fin_maps.v 69.8 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 5 6 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** Finite maps associate data to keys. This file defines an interface for finite maps and collects some theory on it. Most importantly, it proves useful induction principles for finite maps and implements the tactic [simplify_map_equality] to simplify goals involving finite maps. *) `````` Robbert Krebbers committed Feb 13, 2016 7 8 ``````From Coq Require Import Permutation. From prelude Require Export relations vector orders. `````` Robbert Krebbers committed Nov 11, 2015 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 `````` (** * Axiomatization of finite maps *) (** We require Leibniz equality to be extensional on finite maps. This of course limits the space of finite map implementations, but since we are mainly interested in finite maps with numbers as indexes, we do not consider this to be a serious limitation. The main application of finite maps is to implement the memory, where extensionality of Leibniz equality is very important for a convenient use in the assertions of our axiomatic semantics. *) (** Finiteness is axiomatized by requiring that each map can be translated to an association list. The translation to association lists is used to prove well founded recursion on finite maps. *) (** Finite map implementations are required to implement the [merge] function which enables us to give a generic implementation of [union_with], [intersection_with], and [difference_with]. *) Class FinMapToList K A M := map_to_list: M → list (K * A). Class FinMap K M `{FMap M, ∀ A, Lookup K A (M A), ∀ A, Empty (M A), ∀ A, PartialAlter K A (M A), OMap M, Merge M, ∀ A, FinMapToList K A (M A), ∀ i j : K, Decision (i = j)} := { map_eq {A} (m1 m2 : M A) : (∀ i, m1 !! i = m2 !! i) → m1 = m2; lookup_empty {A} i : (∅ : M A) !! i = None; lookup_partial_alter {A} f (m : M A) i : partial_alter f i m !! i = f (m !! i); lookup_partial_alter_ne {A} f (m : M A) i j : i ≠ j → partial_alter f i m !! j = m !! j; lookup_fmap {A B} (f : A → B) (m : M A) i : (f <\$> m) !! i = f <\$> m !! i; NoDup_map_to_list {A} (m : M A) : NoDup (map_to_list m); elem_of_map_to_list {A} (m : M A) i x : (i,x) ∈ map_to_list m ↔ m !! i = Some x; lookup_omap {A B} (f : A → option B) m i : omap f m !! i = m !! i ≫= f; lookup_merge {A B C} (f : option A → option B → option C) `{!PropHolds (f None None = None)} m1 m2 i : merge f m1 m2 !! i = f (m1 !! i) (m2 !! i) }. (** * Derived operations *) (** All of the following functions are defined in a generic way for arbitrary finite map implementations. These generic implementations do not cause a significant performance loss to make including them in the finite map interface worthwhile. *) Instance map_insert `{PartialAlter K A M} : Insert K A M := λ i x, partial_alter (λ _, Some x) i. Instance map_alter `{PartialAlter K A M} : Alter K A M := λ f, partial_alter (fmap f). Instance map_delete `{PartialAlter K A M} : Delete K M := partial_alter (λ _, None). Instance map_singleton `{PartialAlter K A M, Empty M} : `````` Robbert Krebbers committed Dec 21, 2015 59 `````` SingletonM K A M := λ i x, <[i:=x]> ∅. `````` Robbert Krebbers committed Nov 11, 2015 60 61 62 63 64 65 66 67 68 69 70 71 72 73 `````` Definition map_of_list `{Insert K A M, Empty M} : list (K * A) → M := fold_right (λ p, <[p.1:=p.2]>) ∅. Definition map_of_collection `{Elements K C, Insert K A M, Empty M} (f : K → option A) (X : C) : M := map_of_list (omap (λ i, (i,) <\$> f i) (elements X)). Instance map_union_with `{Merge M} {A} : UnionWith A (M A) := λ f, merge (union_with f). Instance map_intersection_with `{Merge M} {A} : IntersectionWith A (M A) := λ f, merge (intersection_with f). Instance map_difference_with `{Merge M} {A} : DifferenceWith A (M A) := λ f, merge (difference_with f). `````` Robbert Krebbers committed Nov 18, 2015 74 75 ``````Instance map_equiv `{∀ A, Lookup K A (M A), Equiv A} : Equiv (M A) | 18 := λ m1 m2, ∀ i, m1 !! i ≡ m2 !! i. `````` Robbert Krebbers committed Nov 11, 2015 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 `````` (** The relation [intersection_forall R] on finite maps describes that the relation [R] holds for each pair in the intersection. *) Definition map_Forall `{Lookup K A M} (P : K → A → Prop) : M → Prop := λ m, ∀ i x, m !! i = Some x → P i x. Definition map_relation `{∀ A, Lookup K A (M A)} {A B} (R : A → B → Prop) (P : A → Prop) (Q : B → Prop) (m1 : M A) (m2 : M B) : Prop := ∀ i, option_relation R P Q (m1 !! i) (m2 !! i). Definition map_included `{∀ A, Lookup K A (M A)} {A} (R : relation A) : relation (M A) := map_relation R (λ _, False) (λ _, True). Definition map_disjoint `{∀ A, Lookup K A (M A)} {A} : relation (M A) := map_relation (λ _ _, False) (λ _, True) (λ _, True). Infix "⊥ₘ" := map_disjoint (at level 70) : C_scope. Hint Extern 0 (_ ⊥ₘ _) => symmetry; eassumption. Notation "( m ⊥ₘ.)" := (map_disjoint m) (only parsing) : C_scope. Notation "(.⊥ₘ m )" := (λ m2, m2 ⊥ₘ m) (only parsing) : C_scope. Instance map_subseteq `{∀ A, Lookup K A (M A)} {A} : SubsetEq (M A) := map_included (=). (** The union of two finite maps only has a meaningful definition for maps that are disjoint. However, as working with partial functions is inconvenient in Coq, we define the union as a total function. In case both finite maps have a value at the same index, we take the value of the first map. *) Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x). Instance map_intersection `{Merge M} {A} : Intersection (M A) := intersection_with (λ x _, Some x). (** The difference operation removes all values from the first map whose index contains a value in the second map as well. *) Instance map_difference `{Merge M} {A} : Difference (M A) := difference_with (λ _ _, None). (** A stronger variant of map that allows the mapped function to use the index of the elements. Implemented by conversion to lists, so not very efficient. *) Definition map_imap `{∀ A, Insert K A (M A), ∀ A, Empty (M A), ∀ A, FinMapToList K A (M A)} {A B} (f : K → A → option B) (m : M A) : M B := map_of_list (omap (λ ix, (fst ix,) <\$> curry f ix) (map_to_list m)). (** * Theorems *) Section theorems. Context `{FinMap K M}. (** ** Setoids *) Section setoid. `````` Robbert Krebbers committed Nov 18, 2015 120 121 `````` Context `{Equiv A} `{!Equivalence ((≡) : relation A)}. Global Instance map_equivalence : Equivalence ((≡) : relation (M A)). `````` Robbert Krebbers committed Nov 11, 2015 122 123 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 124 125 `````` - by intros m i. - by intros m1 m2 ? i. `````` Ralf Jung committed Feb 20, 2016 126 `````` - by intros m1 m2 m3 ?? i; trans (m2 !! i). `````` Robbert Krebbers committed Nov 11, 2015 127 128 129 130 131 `````` Qed. Global Instance lookup_proper (i : K) : Proper ((≡) ==> (≡)) (lookup (M:=M A) i). Proof. by intros m1 m2 Hm. Qed. Global Instance partial_alter_proper : `````` Robbert Krebbers committed Nov 18, 2015 132 `````` Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (partial_alter (M:=M A)). `````` Robbert Krebbers committed Nov 11, 2015 133 134 135 136 137 138 139 140 `````` Proof. by intros f1 f2 Hf i ? <- m1 m2 Hm j; destruct (decide (i = j)) as [->|]; rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne by done; try apply Hf; apply lookup_proper. Qed. Global Instance insert_proper (i : K) : Proper ((≡) ==> (≡) ==> (≡)) (insert (M:=M A) i). Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed. `````` Robbert Krebbers committed Dec 21, 2015 141 142 143 `````` Global Instance singleton_proper k : Proper ((≡) ==> (≡)) (singletonM k : A → M A). Proof. by intros ???; apply insert_proper. Qed. `````` Robbert Krebbers committed Nov 11, 2015 144 145 146 147 148 149 150 151 152 153 154 155 `````` Global Instance delete_proper (i : K) : Proper ((≡) ==> (≡)) (delete (M:=M A) i). Proof. by apply partial_alter_proper; [constructor|]. Qed. Global Instance alter_proper : Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (alter (A:=A) (M:=M A)). Proof. intros ?? Hf; apply partial_alter_proper. by destruct 1; constructor; apply Hf. Qed. Lemma merge_ext f g `{!PropHolds (f None None = None), !PropHolds (g None None = None)} : ((≡) ==> (≡) ==> (≡))%signature f g → `````` Robbert Krebbers committed Nov 18, 2015 156 `````` ((≡) ==> (≡) ==> (≡))%signature (merge (M:=M) f) (merge g). `````` Robbert Krebbers committed Nov 11, 2015 157 158 159 160 `````` Proof. by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf. Qed. Global Instance union_with_proper : `````` Robbert Krebbers committed Nov 18, 2015 161 `````` Proper (((≡) ==> (≡) ==> (≡)) ==> (≡) ==> (≡) ==>(≡)) (union_with (M:=M A)). `````` Robbert Krebbers committed Nov 11, 2015 162 163 164 `````` Proof. intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto. by do 2 destruct 1; first [apply Hf | constructor]. `````` Robbert Krebbers committed Feb 14, 2016 165 `````` Qed. `````` Robbert Krebbers committed Nov 11, 2015 166 167 `````` Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A). Proof. `````` Robbert Krebbers committed Dec 15, 2015 168 169 `````` intros m1 m2 Hm; apply map_eq; intros i. by unfold_leibniz; apply lookup_proper. `````` Robbert Krebbers committed Nov 11, 2015 170 `````` Qed. `````` Robbert Krebbers committed Nov 18, 2015 171 172 173 174 175 176 177 178 179 180 `````` Lemma map_equiv_empty (m : M A) : m ≡ ∅ ↔ m = ∅. Proof. split; [intros Hm; apply map_eq; intros i|by intros ->]. by rewrite lookup_empty, <-equiv_None, Hm, lookup_empty. Qed. Lemma map_equiv_lookup (m1 m2 : M A) i x : m1 ≡ m2 → m1 !! i = Some x → ∃ y, m2 !! i = Some y ∧ x ≡ y. Proof. intros Hm ?. destruct (equiv_Some (m1 !! i) (m2 !! i) x) as (y&?&?); eauto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 ``````End setoid. (** ** General properties *) Lemma map_eq_iff {A} (m1 m2 : M A) : m1 = m2 ↔ ∀ i, m1 !! i = m2 !! i. Proof. split. by intros ->. apply map_eq. Qed. Lemma map_subseteq_spec {A} (m1 m2 : M A) : m1 ⊆ m2 ↔ ∀ i x, m1 !! i = Some x → m2 !! i = Some x. Proof. unfold subseteq, map_subseteq, map_relation. split; intros Hm i; specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver. Qed. Global Instance: EmptySpec (M A). Proof. intros A m. rewrite !map_subseteq_spec. intros i x. by rewrite lookup_empty. Qed. Global Instance: ∀ {A} (R : relation A), PreOrder R → PreOrder (map_included R). Proof. split; [intros m i; by destruct (m !! i); simpl|]. intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i). `````` Robbert Krebbers committed Feb 17, 2016 201 `````` destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_eq/=; `````` Ralf Jung committed Feb 20, 2016 202 `````` done || etrans; eauto. `````` Robbert Krebbers committed Nov 11, 2015 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 ``````Qed. Global Instance: PartialOrder ((⊆) : relation (M A)). Proof. split; [apply _|]. intros m1 m2; rewrite !map_subseteq_spec. intros; apply map_eq; intros i; apply option_eq; naive_solver. Qed. Lemma lookup_weaken {A} (m1 m2 : M A) i x : m1 !! i = Some x → m1 ⊆ m2 → m2 !! i = Some x. Proof. rewrite !map_subseteq_spec. auto. Qed. Lemma lookup_weaken_is_Some {A} (m1 m2 : M A) i : is_Some (m1 !! i) → m1 ⊆ m2 → is_Some (m2 !! i). Proof. inversion 1. eauto using lookup_weaken. Qed. Lemma lookup_weaken_None {A} (m1 m2 : M A) i : m2 !! i = None → m1 ⊆ m2 → m1 !! i = None. Proof. rewrite map_subseteq_spec, !eq_None_not_Some. intros Hm2 Hm [??]; destruct Hm2; eauto. Qed. Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y : m1 !! i = Some x → m1 ⊆ m2 → m2 !! i = Some y → x = y. Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed. Lemma lookup_ne {A} (m : M A) i j : m !! i ≠ m !! j → i ≠ j. Proof. congruence. Qed. Lemma map_empty {A} (m : M A) : (∀ i, m !! i = None) → m = ∅. Proof. intros Hm. apply map_eq. intros. by rewrite Hm, lookup_empty. Qed. Lemma lookup_empty_is_Some {A} i : ¬is_Some ((∅ : M A) !! i). Proof. rewrite lookup_empty. by inversion 1. Qed. Lemma lookup_empty_Some {A} i (x : A) : ¬∅ !! i = Some x. Proof. by rewrite lookup_empty. Qed. Lemma map_subset_empty {A} (m : M A) : m ⊄ ∅. Proof. intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty. Qed. `````` Robbert Krebbers committed Jan 14, 2016 237 238 ``````Lemma map_fmap_empty {A B} (f : A → B) : f <\$> (∅ : M A) = ∅. Proof. by apply map_eq; intros i; rewrite lookup_fmap, !lookup_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 `````` (** ** Properties of the [partial_alter] operation *) Lemma partial_alter_ext {A} (f g : option A → option A) (m : M A) i : (∀ x, m !! i = x → f x = g x) → partial_alter f i m = partial_alter g i m. Proof. intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?]; rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto. Qed. Lemma partial_alter_compose {A} f g (m : M A) i: partial_alter (f ∘ g) i m = partial_alter f i (partial_alter g i m). Proof. intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?]; rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne. Qed. Lemma partial_alter_commute {A} f g (m : M A) i j : i ≠ j → partial_alter f i (partial_alter g j m) = partial_alter g j (partial_alter f i m). Proof. intros. apply map_eq; intros jj. destruct (decide (jj = j)) as [->|?]. { by rewrite lookup_partial_alter_ne, !lookup_partial_alter, lookup_partial_alter_ne. } destruct (decide (jj = i)) as [->|?]. `````` Robbert Krebbers committed Feb 17, 2016 261 `````` - by rewrite lookup_partial_alter, `````` Robbert Krebbers committed Nov 11, 2015 262 `````` !lookup_partial_alter_ne, lookup_partial_alter by congruence. `````` Robbert Krebbers committed Feb 17, 2016 263 `````` - by rewrite !lookup_partial_alter_ne by congruence. `````` Robbert Krebbers committed Nov 11, 2015 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 ``````Qed. Lemma partial_alter_self_alt {A} (m : M A) i x : x = m !! i → partial_alter (λ _, x) i m = m. Proof. intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|]; rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne. Qed. Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m. Proof. by apply partial_alter_self_alt. Qed. Lemma partial_alter_subseteq {A} f (m : M A) i : m !! i = None → m ⊆ partial_alter f i m. Proof. rewrite map_subseteq_spec. intros Hi j x Hj. rewrite lookup_partial_alter_ne; congruence. Qed. Lemma partial_alter_subset {A} f (m : M A) i : m !! i = None → is_Some (f (m !! i)) → m ⊂ partial_alter f i m. Proof. intros Hi Hfi. split; [by apply partial_alter_subseteq|]. rewrite !map_subseteq_spec. inversion Hfi as [x Hx]. intros Hm. apply (Some_ne_None x). rewrite <-(Hm i x); [done|]. by rewrite lookup_partial_alter. Qed. (** ** Properties of the [alter] operation *) Lemma alter_ext {A} (f g : A → A) (m : M A) i : (∀ x, m !! i = Some x → f x = g x) → alter f i m = alter g i m. `````` Robbert Krebbers committed Feb 17, 2016 291 ``````Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 ``````Lemma lookup_alter {A} (f : A → A) m i : alter f i m !! i = f <\$> m !! i. Proof. unfold alter. apply lookup_partial_alter. Qed. Lemma lookup_alter_ne {A} (f : A → A) m i j : i ≠ j → alter f i m !! j = m !! j. Proof. unfold alter. apply lookup_partial_alter_ne. Qed. Lemma alter_compose {A} (f g : A → A) (m : M A) i: alter (f ∘ g) i m = alter f i (alter g i m). Proof. unfold alter, map_alter. rewrite <-partial_alter_compose. apply partial_alter_ext. by intros [?|]. Qed. Lemma alter_commute {A} (f g : A → A) (m : M A) i j : i ≠ j → alter f i (alter g j m) = alter g j (alter f i m). Proof. apply partial_alter_commute. Qed. Lemma lookup_alter_Some {A} (f : A → A) m i j y : alter f i m !! j = Some y ↔ (i = j ∧ ∃ x, m !! j = Some x ∧ y = f x) ∨ (i ≠ j ∧ m !! j = Some y). Proof. destruct (decide (i = j)) as [->|?]. `````` Robbert Krebbers committed Feb 17, 2016 310 `````` - rewrite lookup_alter. naive_solver (simplify_option_eq; eauto). `````` Robbert Krebbers committed Feb 17, 2016 311 `````` - rewrite lookup_alter_ne by done. naive_solver. `````` Robbert Krebbers committed Nov 11, 2015 312 313 314 315 316 317 318 319 320 321 322 ``````Qed. Lemma lookup_alter_None {A} (f : A → A) m i j : alter f i m !! j = None ↔ m !! j = None. Proof. by destruct (decide (i = j)) as [->|?]; rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne. Qed. Lemma alter_id {A} (f : A → A) m i : (∀ x, m !! i = Some x → f x = x) → alter f i m = m. Proof. intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?]. `````` Robbert Krebbers committed Feb 17, 2016 323 `````` { rewrite lookup_alter; destruct (m !! j); f_equal/=; auto. } `````` Robbert Krebbers committed Nov 11, 2015 324 325 326 327 328 329 330 331 332 333 334 335 `````` by rewrite lookup_alter_ne by done. Qed. (** ** Properties of the [delete] operation *) Lemma lookup_delete {A} (m : M A) i : delete i m !! i = None. Proof. apply lookup_partial_alter. Qed. Lemma lookup_delete_ne {A} (m : M A) i j : i ≠ j → delete i m !! j = m !! j. Proof. apply lookup_partial_alter_ne. Qed. Lemma lookup_delete_Some {A} (m : M A) i j y : delete i m !! j = Some y ↔ i ≠ j ∧ m !! j = Some y. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 336 `````` - destruct (decide (i = j)) as [->|?]; `````` Robbert Krebbers committed Nov 11, 2015 337 `````` rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence. `````` Robbert Krebbers committed Feb 17, 2016 338 `````` - intros [??]. by rewrite lookup_delete_ne. `````` Robbert Krebbers committed Nov 11, 2015 339 ``````Qed. `````` Robbert Krebbers committed Jan 16, 2016 340 341 342 ``````Lemma lookup_delete_is_Some {A} (m : M A) i j : is_Some (delete i m !! j) ↔ i ≠ j ∧ is_Some (m !! j). Proof. unfold is_Some; setoid_rewrite lookup_delete_Some; naive_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 343 344 345 346 347 348 349 350 ``````Lemma lookup_delete_None {A} (m : M A) i j : delete i m !! j = None ↔ i = j ∨ m !! j = None. Proof. destruct (decide (i = j)) as [->|?]; rewrite ?lookup_delete, ?lookup_delete_ne; tauto. Qed. Lemma delete_empty {A} i : delete i (∅ : M A) = ∅. Proof. rewrite <-(partial_alter_self ∅) at 2. by rewrite lookup_empty. Qed. `````` Robbert Krebbers committed Feb 17, 2016 351 ``````Lemma delete_singleton {A} i (x : A) : delete i {[i := x]} = ∅. `````` Robbert Krebbers committed Nov 11, 2015 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 ``````Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed. Lemma delete_commute {A} (m : M A) i j : delete i (delete j m) = delete j (delete i m). Proof. destruct (decide (i = j)). by subst. by apply partial_alter_commute. Qed. Lemma delete_insert_ne {A} (m : M A) i j x : i ≠ j → delete i (<[j:=x]>m) = <[j:=x]>(delete i m). Proof. intro. by apply partial_alter_commute. Qed. Lemma delete_notin {A} (m : M A) i : m !! i = None → delete i m = m. Proof. intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?]; rewrite ?lookup_delete, ?lookup_delete_ne. Qed. Lemma delete_partial_alter {A} (m : M A) i f : m !! i = None → delete i (partial_alter f i m) = m. Proof. intros. unfold delete, map_delete. rewrite <-partial_alter_compose. unfold compose. by apply partial_alter_self_alt. Qed. Lemma delete_insert {A} (m : M A) i x : m !! i = None → delete i (<[i:=x]>m) = m. Proof. apply delete_partial_alter. Qed. Lemma insert_delete {A} (m : M A) i x : m !! i = Some x → <[i:=x]>(delete i m) = m. Proof. intros Hmi. unfold delete, map_delete, insert, map_insert. rewrite <-partial_alter_compose. unfold compose. rewrite <-Hmi. by apply partial_alter_self_alt. Qed. Lemma delete_subseteq {A} (m : M A) i : delete i m ⊆ m. Proof. rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto. Qed. Lemma delete_subseteq_compat {A} (m1 m2 : M A) i : m1 ⊆ m2 → delete i m1 ⊆ delete i m2. Proof. rewrite !map_subseteq_spec. intros ? j x. rewrite !lookup_delete_Some. intuition eauto. Qed. Lemma delete_subset_alt {A} (m : M A) i x : m !! i = Some x → delete i m ⊂ m. Proof. split; [apply delete_subseteq|]. rewrite !map_subseteq_spec. intros Hi. apply (None_ne_Some x). by rewrite <-(lookup_delete m i), (Hi i x). Qed. Lemma delete_subset {A} (m : M A) i : is_Some (m !! i) → delete i m ⊂ m. Proof. inversion 1. eauto using delete_subset_alt. Qed. (** ** Properties of the [insert] operation *) Lemma lookup_insert {A} (m : M A) i x : <[i:=x]>m !! i = Some x. Proof. unfold insert. apply lookup_partial_alter. Qed. Lemma lookup_insert_rev {A} (m : M A) i x y : <[i:=x]>m !! i = Some y → x = y. Proof. rewrite lookup_insert. congruence. Qed. Lemma lookup_insert_ne {A} (m : M A) i j x : i ≠ j → <[i:=x]>m !! j = m !! j. Proof. unfold insert. apply lookup_partial_alter_ne. Qed. `````` Robbert Krebbers committed Feb 14, 2016 406 407 ``````Lemma insert_insert {A} (m : M A) i x y : <[i:=x]>(<[i:=y]>m) = <[i:=x]>m. Proof. unfold insert, map_insert. by rewrite <-partial_alter_compose. Qed. `````` Robbert Krebbers committed Nov 11, 2015 408 409 410 411 412 413 414 ``````Lemma insert_commute {A} (m : M A) i j x y : i ≠ j → <[i:=x]>(<[j:=y]>m) = <[j:=y]>(<[i:=x]>m). Proof. apply partial_alter_commute. Qed. Lemma lookup_insert_Some {A} (m : M A) i j x y : <[i:=x]>m !! j = Some y ↔ (i = j ∧ x = y) ∨ (i ≠ j ∧ m !! j = Some y). Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 415 `````` - destruct (decide (i = j)) as [->|?]; `````` Robbert Krebbers committed Nov 11, 2015 416 `````` rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence. `````` Robbert Krebbers committed Feb 17, 2016 417 `````` - intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne. `````` Robbert Krebbers committed Nov 11, 2015 418 ``````Qed. `````` Robbert Krebbers committed Jan 16, 2016 419 420 421 ``````Lemma lookup_insert_is_Some {A} (m : M A) i j x : is_Some (<[i:=x]>m !! j) ↔ i = j ∨ i ≠ j ∧ is_Some (m !! j). Proof. unfold is_Some; setoid_rewrite lookup_insert_Some; naive_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 ``````Lemma lookup_insert_None {A} (m : M A) i j x : <[i:=x]>m !! j = None ↔ m !! j = None ∧ i ≠ j. Proof. split; [|by intros [??]; rewrite lookup_insert_ne]. destruct (decide (i = j)) as [->|]; rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence. Qed. Lemma insert_id {A} (m : M A) i x : m !! i = Some x → <[i:=x]>m = m. Proof. intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|]; by rewrite ?lookup_insert, ?lookup_insert_ne by done. Qed. Lemma insert_included {A} R `{!Reflexive R} (m : M A) i x : (∀ y, m !! i = Some y → R y x) → map_included R m (<[i:=x]>m). Proof. intros ? j; destruct (decide (i = j)) as [->|]. `````` Robbert Krebbers committed Feb 17, 2016 438 439 `````` - rewrite lookup_insert. destruct (m !! j); simpl; eauto. - rewrite lookup_insert_ne by done. by destruct (m !! j); simpl. `````` Robbert Krebbers committed Nov 11, 2015 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 ``````Qed. Lemma insert_subseteq {A} (m : M A) i x : m !! i = None → m ⊆ <[i:=x]>m. Proof. apply partial_alter_subseteq. Qed. Lemma insert_subset {A} (m : M A) i x : m !! i = None → m ⊂ <[i:=x]>m. Proof. intro. apply partial_alter_subset; eauto. Qed. Lemma insert_subseteq_r {A} (m1 m2 : M A) i x : m1 !! i = None → m1 ⊆ m2 → m1 ⊆ <[i:=x]>m2. Proof. rewrite !map_subseteq_spec. intros ?? j ?. destruct (decide (j = i)) as [->|?]; [congruence|]. rewrite lookup_insert_ne; auto. Qed. Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x : m1 !! i = None → <[i:=x]> m1 ⊆ m2 → m1 ⊆ delete i m2. Proof. rewrite !map_subseteq_spec. intros Hi Hix j y Hj. destruct (decide (i = j)) as [->|]; [congruence|]. rewrite lookup_delete_ne by done. apply Hix; by rewrite lookup_insert_ne by done. Qed. Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x : m1 !! i = Some x → delete i m1 ⊆ m2 → m1 ⊆ <[i:=x]> m2. Proof. rewrite !map_subseteq_spec. intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?]. `````` Robbert Krebbers committed Feb 17, 2016 465 466 `````` - rewrite lookup_insert. congruence. - rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne. `````` Robbert Krebbers committed Nov 11, 2015 467 468 469 470 471 472 473 474 475 476 477 478 ``````Qed. Lemma insert_delete_subset {A} (m1 m2 : M A) i x : m1 !! i = None → <[i:=x]> m1 ⊂ m2 → m1 ⊂ delete i m2. Proof. intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|]. contradict Hm21. apply delete_insert_subseteq; auto. eapply lookup_weaken, Hm12. by rewrite lookup_insert. Qed. Lemma insert_subset_inv {A} (m1 m2 : M A) i x : m1 !! i = None → <[i:=x]> m1 ⊂ m2 → ∃ m2', m2 = <[i:=x]>m2' ∧ m1 ⊂ m2' ∧ m2' !! i = None. Proof. `````` Robbert Krebbers committed Feb 19, 2016 479 `````` intros Hi Hm1m2. exists (delete i m2). split_and?. `````` Robbert Krebbers committed Feb 17, 2016 480 `````` - rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto. `````` Robbert Krebbers committed Nov 11, 2015 481 `````` by rewrite lookup_insert. `````` Robbert Krebbers committed Feb 17, 2016 482 483 `````` - eauto using insert_delete_subset. - by rewrite lookup_delete. `````` Robbert Krebbers committed Nov 11, 2015 484 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 485 ``````Lemma insert_empty {A} i (x : A) : <[i:=x]>∅ = {[i := x]}. `````` Robbert Krebbers committed Nov 11, 2015 486 487 488 489 ``````Proof. done. Qed. (** ** Properties of the singleton maps *) Lemma lookup_singleton_Some {A} i j (x y : A) : `````` Robbert Krebbers committed Feb 17, 2016 490 `````` {[i := x]} !! j = Some y ↔ i = j ∧ x = y. `````` Robbert Krebbers committed Nov 11, 2015 491 ``````Proof. `````` Robbert Krebbers committed Dec 21, 2015 492 `````` rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence. `````` Robbert Krebbers committed Nov 11, 2015 493 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 494 ``````Lemma lookup_singleton_None {A} i j (x : A) : {[i := x]} !! j = None ↔ i ≠ j. `````` Robbert Krebbers committed Dec 21, 2015 495 ``````Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; tauto. Qed. `````` Robbert Krebbers committed Feb 17, 2016 496 ``````Lemma lookup_singleton {A} i (x : A) : {[i := x]} !! i = Some x. `````` Robbert Krebbers committed Nov 11, 2015 497 ``````Proof. by rewrite lookup_singleton_Some. Qed. `````` Robbert Krebbers committed Feb 17, 2016 498 ``````Lemma lookup_singleton_ne {A} i j (x : A) : i ≠ j → {[i := x]} !! j = None. `````` Robbert Krebbers committed Nov 11, 2015 499 ``````Proof. by rewrite lookup_singleton_None. Qed. `````` Robbert Krebbers committed Feb 17, 2016 500 ``````Lemma map_non_empty_singleton {A} i (x : A) : {[i := x]} ≠ ∅. `````` Robbert Krebbers committed Nov 11, 2015 501 502 503 504 ``````Proof. intros Hix. apply (f_equal (!! i)) in Hix. by rewrite lookup_empty, lookup_singleton in Hix. Qed. `````` Robbert Krebbers committed Feb 17, 2016 505 ``````Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i := x]} = {[i := y]}. `````` Robbert Krebbers committed Nov 11, 2015 506 ``````Proof. `````` Robbert Krebbers committed Dec 21, 2015 507 `````` unfold singletonM, map_singleton, insert, map_insert. `````` Robbert Krebbers committed Nov 11, 2015 508 509 `````` by rewrite <-partial_alter_compose. Qed. `````` Robbert Krebbers committed Feb 17, 2016 510 ``````Lemma alter_singleton {A} (f : A → A) i x : alter f i {[i := x]} = {[i := f x]}. `````` Robbert Krebbers committed Nov 11, 2015 511 512 ``````Proof. intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?]. `````` Robbert Krebbers committed Feb 17, 2016 513 514 `````` - by rewrite lookup_alter, !lookup_singleton. - by rewrite lookup_alter_ne, !lookup_singleton_ne. `````` Robbert Krebbers committed Nov 11, 2015 515 516 ``````Qed. Lemma alter_singleton_ne {A} (f : A → A) i j x : `````` Robbert Krebbers committed Feb 17, 2016 517 `````` i ≠ j → alter f i {[j := x]} = {[j := x]}. `````` Robbert Krebbers committed Nov 11, 2015 518 519 520 521 522 523 524 525 526 527 ``````Proof. intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?]; rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done. Qed. (** ** Properties of the map operations *) Lemma fmap_empty {A B} (f : A → B) : f <\$> ∅ = ∅. Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed. Lemma omap_empty {A B} (f : A → option B) : omap f ∅ = ∅. Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed. `````` Robbert Krebbers committed Feb 14, 2016 528 529 530 ``````Lemma fmap_insert {A B} (f: A → B) m i x: f <\$> <[i:=x]>m = <[i:=f x]>(f <\$> m). Proof. apply map_eq; intros i'; destruct (decide (i' = i)) as [->|]. `````` Robbert Krebbers committed Feb 17, 2016 531 532 `````` - by rewrite lookup_fmap, !lookup_insert. - by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done. `````` Robbert Krebbers committed Feb 14, 2016 533 534 535 536 537 ``````Qed. Lemma omap_insert {A B} (f : A → option B) m i x y : f x = Some y → omap f (<[i:=x]>m) = <[i:=y]>(omap f m). Proof. intros; apply map_eq; intros i'; destruct (decide (i' = i)) as [->|]. `````` Robbert Krebbers committed Feb 17, 2016 538 539 `````` - by rewrite lookup_omap, !lookup_insert. - by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done. `````` Robbert Krebbers committed Feb 14, 2016 540 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 541 ``````Lemma map_fmap_singleton {A B} (f : A → B) i x : f <\$> {[i := x]} = {[i := f x]}. `````` Robbert Krebbers committed Feb 14, 2016 542 543 544 ``````Proof. by unfold singletonM, map_singleton; rewrite fmap_insert, map_fmap_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 545 ``````Lemma omap_singleton {A B} (f : A → option B) i x y : `````` Robbert Krebbers committed Feb 17, 2016 546 `````` f x = Some y → omap f {[ i := x ]} = {[ i := y ]}. `````` Robbert Krebbers committed Nov 11, 2015 547 ``````Proof. `````` Robbert Krebbers committed Feb 14, 2016 548 549 `````` intros. unfold singletonM, map_singleton. by erewrite omap_insert, omap_empty by eauto. `````` Robbert Krebbers committed Nov 11, 2015 550 551 552 553 554 555 ``````Qed. Lemma map_fmap_id {A} (m : M A) : id <\$> m = m. Proof. apply map_eq; intros i; by rewrite lookup_fmap, option_fmap_id. Qed. Lemma map_fmap_compose {A B C} (f : A → B) (g : B → C) (m : M A) : g ∘ f <\$> m = g <\$> f <\$> m. Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed. `````` Robbert Krebbers committed Dec 15, 2015 556 557 558 559 560 561 ``````Lemma map_fmap_setoid_ext `{Equiv A, Equiv B} (f1 f2 : A → B) m : (∀ i x, m !! i = Some x → f1 x ≡ f2 x) → f1 <\$> m ≡ f2 <\$> m. Proof. intros Hi i; rewrite !lookup_fmap. destruct (m !! i) eqn:?; constructor; eauto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 562 563 564 565 566 567 ``````Lemma map_fmap_ext {A B} (f1 f2 : A → B) m : (∀ i x, m !! i = Some x → f1 x = f2 x) → f1 <\$> m = f2 <\$> m. Proof. intros Hi; apply map_eq; intros i; rewrite !lookup_fmap. by destruct (m !! i) eqn:?; simpl; erewrite ?Hi by eauto. Qed. `````` Robbert Krebbers committed Feb 13, 2016 568 569 570 571 572 573 ``````Lemma omap_ext {A B} (f1 f2 : A → option B) m : (∀ i x, m !! i = Some x → f1 x = f2 x) → omap f1 m = omap f2 m. Proof. intros Hi; apply map_eq; intros i; rewrite !lookup_omap. by destruct (m !! i) eqn:?; simpl; erewrite ?Hi by eauto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 574 575 576 577 578 579 580 581 582 583 584 585 `````` (** ** Properties of conversion to lists *) Lemma map_to_list_unique {A} (m : M A) i x y : (i,x) ∈ map_to_list m → (i,y) ∈ map_to_list m → x = y. Proof. rewrite !elem_of_map_to_list. congruence. Qed. Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1). Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed. Lemma elem_of_map_of_list_1_help {A} (l : list (K * A)) i x : (i,x) ∈ l → (∀ y, (i,y) ∈ l → y = x) → map_of_list l !! i = Some x. Proof. induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|]. setoid_rewrite elem_of_cons. `````` Robbert Krebbers committed Feb 17, 2016 586 `````` intros [?|?] Hdup; simplify_eq; [by rewrite lookup_insert|]. `````` Robbert Krebbers committed Nov 11, 2015 587 `````` destruct (decide (i = j)) as [->|]. `````` Robbert Krebbers committed Feb 17, 2016 588 589 `````` - rewrite lookup_insert; f_equal; eauto. - rewrite lookup_insert_ne by done; eauto. `````` Robbert Krebbers committed Nov 11, 2015 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 ``````Qed. Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x : NoDup (l.*1) → (i,x) ∈ l → map_of_list l !! i = Some x. Proof. intros ? Hx; apply elem_of_map_of_list_1_help; eauto using NoDup_fmap_fst. intros y; revert Hx. rewrite !elem_of_list_lookup; intros [i' Hi'] [j' Hj']. cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i; by rewrite ?list_lookup_fmap, ?Hi', ?Hj'. Qed. Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x : map_of_list l !! i = Some x → (i,x) ∈ l. Proof. induction l as [|[j y] l IH]; simpl; [by rewrite lookup_empty|]. rewrite elem_of_cons. destruct (decide (i = j)) as [->|]; rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence. Qed. Lemma elem_of_map_of_list {A} (l : list (K * A)) i x : NoDup (l.*1) → (i,x) ∈ l ↔ map_of_list l !! i = Some x. Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed. Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i : i ∉ l.*1 → map_of_list l !! i = None. Proof. rewrite elem_of_list_fmap, eq_None_not_Some. intros Hi [x ?]; destruct Hi. exists (i,x); simpl; auto using elem_of_map_of_list_2. Qed. Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i : map_of_list l !! i = None → i ∉ l.*1. Proof. induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|]. `````` Robbert Krebbers committed Feb 17, 2016 619 `````` rewrite elem_of_cons. destruct (decide (i = j)); simplify_eq. `````` Robbert Krebbers committed Feb 17, 2016 620 621 `````` - by rewrite lookup_insert. - by rewrite lookup_insert_ne; intuition. `````` Robbert Krebbers committed Nov 11, 2015 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 ``````Qed. Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i : i ∉ l.*1 ↔ map_of_list l !! i = None. Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed. Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) : NoDup (l1.*1) → l1 ≡ₚ l2 → map_of_list l1 = map_of_list l2. Proof. intros ? Hperm. apply map_eq. intros i. apply option_eq. intros x. by rewrite <-!elem_of_map_of_list; rewrite <-?Hperm. Qed. Lemma map_of_list_inj {A} (l1 l2 : list (K * A)) : NoDup (l1.*1) → NoDup (l2.*1) → map_of_list l1 = map_of_list l2 → l1 ≡ₚ l2. Proof. intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst). intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2. Qed. Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m. Proof. apply map_eq. intros i. apply option_eq. intros x. by rewrite <-elem_of_map_of_list, elem_of_map_to_list by auto using NoDup_fst_map_to_list. Qed. Lemma map_to_of_list {A} (l : list (K * A)) : NoDup (l.*1) → map_to_list (map_of_list l) ≡ₚ l. Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed. Lemma map_to_list_inj {A} (m1 m2 : M A) : map_to_list m1 ≡ₚ map_to_list m2 → m1 = m2. Proof. intros. rewrite <-(map_of_to_list m1), <-(map_of_to_list m2). auto using map_of_list_proper, NoDup_fst_map_to_list. Qed. Lemma map_to_of_list_flip {A} (m1 : M A) l2 : map_to_list m1 ≡ₚ l2 → m1 = map_of_list l2. Proof. intros. rewrite <-(map_of_to_list m1). auto using map_of_list_proper, NoDup_fst_map_to_list. Qed. Lemma map_to_list_empty {A} : map_to_list ∅ = @nil (K * A). Proof. apply elem_of_nil_inv. intros [i x]. rewrite elem_of_map_to_list. apply lookup_empty_Some. Qed. Lemma map_to_list_insert {A} (m : M A) i x : m !! i = None → map_to_list (<[i:=x]>m) ≡ₚ (i,x) :: map_to_list m. Proof. intros. apply map_of_list_inj; csimpl. `````` Robbert Krebbers committed Feb 17, 2016 668 669 `````` - apply NoDup_fst_map_to_list. - constructor; auto using NoDup_fst_map_to_list. `````` Robbert Krebbers committed Nov 11, 2015 670 671 `````` rewrite elem_of_list_fmap. intros [[??] [? Hlookup]]; subst; simpl in *. rewrite elem_of_map_to_list in Hlookup. congruence. `````` Robbert Krebbers committed Feb 17, 2016 672 `````` - by rewrite !map_of_to_list. `````` Robbert Krebbers committed Nov 11, 2015 673 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 674 675 676 677 678 679 ``````Lemma map_to_list_contains {A} (m1 m2 : M A) : m1 ⊆ m2 → map_to_list m1 `contains` map_to_list m2. Proof. intros; apply NoDup_contains; auto using NoDup_map_to_list. intros [i x]. rewrite !elem_of_map_to_list; eauto using lookup_weaken. Qed. `````` Robbert Krebbers committed Nov 11, 2015 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 ``````Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = ∅. Proof. done. Qed. Lemma map_of_list_cons {A} (l : list (K * A)) i x : map_of_list ((i, x) :: l) = <[i:=x]>(map_of_list l). Proof. done. Qed. Lemma map_to_list_empty_inv_alt {A} (m : M A) : map_to_list m ≡ₚ [] → m = ∅. Proof. rewrite <-map_to_list_empty. apply map_to_list_inj. Qed. Lemma map_to_list_empty_inv {A} (m : M A) : map_to_list m = [] → m = ∅. Proof. intros Hm. apply map_to_list_empty_inv_alt. by rewrite Hm. Qed. Lemma map_to_list_insert_inv {A} (m : M A) l i x : map_to_list m ≡ₚ (i,x) :: l → m = <[i:=x]>(map_of_list l). Proof. intros Hperm. apply map_to_list_inj. assert (i ∉ l.