fin_maps.v 71.1 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 5 ``````(* 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 `````` Ralf Jung committed Feb 21, 2016 6 ``````[simplify_map_eq] to simplify goals involving finite maps. *) `````` Robbert Krebbers committed Feb 13, 2016 7 ``````From Coq Require Import Permutation. `````` Robbert Krebbers committed Jul 22, 2016 8 ``````From iris.prelude Require Export relations orders vector. `````` 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 `````` (** * 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), `````` Robbert Krebbers committed Sep 20, 2016 30 `````` EqDecision K} := { `````` Robbert Krebbers committed Nov 11, 2015 31 32 33 34 35 36 37 38 39 40 41 `````` 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; `````` Robbert Krebbers committed May 27, 2016 42 `````` lookup_merge {A B C} (f: option A → option B → option C) `{!DiagNone f} m1 m2 i : `````` Robbert Krebbers committed Nov 11, 2015 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 `````` 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 58 `````` SingletonM K A M := λ i x, <[i:=x]> ∅. `````` Robbert Krebbers committed Nov 11, 2015 59 60 61 62 63 64 65 66 67 68 69 70 71 72 `````` 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 73 74 ``````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 75 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 `````` (** 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 119 120 `````` Context `{Equiv A} `{!Equivalence ((≡) : relation A)}. Global Instance map_equivalence : Equivalence ((≡) : relation (M A)). `````` Robbert Krebbers committed Nov 11, 2015 121 122 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 123 124 `````` - by intros m i. - by intros m1 m2 ? i. `````` Ralf Jung committed Feb 20, 2016 125 `````` - by intros m1 m2 m3 ?? i; trans (m2 !! i). `````` Robbert Krebbers committed Nov 11, 2015 126 127 128 129 130 `````` 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 131 `````` Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (partial_alter (M:=M A)). `````` Robbert Krebbers committed Nov 11, 2015 132 133 134 135 136 137 138 139 `````` 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 140 141 142 `````` Global Instance singleton_proper k : Proper ((≡) ==> (≡)) (singletonM k : A → M A). Proof. by intros ???; apply insert_proper. Qed. `````` Robbert Krebbers committed Nov 11, 2015 143 144 145 146 147 148 149 150 151 `````` 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. `````` Robbert Krebbers committed May 27, 2016 152 `````` Lemma merge_ext f g `{!DiagNone f, !DiagNone g} : `````` Robbert Krebbers committed Nov 11, 2015 153 `````` ((≡) ==> (≡) ==> (≡))%signature f g → `````` Robbert Krebbers committed Nov 18, 2015 154 `````` ((≡) ==> (≡) ==> (≡))%signature (merge (M:=M) f) (merge g). `````` Robbert Krebbers committed Nov 11, 2015 155 156 157 158 `````` 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 159 `````` Proper (((≡) ==> (≡) ==> (≡)) ==> (≡) ==> (≡) ==>(≡)) (union_with (M:=M A)). `````` Robbert Krebbers committed Nov 11, 2015 160 161 162 `````` 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 163 `````` Qed. `````` Robbert Krebbers committed Nov 11, 2015 164 165 `````` Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A). Proof. `````` Robbert Krebbers committed Dec 15, 2015 166 167 `````` intros m1 m2 Hm; apply map_eq; intros i. by unfold_leibniz; apply lookup_proper. `````` Robbert Krebbers committed Nov 11, 2015 168 `````` Qed. `````` Robbert Krebbers committed Nov 18, 2015 169 170 171 172 173 `````` 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. `````` Robbert Krebbers committed Mar 21, 2016 174 `````` Lemma map_equiv_lookup_l (m1 m2 : M A) i x : `````` Robbert Krebbers committed Nov 18, 2015 175 `````` m1 ≡ m2 → m1 !! i = Some x → ∃ y, m2 !! i = Some y ∧ x ≡ y. `````` Robbert Krebbers committed Mar 21, 2016 176 `````` Proof. generalize (equiv_Some_inv_l (m1 !! i) (m2 !! i) x); naive_solver. Qed. `````` Robbert Krebbers committed Jun 14, 2016 177 178 179 180 181 `````` Global Instance map_fmap_proper `{Equiv B} (f : A → B) : Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (fmap (M:=M) f). Proof. intros ? m m' ? k; rewrite !lookup_fmap. by apply option_fmap_proper. Qed. `````` Robbert Krebbers committed Nov 11, 2015 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 ``````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: ∀ {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 197 `````` destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_eq/=; `````` Ralf Jung committed Feb 20, 2016 198 `````` done || etrans; eauto. