collections.v 33.3 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. *) (** This file collects definitions and theorems on collections. Most importantly, it implements some tactics to automatically solve goals involving collections. *) `````` Robbert Krebbers committed Mar 10, 2016 6 ``````From iris.prelude Require Export base tactics orders. `````` Robbert Krebbers committed Nov 11, 2015 7 `````` `````` Robbert Krebbers committed Mar 23, 2016 8 9 ``````Instance collection_disjoint `{ElemOf A C} : Disjoint C := λ X Y, ∀ x, x ∈ X → x ∈ Y → False. `````` Robbert Krebbers committed Nov 11, 2015 10 11 ``````Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y, ∀ x, x ∈ X → x ∈ Y. `````` Robbert Krebbers committed Mar 23, 2016 12 ``````Typeclasses Opaque collection_disjoint collection_subseteq. `````` Robbert Krebbers committed Nov 11, 2015 13 14 15 16 `````` (** * Basic theorems *) Section simple_collection. Context `{SimpleCollection A C}. `````` Robbert Krebbers committed Nov 18, 2015 17 18 `````` Implicit Types x y : A. Implicit Types X Y : C. `````` Robbert Krebbers committed Nov 11, 2015 19 20 21 22 23 24 25 26 27 28 29 `````` Lemma elem_of_empty x : x ∈ ∅ ↔ False. Proof. split. apply not_elem_of_empty. done. Qed. Lemma elem_of_union_l x X Y : x ∈ X → x ∈ X ∪ Y. Proof. intros. apply elem_of_union. auto. Qed. Lemma elem_of_union_r x X Y : x ∈ Y → x ∈ X ∪ Y. Proof. intros. apply elem_of_union. auto. Qed. Global Instance: EmptySpec C. Proof. firstorder auto. Qed. Global Instance: JoinSemiLattice C. Proof. firstorder auto. Qed. `````` Robbert Krebbers committed Feb 15, 2016 30 31 `````` Global Instance: AntiSymm (≡) (@collection_subseteq A C _). Proof. done. Qed. `````` Robbert Krebbers committed Nov 11, 2015 32 33 34 35 36 37 38 39 40 `````` Lemma elem_of_subseteq X Y : X ⊆ Y ↔ ∀ x, x ∈ X → x ∈ Y. Proof. done. Qed. Lemma elem_of_equiv X Y : X ≡ Y ↔ ∀ x, x ∈ X ↔ x ∈ Y. Proof. firstorder. Qed. Lemma elem_of_equiv_alt X Y : X ≡ Y ↔ (∀ x, x ∈ X → x ∈ Y) ∧ (∀ x, x ∈ Y → x ∈ X). Proof. firstorder. Qed. Lemma elem_of_equiv_empty X : X ≡ ∅ ↔ ∀ x, x ∉ X. Proof. firstorder. Qed. `````` Robbert Krebbers committed Mar 23, 2016 41 42 43 `````` Lemma elem_of_disjoint X Y : X ⊥ Y ↔ ∀ x, x ∈ X → x ∈ Y → False. Proof. done. Qed. `````` Robbert Krebbers committed May 31, 2016 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 `````` Global Instance disjoint_sym : Symmetric (@disjoint C _). Proof. intros ??. rewrite !elem_of_disjoint; naive_solver. Qed. Lemma disjoint_empty_l Y : ∅ ⊥ Y. Proof. rewrite elem_of_disjoint; intros x; by rewrite elem_of_empty. Qed. Lemma disjoint_empty_r X : X ⊥ ∅. Proof. rewrite (symmetry_iff _); apply disjoint_empty_l. Qed. Lemma disjoint_singleton_l x Y : {[ x ]} ⊥ Y ↔ x ∉ Y. Proof. rewrite elem_of_disjoint; setoid_rewrite elem_of_singleton; naive_solver. Qed. Lemma disjoint_singleton_r y X : X ⊥ {[ y ]} ↔ y ∉ X. Proof. rewrite (symmetry_iff (⊥)). apply disjoint_singleton_l. Qed. Lemma disjoint_union_l X1 X2 Y : X1 ∪ X2 ⊥ Y ↔ X1 ⊥ Y ∧ X2 ⊥ Y. Proof. rewrite !elem_of_disjoint; setoid_rewrite elem_of_union; naive_solver. Qed. Lemma disjoint_union_r X Y1 Y2 : X ⊥ Y1 ∪ Y2 ↔ X ⊥ Y1 ∧ X ⊥ Y2. Proof. rewrite !