity_union, multiplicity_empty. Qed. `````` Robbert Krebbers committed Feb 21, 2019 154 ``````Global Instance gmultiset_union_right_id : RightId (=@{gmultiset A}) ∅ (∪). `````` Robbert Krebbers committed Nov 15, 2016 155 ``````Proof. intros X. by rewrite (comm_L (∪)), (left_id_L _ _). Qed. `````` Robbert Krebbers committed Feb 21, 2019 156 157 158 159 ``````Global Instance gmultiset_union_idemp : IdemP (=@{gmultiset A}) (∪). Proof. intros X. apply gmultiset_eq; intros x. rewrite !multiplicity_union; lia. Qed. `````` Robbert Krebbers committed Nov 15, 2016 160 `````` `````` Robbert Krebbers committed Feb 21, 2019 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 ``````(** 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. `````` Robbert Krebbers committed Feb 21, 2019 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 ``````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. `````` Robbert Krebbers committed Feb 21, 2019 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 ``````(** 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}) ∅ (⊎). Proof. intros X. by rewrite (comm_L (∪)), (left_id_L _ _). Qed. Global Instance gmultiset_disj_union_inj_1 X : Inj (=) (=) (X ⊎). `````` Robbert Krebbers committed Nov 15, 2016 215 216 ``````Proof. intros Y1 Y2. rewrite !gmultiset_eq. intros HX x; generalize (HX x). `````` Robbert Krebbers committed Feb 21, 2019 217 `````` rewrite !multiplicity_disj_union. lia. `````` Robbert Krebbers committed Nov 15, 2016 218 ``````Qed. `````` Robbert Krebbers committed Feb 21, 2019 219 ``````Global Instance gmultiset_disj_union_inj_2 X : Inj (=) (=) (⊎ X). `````` Robbert Krebbers committed Nov 15, 2016 220 221 ``````Proof. intros Y1 Y2. rewrite <-!(comm_L _ X). apply (inj _). Qed. `````` Robbert Krebbers committed Feb 21, 2019 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 ``````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. `````` Robbert Krebbers committed Feb 21, 2019 240 ``````(** Misc *) `````` 241 ``````Lemma gmultiset_non_empty_singleton x : {[ x ]} ≠@{gmultiset A} ∅. `````` Robbert Krebbers committed Nov 15, 2016 242 ``````Proof. `````` Robbert Krebbers committed Nov 19, 2016 243 244 `````` rewrite gmultiset_eq. intros Hx; generalize (Hx x). by rewrite multiplicity_singleton, multiplicity_empty. `````` Robbert Krebbers committed Nov 18, 2016 245 246 ``````Qed. `````` Robbert Krebbers committed Nov 15, 2016 247 248 249 250 251 252 253 254 ``````(* 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. `````` 255 256 257 `````` intros; apply (f_equal GMultiSet). destruct (map_to_list X) as [|[]] eqn:?. - by apply map_to_list_empty_inv. - naive_solver. `````` Robbert Krebbers committed Nov 15, 2016 258 259 260 261 262 263 264 265 266 267 ``````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. `````` Robbert Krebbers committed Feb 21, 2019 268 269 ``````Lemma gmultiset_elements_disj_union X Y : elements (X ⊎ Y) ≡ₚ elements X ++ elements Y. `````` Robbert Krebbers committed Nov 15, 2016 270 271 272 273 ``````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. `````` Ralf Jung committed Dec 06, 2016 274 `````` { by rewrite (left_id_L _ _ Y), map_to_list_empty. } `````` Robbert Krebbers committed Nov 15, 2016 275 276 277 278 279 280 `````` 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. `````` Robbert