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````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 233 234 ``````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 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 `````` (** ** 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 257 `````` - by rewrite lookup_partial_alter, `````` Robbert Krebbers committed Nov 11, 2015 258 `````` !lookup_partial_alter_ne, lookup_partial_alter by congruence. `````` Robbert Krebbers committed Feb 17, 2016 259 `````` - by rewrite !lookup_partial_alter_ne by congruence. `````` Robbert Krebbers committed Nov 11, 2015 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 ``````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 287 ``````Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 ``````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 306 `````` - rewrite lookup_alter. naive_solver (simplify_option_eq; eauto). `````` Robbert Krebbers committed Feb 17, 2016 307 `````` - rewrite lookup_alter_ne by done. naive_solver. `````` Robbert Krebbers committed Nov 11, 2015 308 309 310 311 312 313 314 315 316 317 318 ``````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 319 `````` { rewrite lookup_alter; destruct (m !! j); f_equal/=; auto. } `````` Robbert Krebbers committed Nov 11, 2015 320 321 322 323 324 325 326 327 328 329 330 331 `````` 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 332 `````` - destruct (decide (i = j)) as [->|?]; `````` Robbert Krebbers committed Nov 11, 2015 333 `````` rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence. `````` Robbert Krebbers committed Feb 17, 2016 334 `````` - intros [??]. by rewrite lookup_delete_ne. `````` Robbert Krebbers committed Nov 11, 2015 335 ``````Qed. `````` Robbert Krebbers committed Jan 16, 2016 336 337 338 ``````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 339 340 341 342 343 344 345 346 ``````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 347 ``````Lemma delete_singleton {A} i (x : A) : delete i {[i := x]} = ∅. `````` Robbert Krebbers committed Nov 11, 2015 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 ``````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. `````` Robbert Krebbers committed May 31, 2016 369 370 ``````Lemma insert_delete {A} (m : M A) i x : <[i:=x]>(delete i m) = <[i:=x]> m. Proof. symmetry; apply (partial_alter_compose (λ _, Some x)). Qed. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````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 397 398 ``````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 399 400 401 402 403 404 405 ``````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 406 `````` - destruct (decide (i = j)) as [->|?]; `````` Robbert Krebbers committed Nov 11, 2015 407 `````` rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence. `````` Robbert Krebbers committed Feb 17, 2016 408 `````` - intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne. `````` Robbert Krebbers committed Nov 11, 2015 409 ``````Qed. `````` Robbert Krebbers committed Jan 16, 2016 410 411 412 ``````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 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 ``````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 429 430 `````` - 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 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 ``````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 456 457 `````` - rewrite