*1 ∧ NoDup (l.*1)) as []. { rewrite <-NoDup_cons. change (NoDup (((i,x)::l).*1)). rewrite <-Hperm. auto using NoDup_fst_map_to_list. } rewrite Hperm, map_to_list_insert, map_to_of_list; auto using not_elem_of_map_of_list_1. Qed. Lemma map_choose {A} (m : M A) : m ≠ ∅ → ∃ i x, m !! i = Some x. Proof. intros Hemp. destruct (map_to_list m) as [|[i x] l] eqn:Hm. { destruct Hemp; eauto using map_to_list_empty_inv. } `````` Robbert Krebbers committed Nov 11, 2015 703 `````` exists i, x. rewrite <-elem_of_map_to_list, Hm. by left. `````` Robbert Krebbers committed Nov 11, 2015 704 705 706 707 708 709 710 ``````Qed. (** Properties of the imap function *) Lemma lookup_imap {A B} (f : K → A → option B) m i : map_imap f m !! i = m !! i ≫= f i. Proof. unfold map_imap; destruct (m !! i ≫= f i) as [y|] eqn:Hi; simpl. `````` Robbert Krebbers committed Feb 17, 2016 711 `````` - destruct (m !! i) as [x|] eqn:?; simplify_eq/=. `````` Robbert Krebbers committed Nov 11, 2015 712 713 `````` apply elem_of_map_of_list_1_help. { apply elem_of_list_omap; exists (i,x); split; `````` Robbert Krebbers committed Feb 17, 2016 714 `````` [by apply elem_of_map_to_list|by simplify_option_eq]. } `````` Robbert Krebbers committed Nov 11, 2015 715 `````` intros y'; rewrite elem_of_list_omap; intros ([i' x']&Hi'&?). `````` Robbert Krebbers committed Feb 17, 2016 716 `````` by rewrite elem_of_map_to_list in Hi'; simplify_option_eq. `````` Robbert Krebbers committed Feb 17, 2016 717 `````` - apply not_elem_of_map_of_list; rewrite elem_of_list_fmap. `````` Robbert Krebbers committed Feb 17, 2016 718 `````` intros ([i' x]&->&Hi'); simplify_eq/=. `````` Robbert Krebbers committed Nov 11, 2015 719 `````` rewrite elem_of_list_omap in Hi'; destruct Hi' as ([j y]&Hj&?). `````` Robbert Krebbers committed Feb 17, 2016 720 `````` rewrite elem_of_map_to_list in Hj; simplify_option_eq. `````` Robbert Krebbers committed Nov 11, 2015 721 722 723 724 725 726 727 728 729 730 731 ``````Qed. (** ** Properties of conversion from collections *) Lemma lookup_map_of_collection {A} `{FinCollection K C} (f : K → option A) X i x : map_of_collection f X !! i = Some x ↔ i ∈ X ∧ f i = Some x. Proof. assert (NoDup (fst <\$> omap (λ i, (i,) <\$> f i) (elements X))). { induction (NoDup_elements X) as [|i' l]; csimpl; [constructor|]. destruct (f i') as [x'|]; csimpl; auto; constructor; auto. rewrite elem_of_list_fmap. setoid_rewrite elem_of_list_omap. `````` Robbert Krebbers committed Feb 17, 2016 732 `````` by intros (?&?&?&?&?); simplify_option_eq. } `````` Robbert Krebbers committed Nov 11, 2015 733 734 `````` unfold map_of_collection; rewrite <-elem_of_map_of_list by done. rewrite elem_of_list_omap. setoid_rewrite elem_of_elements; split. `````` Robbert Krebbers committed Feb 17, 2016 735 736 `````` - intros (?&?&?); simplify_option_eq; eauto. - intros [??]; exists i; simplify_option_eq; eauto. `````` Robbert Krebbers committed Nov 11, 2015 737 738 739 740 741 742 743 744 745 746 747 748 749 ``````Qed. (** ** Induction principles *) Lemma map_ind {A} (P : M A → Prop) : P ∅ → (∀ i x m, m !! i = None → P m → P (<[i:=x]>m)) → ∀ m, P m. Proof. intros ? Hins. cut (∀ l, NoDup (l.*1) → ∀ m, map_to_list m ≡ₚ l → P m). { intros help m. apply (help (map_to_list m)); auto using NoDup_fst_map_to_list. } induction l as [|[i x] l IH]; intros Hnodup m Hml. { apply map_to_list_empty_inv_alt in Hml. by subst. } inversion_clear Hnodup. apply map_to_list_insert_inv in Hml; subst m. apply Hins. `````` Robbert Krebbers committed Feb 17, 2016 750 751 `````` - by apply not_elem_of_map_of_list_1. - apply IH; auto using map_to_of_list. `````` Robbert Krebbers committed Nov 11, 2015 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 ``````Qed. Lemma map_to_list_length {A} (m1 m2 : M A) : m1 ⊂ m2 → length (map_to_list m1) < length (map_to_list m2). Proof. revert m2. induction m1 as [|i x m ? IH] using map_ind. { intros m2 Hm2. rewrite map_to_list_empty. simpl. apply neq_0_lt. intros Hlen. symmetry in Hlen. apply nil_length_inv, map_to_list_empty_inv in Hlen. rewrite Hlen in Hm2. destruct (irreflexivity (⊂) ∅ Hm2). } intros m2 Hm2. destruct (insert_subset_inv m m2 i x) as (m2'&?&?&?); auto; subst. rewrite !map_to_list_insert; simpl; auto with arith. Qed. Lemma map_wf {A} : wf (strict (@subseteq (M A) _)). Proof. apply (wf_projected (<) (length ∘ map_to_list)). `````` Robbert Krebbers committed Feb 17, 2016 768 769 `````` - by apply map_to_list_length. - by apply lt_wf. `````` Robbert Krebbers committed Nov 11, 2015 770 771 772 773 774 775 776 777 778 ``````Qed. (** ** Properties of the [map_Forall] predicate *) Section map_Forall. Context {A} (P : K → A → Prop). Lemma map_Forall_to_list m : map_Forall P m ↔ Forall (curry P) (map_to_list m). Proof. rewrite Forall_forall. split. `````` Robbert Krebbers committed Feb 17, 2016 779 780 `````` - intros Hforall [i x]. rewrite elem_of_map_to_list. by apply (Hforall i x). - intros Hforall i x. rewrite <-elem_of_map_to_list. by apply (Hforall (i,x)). `````` Robbert Krebbers committed Nov 11, 2015 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 ``````Qed. Lemma map_Forall_empty : map_Forall P ∅. Proof. intros i x. by rewrite lookup_empty. Qed. Lemma map_Forall_impl (Q : K → A → Prop) m : map_Forall P m → (∀ i x, P i x → Q i x) → map_Forall Q m. Proof. unfold map_Forall; naive_solver. Qed. Lemma map_Forall_insert_11 m i x : map_Forall P (<[i:=x]>m) → P i x. Proof. intros Hm. by apply Hm; rewrite lookup_insert. Qed. Lemma map_Forall_insert_12 m i x : m !! i = None → map_Forall P (<[i:=x]>m) → map_Forall P m. Proof. intros ? Hm j y ?; apply Hm. by rewrite lookup_insert_ne by congruence. Qed. Lemma map_Forall_insert_2 m i x : P i x → map_Forall P m → map_Forall P (<[i:=x]>m). Proof. intros ?? j y; rewrite lookup_insert_Some; naive_solver. Qed. Lemma map_Forall_insert m i x : m !! i = None → map_Forall P (<[i:=x]>m) ↔ P i x ∧ map_Forall P m. Proof. naive_solver eauto using map_Forall_insert_11, map_Forall_insert_12, map_Forall_insert_2. Qed. Lemma map_Forall_ind (Q : M A → Prop) : Q ∅ → (∀ m i x, m !! i = None → P i x → map_Forall P m → Q m → Q (<[i:=x]>m)) → ∀ m, map_Forall P m → Q m. Proof. intros Hnil Hinsert m. induction m using map_ind; auto. rewrite map_Forall_insert by done; intros [??]; eauto. Qed. Context `{∀ i x, Decision (P i x)}. Global Instance map_Forall_dec m : Decision (map_Forall P m). Proof. refine (cast_if (decide (Forall (curry P) (map_to_list m)))); by rewrite map_Forall_to_list. Defined. Lemma map_not_Forall (m : M A) : ¬map_Forall P m ↔ ∃ i x, m !! i = Some x ∧ ¬P i x. Proof. split; [|intros (i&x&?&?) Hm; specialize (Hm i x); tauto]. rewrite map_Forall_to_list. intros Hm. apply (not_Forall_Exists _), Exists_exists in Hm. `````` Robbert Krebbers committed Nov 11, 2015 824 `````` destruct Hm as ([i x]&?&?). exists i, x. by rewrite <-elem_of_map_to_list. `````` Robbert Krebbers committed Nov 11, 2015 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 ``````Qed. End map_Forall. (** ** Properties of the [merge] operation *) Section merge. Context {A} (f : option A → option A → option A). Context `{!PropHolds (f None None = None)}. Global Instance: LeftId (=) None f → LeftId (=) ∅ (merge f). Proof. intros ??. apply map_eq. intros. by rewrite !