Krebbers committed Dec 05, 2016 281 282 `````` rewrite (assoc_L _). f_equiv. rewrite (comm _); simpl. by rewrite replicate_plus, Permutation_middle. `````` Robbert Krebbers committed Nov 15, 2016 283 284 285 286 287 288 289 290 291 292 293 `````` - 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 committed Jun 20, 2018 294 `````` - intros. destruct (X !! x) as [n|] eqn:Hx; [|lia]. `````` Robbert Krebbers committed Nov 15, 2016 295 `````` exists (x,n); split; [|by apply elem_of_map_to_list]. `````` Ralf Jung committed Jun 20, 2018 296 `````` apply elem_of_replicate; auto with lia. `````` Robbert Krebbers committed Nov 15, 2016 297 ``````Qed. `````` Robbert Krebbers committed Nov 22, 2016 298 299 300 301 ``````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 committed Jun 20, 2018 302 `````` destruct (X !! x); naive_solver lia. `````` Robbert Krebbers committed Nov 22, 2016 303 ``````Qed. `````` Robbert Krebbers committed Nov 15, 2016 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 `````` (* 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. `````` Robbert Krebbers committed Feb 21, 2019 339 ``````Lemma gmultiset_size_disj_union X Y : size (X ⊎ Y) = size X + size Y. `````` Robbert Krebbers committed Nov 15, 2016 340 341 ``````Proof. unfold size, gmultiset_size; simpl. `````` Robbert Krebbers committed Feb 21, 2019 342 `````` by rewrite gmultiset_elements_disj_union, app_length. `````` Robbert Krebbers committed Nov 15, 2016 343 ``````Qed. `````` Robbert Krebbers committed Nov 19, 2016 344 345 `````` (* Order stuff *) `````` Robbert Krebbers committed Apr 05, 2018 346 ``````Global Instance gmultiset_po : PartialOrder (⊆@{gmultiset A}). `````` Robbert Krebbers committed Nov 19, 2016 347 348 349 350 351 352 353 ``````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. `````` Robbert Krebbers committed Nov 21, 2016 354 355 356 357 358 ``````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 committed Jun 20, 2018 359 `````` destruct (gmultiset_car X !! x), (gmultiset_car Y !! x); naive_solver lia. `````` Robbert Krebbers committed Nov 21, 2016 360 ``````Qed. `````` Robbert Krebbers committed Apr 05, 2018 361 ``````Global Instance gmultiset_subseteq_dec : RelDecision (⊆@{gmultiset A}). `````` Robbert Krebbers committed Nov 21, 2016 362 ``````Proof. `````` 363 `````` refine (λ X Y, cast_if (decide (map_relation (≤) `````` Robbert Krebbers committed Nov 21, 2016 364 365 366 367 `````` (λ _, False) (λ _, True) (gmultiset_car X) (gmultiset_car Y)))); by rewrite gmultiset_subseteq_alt. Defined. `````` Robbert Krebbers committed Nov 19, 2016 368 369 ``````Lemma gmultiset_subset_subseteq X Y : X ⊂ Y → X ⊆ Y. Proof. apply strict_include. Qed. `````` Tej Chajed committed Nov 28, 2018 370 ``````Hint Resolve gmultiset_subset_subseteq : core. `````` Robbert Krebbers committed Nov 19, 2016 371 372 `````` Lemma gmultiset_empty_subseteq X : ∅ ⊆ X. `````` Ralf Jung committed Jun 20, 2018 373 ``````Proof. intros x. rewrite multiplicity_empty. lia. Qed. `````` Robbert Krebbers committed Nov 19, 2016 374 375 `````` Lemma gmultiset_union_subseteq_l X Y : X ⊆ X ∪ Y. `````` Ralf Jung committed Jun 20, 2018 376 ``````Proof. intros x. rewrite multiplicity_union. lia. Qed. `````` Robbert Krebbers committed Nov 19, 2016 377 ``````Lemma gmultiset_union_subseteq_r X Y : Y ⊆ X ∪ Y. `````` Ralf Jung committed Jun 20, 2018 378 ``````Proof. intros x. rewrite multiplicity_union. lia. Qed. `````` Robbert Krebbers committed Mar 09, 2017 379 ``````Lemma gmultiset_union_mono X1 X2 Y1 Y2 : X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ∪ Y1 ⊆ X2 ∪ Y2. `````` Robbert Krebbers committed Feb 21, 2019 380 381 382 383 ``````Proof. intros HX HY x. rewrite !multiplicity_union. specialize (HX x); specialize (HY x); lia. Qed. `````` Robbert Krebbers committed Mar 09, 2017 384 385 386 387 ``````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 committed Nov 19, 2016 388 `````` `````` Robbert Krebbers committed Feb 21, 2019 389 390 391 392 393 394 395 396 397 398 399 ``````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 committed Nov 19, 2016 400 ``````Lemma gmultiset_subset X Y : X ⊆ Y → size X < size Y → X ⊂ Y. `````` Ralf Jung committed Jun 20, 2018 401 ``````Proof. intros. apply strict_spec_alt; split; naive_solver auto with lia. Qed. `````` Robbert Krebbers committed Feb 21, 2019 402 ``````Lemma gmultiset_disj_union_subset_l X Y : Y ≠ ∅ → X ⊂ X ⊎ Y. `````` Robbert Krebbers committed Nov 19, 2016 403 404 ``````Proof. intros HY%gmultiset_size_non_empty_iff. `````` Robbert Krebbers committed Feb 21, 2019 405 406 `````` apply gmultiset_subset; auto using gmultiset_disj_union_subseteq_l. rewrite gmultiset_size_disj_union; lia. `````` Robbert Krebbers committed Nov 19, 2016 407 ``````Qed. `````` Robbert Krebbers committed Feb 21, 2019 408 409 ``````Lemma gmultiset_union_subset_r X Y : X ≠ ∅ → Y ⊂ X ⊎ Y. Proof. rewrite (comm_L (⊎)). apply gmultiset_disj_union_subset_l. Qed. `````` Robbert Krebbers committed Nov 19, 2016 410 `````` `````` Robbert Krebbers committed Nov 24, 2016 411 ``````Lemma gmultiset_elem_of_singleton_subseteq x X : x ∈ X ↔ {[ x ]} ⊆ X. `````` Robbert Krebbers committed Nov 19, 2016 412 ``````Proof. `````` Robbert Krebbers committed Nov 24, 2016 413 414 `````` rewrite elem_of_multiplicity. split. - intros Hx y; destruct (decide (x = y)) as [->|]. `````` Ralf Jung committed Jun 20, 2018 415 416 417 `````` + rewrite multiplicity_singleton; lia. + rewrite multiplicity_singleton_ne by done; lia. - intros Hx. generalize (Hx x). rewrite multiplicity_singleton. lia. `````` Robbert Krebbers committed Nov 19, 2016 418 419 ``````Qed. `````` Robbert Krebbers committed Nov 24, 2016 420 421 422 ``````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 committed Feb 21, 2019 423 ``````Lemma gmultiset_disj_union_difference X Y : X ⊆ Y → Y = X ⊎ Y ∖ X. `````` Robbert Krebbers committed Nov 19, 2016 424 425 ``````Proof. intros HXY. apply gmultiset_eq; intros x; specialize (HXY x). `````` Robbert Krebbers committed Feb 21, 2019 426 `````` rewrite multiplicity_disj_union, multiplicity_difference; lia. `````` Robbert Krebbers committed Nov 19, 2016 427 ``````Qed. `````` Robbert Krebbers committed Feb 21, 2019 428 ``````Lemma gmultiset_disj_union_difference' x Y : x ∈ Y → Y = {[ x ]} ⊎ Y ∖ {[ x ]}. `````` Robbert Krebbers committed Nov 24, 2016 429 ``````Proof. `````` Robbert Krebbers committed Feb 21, 2019 430 `````` intros. by apply gmultiset_disj_union_difference, `````` Robbert Krebbers committed Nov 24, 2016 431 432 `````` gmultiset_elem_of_singleton_subseteq. Qed. `````` Robbert Krebbers committed Nov 19, 2016 433 `````` `````` Robbert Krebbers