lookup_insert. congruence. - rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne. `````` Robbert Krebbers committed Nov 11, 2015 458 459 460 461 462 463 464 465 466 467 468 469 ``````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 470 `````` intros Hi Hm1m2. exists (delete i m2). split_and?. `````` Robbert Krebbers committed May 31, 2016 471 472 `````` - rewrite insert_delete, insert_id. done. eapply lookup_weaken, strict_include; eauto. by rewrite lookup_insert. `````` Robbert Krebbers committed Feb 17, 2016 473 474 `````` - eauto using insert_delete_subset. - by rewrite lookup_delete. `````` Robbert Krebbers committed Nov 11, 2015 475 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 476 ``````Lemma insert_empty {A} i (x : A) : <[i:=x]>∅ = {[i := x]}. `````` Robbert Krebbers committed Nov 11, 2015 477 ``````Proof. done. Qed. `````` Robbert Krebbers committed Sep 27, 2016 478 479 480 481 ``````Lemma insert_non_empty {A} (m : M A) i x : <[i:=x]>m ≠ ∅. Proof. intros Hi%(f_equal (!! i)). by rewrite lookup_insert, lookup_empty in Hi. Qed. `````` Robbert Krebbers committed Nov 11, 2015 482 483 484 `````` (** ** Properties of the singleton maps *) Lemma lookup_singleton_Some {A} i j (x y : A) : `````` Robbert Krebbers committed Feb 17, 2016 485 `````` {[i := x]} !! j = Some y ↔ i = j ∧ x = y. `````` Robbert Krebbers committed Nov 11, 2015 486 ``````Proof. `````` Robbert Krebbers committed Dec 21, 2015 487 `````` rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence. `````` Robbert Krebbers committed Nov 11, 2015 488 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 489 ``````Lemma lookup_singleton_None {A} i j (x : A) : {[i := x]} !! j = None ↔ i ≠ j. `````` Robbert Krebbers committed Dec 21, 2015 490 ``````Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; tauto. Qed. `````` Robbert Krebbers committed Feb 17, 2016 491 ``````Lemma lookup_singleton {A} i (x : A) : {[i := x]} !! i = Some x. `````` Robbert Krebbers committed Nov 11, 2015 492 ``````Proof. by rewrite lookup_singleton_Some. Qed. `````` Robbert Krebbers committed Feb 17, 2016 493 ``````Lemma lookup_singleton_ne {A} i j (x : A) : i ≠ j → {[i := x]} !! j = None. `````` Robbert Krebbers committed Nov 11, 2015 494 ``````Proof. by rewrite lookup_singleton_None. Qed. `````` Robbert Krebbers committed Feb 17, 2016 495 ``````Lemma map_non_empty_singleton {A} i (x : A) : {[i := x]} ≠ ∅. `````` Robbert Krebbers committed Nov 11, 2015 496 497 498 499 ``````Proof. intros Hix. apply (f_equal (!! i)) in Hix. by rewrite lookup_empty, lookup_singleton in Hix. Qed. `````` Robbert Krebbers committed Feb 17, 2016 500 ``````Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i := x]} = {[i := y]}. `````` Robbert Krebbers committed Nov 11, 2015 501 ``````Proof. `````` Robbert Krebbers committed Dec 21, 2015 502 `````` unfold singletonM, map_singleton, insert, map_insert. `````` Robbert Krebbers committed Nov 11, 2015 503 504 `````` by rewrite <-partial_alter_compose. Qed. `````` Robbert Krebbers committed Feb 17, 2016 505 ``````Lemma alter_singleton {A} (f : A → A) i x : alter f i {[i := x]} = {[i := f x]}. `````` Robbert Krebbers committed Nov 11, 2015 506 507 ``````Proof. intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?]. `````` Robbert Krebbers committed Feb 17, 2016 508 509 `````` - by rewrite lookup_alter, !lookup_singleton. - by rewrite lookup_alter_ne, !lookup_singleton_ne. `````` Robbert Krebbers committed Nov 11, 2015 510 511 ``````Qed. Lemma alter_singleton_ne {A} (f : A → A) i j x : `````` Robbert Krebbers committed Feb 17, 2016 512 `````` i ≠ j → alter f i {[j := x]} = {[j := x]}. `````` Robbert Krebbers committed Nov 11, 2015 513 514 515 516 ``````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. `````` Robbert Krebbers committed Sep 27, 2016 517 518 ``````Lemma singleton_non_empty {A} i (x : A) : {[i:=x]} ≠ ∅. Proof. apply insert_non_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 519 520 521 522 523 524 `````` (** ** 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 525 526 527 ``````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 528 529 `````` - by rewrite lookup_fmap, !lookup_insert. - by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done. `````` Robbert Krebbers committed Feb 14, 2016 530 ``````Qed. `````` Jacques-Henri Jourdan committed Jul 01, 2016 531 532 533 534 535 536 ``````Lemma fmap_delete {A B} (f: A → B) m i: f <\$> delete i m = delete i (f <\$> m). Proof. apply map_eq; intros i'; destruct (decide (i' = i)) as [->|]. - by rewrite lookup_fmap, !lookup_delete. - by rewrite lookup_fmap, !lookup_delete_ne, lookup_fmap by done. Qed. `````` Robbert Krebbers committed Feb 14, 2016 537 538 539 540 ``````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 541 542 `````` - by rewrite lookup_omap, !lookup_insert. - by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done. `````` Robbert Krebbers committed Feb 14, 2016 543 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 544 ``````Lemma map_fmap_singleton {A B} (f : A → B) i x : f <\$> {[i := x]} = {[i := f x]}. `````` Robbert Krebbers committed Feb 14, 2016 545 546 547 ``````Proof. by unfold singletonM, map_singleton; rewrite fmap_insert, map_fmap_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 548 ``````Lemma omap_singleton {A B} (f : A → option B) i x y : `````` Robbert Krebbers committed Feb 17, 2016 549 `````` f x = Some y → omap f {[ i := x ]} = {[ i := y ]}. `````` Robbert Krebbers committed Nov 11, 2015 550 ``````Proof. `````` Robbert Krebbers committed Feb 14, 2016 551 552 `````` intros. unfold singletonM, map_singleton. by erewrite omap_insert, omap_empty by eauto. `````` Robbert Krebbers committed Nov 11, 2015 553 554 555 556 557 558 ``````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 559 560 561 562 563 564 ``````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 565 566 567 568 569 570 ``````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 571 572 573 574 575 576 ``````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 577 578 579 580 581 582 583 584 585 586 587 588 `````` (** ** 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 589 `````` intros [?|?] Hdup; simplify_eq; [by rewrite lookup_insert|]. `````` Robbert Krebbers committed Nov 11, 2015 590 `````` destruct (decide (i = j)) as [->|]. `````` Robbert Krebbers committed Feb 17, 2016 591 592 `````` - rewrite lookup_insert; f_equal; eauto. - rewrite lookup_insert_ne by done; eauto. `````` Robbert Krebbers committed Nov 11, 2015 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 ``````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 622 `````` rewrite elem_of_cons. destruct (decide (i = j)); simplify_eq. `````` Robbert Krebbers committed Feb 17, 2016 623 624 `````` - by rewrite lookup_insert. - by rewrite lookup_insert_ne; intuition. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````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. `````` Robbert Krebbers committed May 31, 2016 662 663 664 665 666 667 668 669 670 671 672 673 674 `````` 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_of_list_fmap {A B} (f : A → B) l : map_of_list (prod_map id f <\$> l) = f <\$> map_of_list l. Proof. induction l as [|[i x] l IH]; csimpl; rewrite ?fmap_empty; auto. rewrite <-map_of_list_cons; simpl. by rewrite IH, <-fmap_insert. Qed. `````` Robbert Krebbers committed Nov 11, 2015 675 676 677 678 679 680 681 682 683 ``````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 684 685 `````` - apply NoDup_fst_map_to_list. - constructor; auto using NoDup_fst_map_to_list. `````` Robbert Krebbers committed Nov 11, 2015 686 687 `````` 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 688 `````` - by rewrite !map_of_to_list. `````` Robbert Krebbers committed Nov 11, 2015 689 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 690 691 692 693 694 695 ``````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 May 31, 2016 696 697 698 699 700 701 702 703 704 705 ``````Lemma map_to_list_fmap {A B} (f : A → B) m : map_to_list (f <\$> m) ≡ₚ prod_map id f <\$> map_to_list m. Proof. assert (NoDup ((prod_map id f <\$> map_to_list m).