(symmetry_iff (⊥) X). apply disjoint_union_l. Qed. `````` Robbert Krebbers committed Nov 11, 2015 63 64 65 66 67 68 69 70 71 72 73 74 75 `````` Lemma collection_positive_l X Y : X ∪ Y ≡ ∅ → X ≡ ∅. Proof. rewrite !elem_of_equiv_empty. setoid_rewrite elem_of_union. naive_solver. Qed. Lemma collection_positive_l_alt X Y : X ≢ ∅ → X ∪ Y ≢ ∅. Proof. eauto using collection_positive_l. Qed. Lemma elem_of_singleton_1 x y : x ∈ {[y]} → x = y. Proof. by rewrite elem_of_singleton. Qed. Lemma elem_of_singleton_2 x y : x = y → x ∈ {[y]}. Proof. by rewrite elem_of_singleton. Qed. Lemma elem_of_subseteq_singleton x X : x ∈ X ↔ {[ x ]} ⊆ X. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 76 77 `````` - intros ??. rewrite elem_of_singleton. by intros ->. - intros Ex. by apply (Ex x), elem_of_singleton. `````` Robbert Krebbers committed Nov 11, 2015 78 `````` Qed. `````` Robbert Krebbers committed Mar 23, 2016 79 `````` `````` Robbert Krebbers committed Nov 18, 2015 80 `````` Global Instance singleton_proper : Proper ((=) ==> (≡)) (singleton (B:=C)). `````` Robbert Krebbers committed Nov 11, 2015 81 `````` Proof. by repeat intro; subst. Qed. `````` Robbert Krebbers committed Nov 18, 2015 82 `````` Global Instance elem_of_proper : `````` Robbert Krebbers committed Mar 23, 2016 83 `````` Proper ((=) ==> (≡) ==> iff) (@elem_of A C _) | 5. `````` Robbert Krebbers committed Nov 11, 2015 84 `````` Proof. intros ???; subst. firstorder. Qed. `````` Ralf Jung committed Mar 23, 2016 85 `````` Global Instance disjoint_proper: Proper ((≡) ==> (≡) ==> iff) (@disjoint C _). `````` Robbert Krebbers committed Mar 23, 2016 86 `````` Proof. intros ??????. by rewrite !elem_of_disjoint; setoid_subst. Qed. `````` Robbert Krebbers committed Nov 11, 2015 87 88 89 `````` Lemma elem_of_union_list Xs x : x ∈ ⋃ Xs ↔ ∃ X, X ∈ Xs ∧ x ∈ X. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 90 `````` - induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|]. `````` Robbert Krebbers committed Nov 11, 2015 91 `````` setoid_rewrite elem_of_cons. apply elem_of_union in HXs. naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 92 `````` - intros [X []]. induction 1; simpl; [by apply elem_of_union_l |]. `````` Robbert Krebbers committed Nov 11, 2015 93 94 `````` intros. apply elem_of_union_r; auto. Qed. `````` Robbert Krebbers committed Nov 18, 2015 95 `````` Lemma non_empty_singleton x : ({[ x ]} : C) ≢ ∅. `````` Robbert Krebbers committed Nov 11, 2015 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 `````` Proof. intros [E _]. by apply (elem_of_empty x), E, elem_of_singleton. Qed. Lemma not_elem_of_singleton x y : x ∉ {[ y ]} ↔ x ≠ y. Proof. by rewrite elem_of_singleton. Qed. Lemma not_elem_of_union x X Y : x ∉ X ∪ Y ↔ x ∉ X ∧ x ∉ Y. Proof. rewrite elem_of_union. tauto. Qed. Section leibniz. Context `{!LeibnizEquiv C}. Lemma elem_of_equiv_L X Y : X = Y ↔ ∀ x, x ∈ X ↔ x ∈ Y. Proof. unfold_leibniz. apply elem_of_equiv. Qed. Lemma elem_of_equiv_alt_L X Y : X = Y ↔ (∀ x, x ∈ X → x ∈ Y) ∧ (∀ x, x ∈ Y → x ∈ X). Proof. unfold_leibniz. apply elem_of_equiv_alt. Qed. Lemma elem_of_equiv_empty_L X : X = ∅ ↔ ∀ x, x ∉ X. Proof. unfold_leibniz. apply elem_of_equiv_empty. Qed. Lemma collection_positive_l_L X Y : X ∪ Y = ∅ → X = ∅. Proof. unfold_leibniz. apply collection_positive_l. Qed. Lemma collection_positive_l_alt_L X Y : X ≠ ∅ → X ∪ Y ≠ ∅. Proof. unfold_leibniz. apply collection_positive_l_alt. Qed. Lemma non_empty_singleton_L x : {[ x ]} ≠ ∅. Proof. unfold_leibniz. apply non_empty_singleton. Qed. End leibniz. Section dec. Context `{∀ X Y : C, Decision (X ⊆ Y)}. Global Instance elem_of_dec_slow (x : A) (X : C) : Decision (x ∈ X) | 100. Proof. refine (cast_if (decide_rel (⊆) {[ x ]} X)); by rewrite elem_of_subseteq_singleton. Defined. End dec. End simple_collection. `````` Robbert Krebbers committed Feb 24, 2016 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 ``````(** * Tactics *) (** The tactic [set_unfold] transforms all occurrences of [(∪)], [(∩)], [(∖)], [(<\$>)], [∅], [{[_]}], [(≡)], and [(⊆)] into logically equivalent propositions involving just [∈]. For example, [A → x ∈ X ∪ ∅] becomes [A → x ∈ X ∨ False]. This transformation is implemented using type classes instead of [rewrite]ing to ensure that we traverse each term at most once. *) Class SetUnfold (P Q : Prop) := { set_unfold : P ↔ Q }. Arguments set_unfold _ _ {_}. Hint Mode SetUnfold + - : typeclass_instances. Class SetUnfoldSimpl (P Q : Prop) := { set_unfold_simpl : SetUnfold P Q }. Hint Extern 0 (SetUnfoldSimpl _ _) => csimpl; constructor : typeclass_instances. Instance set_unfold_fallthrough P : SetUnfold P P | 1000. done. Qed. Definition set_unfold_1 `{SetUnfold P Q} : P → Q := proj1 (set_unfold P Q). Definition set_unfold_2 `{SetUnfold P Q} : Q → P := proj2 (set_unfold P Q). Lemma set_unfold_impl P Q P' Q' : SetUnfold P P' → SetUnfold Q Q' → SetUnfold (P → Q) (P' → Q'). Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed. Lemma set_unfold_and P Q P' Q' : SetUnfold P P' → SetUnfold Q Q' → SetUnfold (P ∧ Q) (P' ∧ Q'). Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed. Lemma set_unfold_or P Q P' Q' : SetUnfold P P' → SetUnfold Q Q' → SetUnfold (P ∨ Q) (P' ∨ Q'). Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed. Lemma set_unfold_iff P Q P' Q' : SetUnfold P P' → SetUnfold Q Q' → SetUnfold (P ↔ Q) (P' ↔ Q'). Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed. Lemma set_unfold_not P P' : SetUnfold P P' → SetUnfold (¬P) (¬P'). Proof. constructor. by rewrite (set_unfold P P'). Qed. Lemma set_unfold_forall {A} (P P' : A → Prop) : (∀ x, SetUnfold (P x) (P' x)) → SetUnfold (∀ x, P x) (∀ x, P' x). Proof. constructor. naive_solver. Qed. Lemma set_unfold_exist {A} (P P' : A → Prop) : (∀ x, SetUnfold (P x) (P' x)) → SetUnfold (∃ x, P x) (∃ x, P' x). Proof. constructor. naive_solver. Qed. (* Avoid too eager application of the above instances (and thus too eager unfolding of type class transparent definitions). *) Hint Extern 0 (SetUnfold (_ → _) _) => class_apply set_unfold_impl : typeclass_instances. Hint Extern 0 (SetUnfold (_ ∧ _) _) => class_apply set_unfold_and : typeclass_instances. Hint Extern 0 (SetUnfold (_ ∨ _) _) => class_apply set_unfold_or : typeclass_instances. Hint Extern 0 (SetUnfold (_ ↔ _) _) => class_apply set_unfold_iff : typeclass_instances. Hint Extern 0 (SetUnfold (¬ _) _) => class_apply set_unfold_not : typeclass_instances. Hint Extern 1 (SetUnfold (∀ _, _) _) => class_apply set_unfold_forall : typeclass_instances. Hint Extern 0 (SetUnfold (∃ _, _) _) => class_apply set_unfold_exist : typeclass_instances. Section set_unfold_simple. Context `{SimpleCollection A C}. Implicit Types x y : A. Implicit Types X Y : C. Global Instance set_unfold_empty x : SetUnfold (x ∈ ∅) False. Proof. constructor; apply elem_of_empty. Qed. Global Instance set_unfold_singleton x y : SetUnfold (x ∈ {[ y ]}) (x = y). Proof. constructor; apply elem_of_singleton. Qed. Global Instance set_unfold_union x X Y P Q : SetUnfold (x ∈ X) P → SetUnfold (x ∈ Y) Q → SetUnfold (x ∈ X ∪ Y) (P ∨ Q). Proof. intros ??; constructor. by rewrite elem_of_union, (set_unfold (x ∈ X) P), (set_unfold (x ∈ Y) Q). Qed. Global Instance set_unfold_equiv_same X : SetUnfold (X ≡ X) True | 1. Proof. done. Qed. Global Instance set_unfold_equiv_empty_l X (P : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (∅ ≡ X) (∀ x, ¬P x) | 5. Proof. intros ?; constructor. rewrite (symmetry_iff equiv), elem_of_equiv_empty; naive_solver. Qed. Global Instance set_unfold_equiv_empty_r (P : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (X ≡ ∅) (∀ x, ¬P x) | 5. Proof. constructor. rewrite elem_of_equiv_empty; naive_solver. Qed. Global Instance set_unfold_equiv (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X ≡ Y) (∀ x, P x ↔ Q x) | 10. Proof. constructor. rewrite elem_of_equiv; naive_solver. Qed. Global Instance set_unfold_subseteq (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X ⊆ Y) (∀ x, P x → Q x). Proof. constructor. rewrite elem_of_subseteq; naive_solver. Qed. Global Instance set_unfold_subset (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X ⊂ Y) ((∀ x, P x → Q x) ∧ ¬∀ x, P x ↔ Q x). Proof. constructor. rewrite subset_spec, elem_of_subseteq, elem_of_equiv. repeat f_equiv; naive_solver. Qed. `````` Robbert Krebbers committed Mar 23, 2016 226 227 228 229 `````` Global Instance set_unfold_disjoint (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X ⊥ Y) (∀ x, P x → Q x → False). Proof. constructor. rewrite elem_of_disjoint. naive_solver. Qed. `````` Robbert Krebbers committed Feb 24, 2016 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 `````` Context `{!LeibnizEquiv C}. Global