(lookup_merge f), lookup_empty, (left_id_L None f). Qed. Global Instance: RightId (=) None f → RightId (=) ∅ (merge f). Proof. intros ??. apply map_eq. intros. by rewrite !(lookup_merge f), lookup_empty, (right_id_L None f). Qed. `````` Robbert Krebbers committed Feb 11, 2016 842 ``````Lemma merge_comm m1 m2 : `````` Robbert Krebbers committed Nov 11, 2015 843 844 845 `````` (∀ i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) → merge f m1 m2 = merge f m2 m1. Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed. `````` Robbert Krebbers committed Feb 11, 2016 846 ``````Global Instance: Comm (=) f → Comm (=) (merge f). `````` Robbert Krebbers committed Nov 11, 2015 847 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 848 `````` intros ???. apply merge_comm. intros. by apply (comm f). `````` Robbert Krebbers committed Nov 11, 2015 849 ``````Qed. `````` Robbert Krebbers committed Feb 11, 2016 850 ``````Lemma merge_assoc m1 m2 m3 : `````` Robbert Krebbers committed Nov 11, 2015 851 852 853 854 `````` (∀ i, f (m1 !! i) (f (m2 !! i) (m3 !! i)) = f (f (m1 !! i) (m2 !! i)) (m3 !! i)) → merge f m1 (merge f m2 m3) = merge f (merge f m1 m2) m3. Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed. `````` Robbert Krebbers committed Feb 11, 2016 855 ``````Global Instance: Assoc (=) f → Assoc (=) (merge f). `````` Robbert Krebbers committed Nov 11, 2015 856 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 857 `````` intros ????. apply merge_assoc. intros. by apply (assoc_L f). `````` Robbert Krebbers committed Nov 11, 2015 858 ``````Qed. `````` Robbert Krebbers committed Feb 11, 2016 859 ``````Lemma merge_idemp m1 : `````` Robbert Krebbers committed Nov 11, 2015 860 861 `````` (∀ i, f (m1 !! i) (m1 !! i) = m1 !! i) → merge f m1 m1 = m1. Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed. `````` Robbert Krebbers committed Feb 11, 2016 862 863 ``````Global Instance: IdemP (=) f → IdemP (=) (merge f). Proof. intros ??. apply merge_idemp. intros. by apply (idemp f). Qed. `````` Robbert Krebbers committed Nov 11, 2015 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 ``````End merge. Section more_merge. Context {A B C} (f : option A → option B → option C). Context `{!PropHolds (f None None = None)}. Lemma merge_Some m1 m2 m : (∀ i, m !! i = f (m1 !! i) (m2 !! i)) ↔ merge f m1 m2 = m. Proof. split; [|intros <-; apply (lookup_merge _) ]. intros Hlookup. apply map_eq; intros. rewrite Hlookup. apply (lookup_merge _). Qed. Lemma merge_empty : merge f ∅ ∅ = ∅. Proof. apply map_eq. intros. by rewrite !(lookup_merge f), !lookup_empty. Qed. Lemma partial_alter_merge g g1 g2 m1 m2 i : g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (g2 (m2 !! i)) → partial_alter g i (merge f m1 m2) = merge f (partial_alter g1 i m1) (partial_alter g2 i m2). Proof. intro. apply map_eq. intros j. destruct (decide (i = j)); subst. `````` Robbert Krebbers committed Feb 17, 2016 883 884 `````` - by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _). - by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _). `````` Robbert Krebbers committed Nov 11, 2015 885 886 887 888 889 890 ``````Qed. Lemma partial_alter_merge_l g g1 m1 m2 i : g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (m2 !! i) → partial_alter g i (merge f m1 m2) = merge f (partial_alter g1 i m1) m2. Proof. intro. apply map_eq. intros j. destruct (decide (i = j)); subst. `````` Robbert Krebbers committed Feb 17, 2016 891 892 `````` - by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _). - by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _). `````` Robbert Krebbers committed Nov 11, 2015 893 894 895 896 897 898 ``````Qed. Lemma partial_alter_merge_r g g2 m1 m2 i : g (f (m1 !! i) (m2 !! i)) = f (m1 !! i) (g2 (m2 !! i)) → partial_alter g i (merge f m1 m2) = merge f m1 (partial_alter g2 i m2). Proof. intro. apply map_eq. intros j. destruct (decide (i = j)); subst. `````` Robbert Krebbers committed Feb 17, 2016 899 900 `````` - by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _). - by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _). `````` Robbert Krebbers committed Nov 11, 2015 901 902 903 904 905 906 ``````Qed. Lemma insert_merge m1 m2 i x y z : f (Some y) (Some z) = Some x → <[i:=x]>(merge f m1 m2) = merge f (<[i:=y]>m1) (<[i:=z]>m2). Proof. by intros; apply partial_alter_merge. Qed. Lemma merge_singleton i x y z : `````` Robbert Krebbers committed Feb 17, 2016 907 `````` f (Some y) (Some z) = Some x → merge f {[i := y]} {[i := z]} = {[i := x]}. `````` Robbert Krebbers committed Nov 11, 2015 908 ``````Proof. `````` Robbert Krebbers committed Dec 21, 2015 909 `````` intros. by erewrite <-!insert_empty, <-insert_merge, merge_empty by eauto. `````` Robbert Krebbers committed Nov 11, 2015 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 ``````Qed. Lemma insert_merge_l m1 m2 i x y : f (Some y) (m2 !! i) = Some x → <[i:=x]>(merge f m1 m2) = merge f (<[i:=y]>m1) m2. Proof. by intros; apply partial_alter_merge_l. Qed. Lemma insert_merge_r m1 m2 i x z : f (m1 !! i) (Some z) = Some x → <[i:=x]>(merge f m1 m2) = merge f m1 (<[i:=z]>m2). Proof. by intros; apply partial_alter_merge_r. Qed. End more_merge. (** ** Properties on the [map_relation] relation *) Section Forall2. Context {A B} (R : A → B → Prop) (P : A → Prop) (Q : B → Prop). Context `{∀ x y, Decision (R x y), ∀ x, Decision (P x), ∀ y, Decision (Q y)}. Let f (mx : option A) (my : option B) : option bool := match mx, my with | Some x, Some y => Some (bool_decide (R x y)) | Some x, None => Some (bool_decide (P x)) | None, Some y => Some (bool_decide (Q y)) | None, None => None end. Lemma map_relation_alt (m1 : M A) (m2 : M B) : map_relation R P Q m1 m2 ↔ map_Forall (λ _, Is_true) (merge f m1 m2). Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 937 `````` - intros Hm i P'; rewrite lookup_merge by done; intros. `````` Robbert Krebbers committed Nov 11, 2015 938 `````` specialize (Hm i). destruct (m1 !! i), (m2 !! i); `````` Robbert Krebbers committed Feb 17, 2016 939 `````` simplify_eq/=; auto using bool_decide_pack. `````` Robbert Krebbers committed Feb 17, 2016 940 `````` - intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done. `````` Robbert Krebbers committed Feb 17, 2016 941 `````` destruct (m1 !! i), (m2 !! i); simplify_eq/=; auto; `````` Robbert Krebbers committed Nov 11, 2015 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 `````` by eapply bool_decide_unpack, Hm. Qed. Global Instance map_relation_dec `{∀ x y, Decision (R x y), ∀ x, Decision (P x), ∀ y, Decision (Q y)} m1 m2 : Decision (map_relation R P Q m1 m2). Proof. refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2)))); abstract by rewrite map_relation_alt. Defined. (** Due to the finiteness of finite maps, we can extract a witness if the relation does not hold. *) Lemma map_not_Forall2 (m1 : M A) (m2 : M B) : ¬map_relation R P Q m1 m2 ↔ ∃ i, (∃ x y, m1 !! i = Some x ∧ m2 !! i = Some y ∧ ¬R x y) ∨ (∃ x, m1 !! i = Some x ∧ m2 !! i = None ∧ ¬P x) ∨ (∃ y, m1 !! i = None ∧ m2 !! i = Some y ∧ ¬Q y). Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 959 `````` - rewrite map_relation_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i. `````` Robbert Krebbers committed Nov 11, 2015 960 961 `````` rewrite lookup_merge in Hm by done. destruct (m1 !! i), (m2 !! i); naive_solver auto 2 using bool_decide_pack. `````` Robbert Krebbers committed Feb 17, 2016 962 `````` - unfold map_relation, option_relation. `````` Robbert Krebbers committed Nov 11, 2015 963 `````` by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm; `````` Robbert Krebbers committed Feb 17, 2016 964 `````` specialize (Hm i); simplify_option_eq. `````` Robbert Krebbers committed Nov 11, 2015 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 ``````Qed. End Forall2. (** ** Properties on the disjoint maps *) Lemma map_disjoint_spec {A} (m1 m2 : M A) : m1 ⊥ₘ m2 ↔ ∀ i x y, m1 !! i = Some x → m2 !! i = Some y → False. Proof. split; intros Hm i; specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver. Qed. Lemma map_disjoint_alt {A} (m1 m2 : M A) : m1 ⊥ₘ m2 ↔ ∀ i, m1 !! i = None ∨ m2 !! i = None. Proof. split; intros Hm1m2 i; specialize (Hm1m2 i); destruct (m1 !! i), (m2 !! i); naive_solver. Qed. Lemma map_not_disjoint {A} (m1 m2 : M A) : ¬m1 ⊥ₘ m2 ↔ ∃ i x1 x2, m1 !! i = Some x1 ∧ m2 !! i = Some x2. Proof. unfold disjoint, map_disjoint. rewrite map_not_Forall2 by solve_decision. split; [|naive_solver]. intros [i[(x&y&?&?&?)|[(x&?&?&[])|(y&?&?&[])]]]; naive_solver. Qed. Global Instance: Symmetric (map_disjoint : relation (M A)). Proof. intros A m1 m2. rewrite !map_disjoint_spec. naive_solver. Qed. Lemma map_disjoint_empty_l {A} (m : M A) : ∅ ⊥ₘ m. Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed. Lemma map_disjoint_empty_r {A} (m : M A) : m ⊥ₘ ∅. Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed. Lemma map_disjoint_weaken {A} (m1 m1' m2 m2' : M A) : m1' ⊥ₘ m2' → m1 ⊆ m1' → m2 ⊆ m2' → m1 ⊥ₘ m2. Proof. rewrite !map_subseteq_spec, !map_disjoint_spec. eauto. Qed. Lemma map_disjoint_weaken_l {A} (m1 m1' m2 : M A) : m1' ⊥ₘ m2 → m1 ⊆ m1' → m1 ⊥ₘ m2. Proof. eauto using map_disjoint_weaken. Qed. Lemma map_disjoint_weaken_r {A} (m1 m2 m2' : M A) : m1 ⊥ₘ m2' → m2 ⊆ m2' → m1 ⊥ₘ m2. Proof. eauto using map_disjoint_weaken. Qed. Lemma map_disjoint_Some_l {A} (m1 m2 : M A) i x: m1 ⊥ₘ m2 → m1 !! i = Some x → m2 !! i = None. Proof. rewrite map_disjoint_spec, eq_None_not_Some. intros ?? [??]; eauto. Qed. Lemma map_disjoint_Some_r {A} (m1 m2 : M A) i x: m1 ⊥ₘ m2 → m2 !! i = Some x → m1 !! i = None. Proof. rewrite (symmetry_iff map_disjoint). apply map_disjoint_Some_l. Qed. `````` Robbert Krebbers committed Feb 17, 2016 1009 ``````Lemma map_disjoint_singleton_l {A} (m: M A) i x : {[i:=x]} ⊥ₘ m ↔ m !! i = None. `````` Robbert Krebbers committed Nov 11, 2015 1010 1011 ``````Proof. split; [|rewrite !map_disjoint_spec]. `````` Robbert Krebbers committed Feb 17, 2016 1012 `````` - intro. apply (map_disjoint_Some_l {[i := x]} _ _ x); `````` Robbert Krebbers committed Nov 11, 2015 1013 `````` auto using lookup_singleton. `````` Robbert Krebbers committed Feb 17, 2016 1014 `````` - intros ? j y1 y2. destruct (decide (i = j)) as [->|]. `````` Robbert Krebbers committed Nov 11, 2015 1015 1016 1017 1018 `````` + rewrite lookup_singleton. intuition congruence. + by rewrite lookup_singleton_ne. Qed. Lemma map_disjoint_singleton_r {A} (m : M A) i x : `````` Robbert Krebbers committed Feb 17, 2016 1019 `````` m ⊥ₘ {[i := x]} ↔ m !! i = None. `````` Robbert Krebbers committed Nov 11, 2015 1020 1021 ``````Proof. by rewrite (symmetry_iff map_disjoint), map_disjoint_singleton_l. Qed. Lemma map_disjoint_singleton_l_2 {A} (m : M A) i x : `````` Robbert Krebbers committed Feb 17, 2016 1022 `````` m !! i = None → {[i := x]} ⊥ₘ m. `````` Robbert Krebbers committed Nov 11, 2015 1023 1024 ``````Proof. by rewrite map_disjoint_singleton_l. Qed. Lemma map_disjoint_singleton_r_2 {A} (m : M A) i x : `````` Robbert Krebbers committed Feb 17, 2016 1025 `````` m !! i = None → m ⊥ₘ {[i := x]}. `````` Robbert Krebbers committed Nov 11, 2015 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 ``````Proof. by rewrite map_disjoint_singleton_r. Qed. Lemma map_disjoint_delete_l {A} (m1 m2 : M A) i : m1 ⊥ₘ m2 → delete i m1 ⊥ₘ m2. Proof. rewrite !map_disjoint_alt. intros Hdisjoint j. destruct (Hdisjoint j); auto. rewrite lookup_delete_None. tauto. Qed. Lemma map_disjoint_delete_r {A} (m1 m2 : M A) i : m1 ⊥ₘ m2 → m1 ⊥ₘ delete i m2. Proof. symmetry. by apply map_disjoint_delete_l. Qed. (** ** Properties of the [union_with] operation *) Section union_with. Context {A} (f : A → A → option A). Lemma lookup_union_with m1 m2 i : union_with f m1 m2 !! i = union_with f (m1 !! i) (m2 !! i). Proof. by rewrite <-(lookup_merge _). Qed. Lemma lookup_union_with_Some m1 m2 i z : union_with f m1 m2 !! i = Some z ↔ (m1 !! i = Some z ∧ m2 !! i = None) ∨ (m1 !! i = None ∧ m2 !! i = Some z) ∨ (∃ x y, m1 !! i = Some x ∧ m2 !! i = Some y ∧ f x y = Some z). Proof. rewrite lookup_union_with. destruct (m1 !! i), (m2 !! i); compute; naive_solver. Qed. Global Instance: LeftId (@eq (M A)) ∅ (union_with f). Proof. unfold union_with, map_union_with. apply _. Qed. Global Instance: RightId (@eq (M A)) ∅ (union_with f). Proof. unfold union_with, map_union_with. apply _. Qed. `````` Robbert Krebbers committed Feb 11, 2016 1055 ``````Lemma union_with_comm m1 m2 : `````` Robbert Krebbers committed Nov 11, 2015 1056 1057 1058 `````` (∀ i x y, m1 !! i = Some x → m2 !! i = Some y → f x y = f y x) → union_with f m1 m2 = union_with f m2 m1. Proof. `````` Robbert Krebbers committed Feb 11, 2016 1059 `````` intros. apply (merge_comm _). intros i. `````` Robbert Krebbers committed Nov 11, 2015 1060 1061 `````` destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 1062 1063 1064 ``````Global Instance: Comm (=) f → Comm (@eq (M A)) (union_with f). Proof. intros ???. apply union_with_comm. eauto. Qed. Lemma union_with_idemp m : `````` Robbert Krebbers committed Nov 11, 2015 1065 1066 `````` (∀ i x, m !! i = Some x → f x x = Some x) → union_with f m m = m. Proof. `````` Robbert Krebbers committed Feb 11, 2016 1067 `````` intros. apply (merge_idemp _). intros i. `````` Robbert Krebbers committed Nov 11, 2015 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 `````` destruct (m !! i) eqn:?; simpl; eauto. Qed. Lemma alter_union_with (g : A → A) m1 m2 i : (∀ x y, m1 !! i = Some x → m2 !! i = Some y → g <\$> f x y = f (g x) (g y)) → alter g i (union_with f m1 m2) = union_with f (alter g i m1) (alter g i m2). Proof. intros. apply (partial_alter_merge _). destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto. Qed. Lemma alter_union_with_l (g : A → A) m1 m2 i : (∀ x y, m1 !! i = Some x → m2 !! i = Some y → g <\$> f x y = f (g x) y) → (∀ y, m1 !! i = None → m2 !! i = Some y → g y = y) → alter g i (union_with f m1 m2) = union_with f (alter g i m1) m2. Proof. intros. apply (partial_alter_merge_l _). `````` Robbert Krebbers committed Feb 17, 2016 1084 `````` destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; f_equal/=; auto. `````` Robbert Krebbers committed Nov 11, 2015 1085 1086 1087 1088 1089 1090 1091 ``````Qed. Lemma alter_union_with_r (g : A → A) m1 m2 i : (∀ x y, m1 !! i = Some x → m2 !! i = Some y → g <\$> f x y = f x (g y)) → (∀ x, m1 !! i = Some x → m2 !! i = None → g x = x) → alter g i (union_with f m1 m2) = union_with f m1 (alter g i m2). Proof. intros. apply (partial_alter_merge_r _). `````` Robbert Krebbers committed Feb 17, 2016 1092 `` destruct (m1 !! i)``