*1)). { erewrite <-list_fmap_compose, (list_fmap_ext _ fst) by done. apply NoDup_fst_map_to_list. } rewrite <-(map_of_to_list m) at 1. by rewrite <-map_of_list_fmap, map_to_of_list. Qed. `````` Robbert Krebbers committed Nov 11, 2015 706 707 708 709 ``````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. `````` Robbert Krebbers committed Sep 27, 2016 710 711 712 713 714 ``````Lemma map_to_list_empty' {A} (m : M A) : map_to_list m = [] ↔ m = ∅. Proof. split. apply map_to_list_empty_inv. intros ->. apply map_to_list_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 715 716 717 718 719 720 721 722 723 724 ``````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. `````` Robbert Krebbers committed May 31, 2016 725 `````` `````` Robbert Krebbers committed Nov 11, 2015 726 727 728 729 ``````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 730 `````` exists i, x. rewrite <-elem_of_map_to_list, Hm. by left. `````` Robbert Krebbers committed Nov 11, 2015 731 732 ``````Qed. `````` Robbert Krebbers committed Sep 28, 2016 733 734 735 736 737 738 ``````Global Instance map_eq_dec_empty {A} (m : M A) : Decision (m = ∅) | 20. Proof. refine (cast_if (decide (elements m = []))); [apply _|by rewrite <-?map_to_list_empty' ..]. Defined. `````` Robbert Krebbers committed Nov 11, 2015 739 740 741 742 743 ``````(** 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 744 `````` - destruct (m !! i) as [x|] eqn:?; simplify_eq/=. `````` Robbert Krebbers committed Nov 11, 2015 745 746 `````` apply elem_of_map_of_list_1_help. { apply elem_of_list_omap; exists (i,x); split; `````` Robbert Krebbers committed Feb 17, 2016 747 `````` [by apply elem_of_map_to_list|by simplify_option_eq]. } `````` Robbert Krebbers committed Nov 11, 2015 748 `````` intros y'; rewrite elem_of_list_omap; intros ([i' x']&Hi'&?). `````` Robbert Krebbers committed Feb 17, 2016 749 `````` by rewrite elem_of_map_to_list in Hi'; simplify_option_eq. `````` Robbert Krebbers committed Feb 17, 2016 750 `````` - apply not_elem_of_map_of_list; rewrite elem_of_list_fmap. `````` Robbert Krebbers committed Feb 17, 2016 751 `````` intros ([i' x]&->&Hi'); simplify_eq/=. `````` Robbert Krebbers committed Nov 11, 2015 752 `````` rewrite elem_of_list_omap in Hi'; destruct Hi' as ([j y]&Hj&?). `````` Robbert Krebbers committed Feb 17, 2016 753 `````` rewrite elem_of_map_to_list in Hj; simplify_option_eq. `````` Robbert Krebbers committed Nov 11, 2015 754 755 756 757 758 759 760 761 762 763 764 ``````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 765 `````` by intros (?&?&?&?&?); simplify_option_eq. } `````` Robbert Krebbers committed Nov 11, 2015 766 767 `````` 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 768 769 `````` - intros (?&?&?); simplify_option_eq; eauto. - intros [??]; exists i; simplify_option_eq; eauto. `````` Robbert Krebbers committed Nov 11, 2015 770 771 772 773 774 775 776 777 778 779 780 781 782 ``````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 783 784 `````` - by apply not_elem_of_map_of_list_1. - apply IH; auto using map_to_of_list. `````` Robbert Krebbers committed Nov 11, 2015 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 ``````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 801 802 `````` - by apply map_to_list_length. - by apply lt_wf. `````` Robbert Krebbers committed Nov 11, 2015 803 804 805 806 807 808 809 810 811 ``````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 812 813 `````` - 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 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 ``````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 857 `````` destruct Hm as ([i x]&?