committed Nov 15, 2016 434 435 ``````Lemma gmultiset_size_difference X Y : Y ⊆ X → size (X ∖ Y) = size X - size Y. Proof. `````` Robbert Krebbers committed Feb 21, 2019 436 437 `````` intros HX%gmultiset_disj_union_difference. rewrite HX at 2; rewrite gmultiset_size_disj_union. lia. `````` Robbert Krebbers committed Nov 15, 2016 438 439 ``````Qed. `````` Robbert Krebbers committed Nov 19, 2016 440 441 442 443 ``````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 committed Jun 20, 2018 444 `````` rewrite multiplicity_difference, multiplicity_empty; lia. `````` Robbert Krebbers committed Nov 19, 2016 445 446 447 448 449 ``````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|]. `````` Robbert Krebbers committed Feb 21, 2019 450 `````` by rewrite <-(gmultiset_disj_union_difference X Y). `````` Robbert Krebbers committed Nov 19, 2016 451 452 ``````Qed. `````` Robbert Krebbers committed Nov 15, 2016 453 ``````(* Mononicity *) `````` Robbert Krebbers committed Jan 31, 2017 454 ``````Lemma gmultiset_elements_submseteq X Y : X ⊆ Y → elements X ⊆+ elements Y. `````` Robbert Krebbers committed Nov 19, 2016 455 ``````Proof. `````` Robbert Krebbers committed Feb 21, 2019 456 `````` intros ->%gmultiset_disj_union_difference. rewrite gmultiset_elements_disj_union. `````` Robbert Krebbers committed Jan 31, 2017 457 `````` by apply submseteq_inserts_r. `````` Robbert Krebbers committed Nov 19, 2016 458 459 ``````Qed. `````` Robbert Krebbers committed Nov 15, 2016 460 ``````Lemma gmultiset_subseteq_size X Y : X ⊆ Y → size X ≤ size Y. `````` Robbert Krebbers committed Jan 31, 2017 461 ``````Proof. intros. by apply submseteq_length, gmultiset_elements_submseteq. Qed. `````` Robbert Krebbers committed Nov 15, 2016 462 463 464 465 `````` Lemma gmultiset_subset_size X Y : X ⊂ Y → size X < size Y. Proof. intros HXY. assert (size (Y ∖ X) ≠ 0). `````` Robbert Krebbers committed Nov 19, 2016 466 `````` { by apply gmultiset_size_non_empty_iff, gmultiset_non_empty_difference. } `````` Robbert Krebbers committed Feb 21, 2019 467 468 `````` rewrite (gmultiset_disj_union_difference X Y), gmultiset_size_disj_union by auto. lia. `````` Robbert Krebbers committed Nov 15, 2016 469 470 471 ``````Qed. (* Well-foundedness *) `````` Robbert Krebbers committed Apr 05, 2018 472 ``````Lemma gmultiset_wf : wf (⊂@{gmultiset A}). `````` Robbert Krebbers committed Nov 15, 2016 473 474 475 ``````Proof. apply (wf_projected (<) size); auto using gmultiset_subset_size, lt_wf. Qed. `````` Robbert Krebbers committed Nov 19, 2016 476 477 `````` Lemma gmultiset_ind (P : gmultiset A → Prop) : `````` Robbert Krebbers committed Feb 21, 2019 478 `````` P ∅ → (∀ x X, P X → P ({[ x ]} ⊎ X)) → ∀ X, P X. `````` Robbert Krebbers committed Nov 19, 2016 479 480 481 ``````Proof. intros Hemp Hinsert X. induction (gmultiset_wf X) as [X _ IH]. destruct (gmultiset_choose_or_empty X) as [[x Hx]| ->]; auto. `````` Robbert Krebbers committed Feb 21, 2019 482 `````` rewrite (gmultiset_disj_union_difference' x X) by done. `````` Robbert Krebbers committed Nov 24, 2016 483 484 `````` apply Hinsert, IH, gmultiset_difference_subset, gmultiset_elem_of_singleton_subseteq; auto using gmultiset_non_empty_singleton. `````` Robbert Krebbers committed Nov 19, 2016 485 ``````Qed. `````` Robbert Krebbers committed Nov 15, 2016 486 ``End lemmas.``