&?). exists i, x. by rewrite <-elem_of_map_to_list. `````` Robbert Krebbers committed Nov 11, 2015 858 859 860 861 862 ``````Qed. End map_Forall. (** ** Properties of the [merge] operation *) Section merge. `````` Robbert Krebbers committed May 27, 2016 863 ``````Context {A} (f : option A → option A → option A) `{!DiagNone f}. `````` Robbert Krebbers committed Nov 11, 2015 864 865 866 867 868 869 870 871 872 873 ``````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 874 ``````Lemma merge_comm m1 m2 : `````` Robbert Krebbers committed Nov 11, 2015 875 876 877 `````` (∀ 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 May 27, 2016 878 879 ``````Global Instance merge_comm' : Comm (=) f → Comm (=) (merge f). Proof. intros ???. apply merge_comm. intros. by apply (comm f). Qed. `````` Robbert Krebbers committed Feb 11, 2016 880 ``````Lemma merge_assoc m1 m2 m3 : `````` Robbert Krebbers committed Nov 11, 2015 881 882 883 884 `````` (∀ 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 May 27, 2016 885 886 ``````Global Instance merge_assoc' : Assoc (=) f → Assoc (=) (merge f). Proof. intros ????. apply merge_assoc. intros. by apply (assoc_L f). Qed. `````` Robbert Krebbers committed Feb 11, 2016 887 ``````Lemma merge_idemp m1 : `````` Robbert Krebbers committed Nov 11, 2015 888 889 `````` (∀ 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 May 27, 2016 890 ``````Global Instance merge_idemp' : IdemP (=) f → IdemP (=) (merge f). `````` Robbert Krebbers committed Feb 11, 2016 891 ``````Proof. intros ??. apply merge_idemp. intros. by apply (idemp f). Qed. `````` Robbert Krebbers committed Nov 11, 2015 892 893 894 ``````End merge. Section more_merge. `````` Robbert Krebbers committed May 27, 2016 895 896 ``````Context {A B C} (f : option A → option B → option C) `{!DiagNone f}. `````` Robbert Krebbers committed Nov 11, 2015 897 898 899 900 901 902 903 904 905 906 907 908 909 910 ``````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 911 912 `````` - 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 913 914 915 916 917 918 ``````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 919 920 `````` - 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 921 922 923 924 925 926 ``````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 927 928 `````` - 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 929 930 931 932 933 934 ``````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 935 `````` f (Some y) (Some z) = Some x → merge f {[i := y]} {[i := z]} = {[i := x]}. `````` Robbert Krebbers committed Nov 11, 2015 936 ``````Proof. `````` Robbert Krebbers committed Dec 21, 2015 937 `````` intros. by erewrite <-!insert_empty, <-insert_merge, merge_empty by eauto. `````` Robbert Krebbers committed Nov 11, 2015 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 ``````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 965 `````` - intros Hm i P'; rewrite lookup_merge by done; intros. `````` Robbert Krebbers committed Nov 11, 2015 966 `````` specialize (Hm i). destruct (m1 !! i), (m2 !! i); `````` Robbert Krebbers committed Feb 17, 2016 967 `````` simplify_eq/=; auto using bool_decide_pack. `````` Robbert Krebbers committed Feb 17, 2016 968 `````` - intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done. `````` Robbert Krebbers committed Feb 17, 2016 969 `````` destruct (m1 !! i), (m2 !! i); simplify_eq/=; auto; `````` Robbert Krebbers committed Nov 11, 2015 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 `````` 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 987 `````` - rewrite map_relation_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i. `````` Robbert Krebbers committed Nov 11, 2015 988 989 `````` 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 990 `````` - unfold map_relation, option_relation. `````` Robbert Krebbers committed Nov 11, 2015 991 `````` by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm; `````` Robbert Krebbers committed Feb 17, 2016 992 `````` specialize (Hm i); simplify_option_eq. `````` Robbert Krebbers committed Nov 11, 2015 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 ``````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. `````` Robbert Krebbers committed May 27, 2016 1016 ``````Global Instance map_disjoint_sym : Symmetric (map_disjoint : relation (M A)). `````` Robbert Krebbers committed Nov 11, 2015 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 ``````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. Pro``````