Instance set_unfold_equiv_same_L X : SetUnfold (X = X) True | 1. Proof. done. Qed. Global Instance set_unfold_equiv_empty_l_L X (P : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (∅ = X) (∀ x, ¬P x) | 5. Proof. constructor. rewrite (symmetry_iff eq), elem_of_equiv_empty_L; naive_solver. Qed. Global Instance set_unfold_equiv_empty_r_L (P : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (X = ∅) (∀ x, ¬P x) | 5. Proof. constructor. rewrite elem_of_equiv_empty_L; naive_solver. Qed. Global Instance set_unfold_equiv_L (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X = Y) (∀ x, P x ↔ Q x) | 10. Proof. constructor. rewrite elem_of_equiv_L; naive_solver. Qed. End set_unfold_simple. Section set_unfold. Context `{Collection A C}. Implicit Types x y : A. Implicit Types X Y : C. Global Instance set_unfold_intersection x X Y P Q : SetUnfold (x ∈ X) P → SetUnfold (x ∈ Y) Q → SetUnfold (x ∈ X ∩ Y) (P ∧ Q). Proof. intros ??; constructor. by rewrite elem_of_intersection, (set_unfold (x ∈ X) P), (set_unfold (x ∈ Y) Q). Qed. Global Instance set_unfold_difference x X Y P Q : SetUnfold (x ∈ X) P → SetUnfold (x ∈ Y) Q → SetUnfold (x ∈ X ∖ Y) (P ∧ ¬Q). Proof. intros ??; constructor. by rewrite elem_of_difference, (set_unfold (x ∈ X) P), (set_unfold (x ∈ Y) Q). Qed. End set_unfold. Section set_unfold_monad. Context `{CollectionMonad M} {A : Type}. Implicit Types x y : A. Global Instance set_unfold_ret x y : SetUnfold (x ∈ mret y) (x = y). Proof. constructor; apply elem_of_ret. Qed. Global Instance set_unfold_bind {B} (f : A → M B) X (P Q : A → Prop) : (∀ y, SetUnfold (y ∈ X) (P y)) → (∀ y, SetUnfold (x ∈ f y) (Q y)) → SetUnfold (x ∈ X ≫= f) (∃ y, Q y ∧ P y). Proof. constructor. rewrite elem_of_bind; naive_solver. Qed. Global Instance set_unfold_fmap {B} (f : A → B) X (P : A → Prop) : (∀ y, SetUnfold (y ∈ X) (P y)) → SetUnfold (x ∈ f <\$> X) (∃ y, x = f y ∧ P y). Proof. constructor. rewrite elem_of_fmap; naive_solver. Qed. Global Instance set_unfold_join (X : M (M A)) (P : M A → Prop) : (∀ Y, SetUnfold (Y ∈ X) (P Y)) → SetUnfold (x ∈ mjoin X) (∃ Y, x ∈ Y ∧ P Y). Proof. constructor. rewrite elem_of_join; naive_solver. Qed. End set_unfold_monad. Ltac set_unfold := let rec unfold_hyps := try match goal with | H : _ |- _ => apply set_unfold_1 in H; revert H; first [unfold_hyps; intros H | intros H; fail 1] end in apply set_unfold_2; unfold_hyps; csimpl in *. (** Since [firstorder] fails or loops on very small goals generated by [set_solver] already. We use the [naive_solver] tactic as a substitute. This tactic either fails or proves the goal. *) Tactic Notation "set_solver" "by" tactic3(tac) := `````` Ralf Jung committed Mar 05, 2016 299 `````` try fast_done; `````` Robbert Krebbers committed Feb 24, 2016 300 301 302 303 304 305 306 307 308 309 310 311 312 `````` intros; setoid_subst; set_unfold; intros; setoid_subst; try match goal with |- _ ∈ _ => apply dec_stable end; naive_solver tac. Tactic Notation "set_solver" "-" hyp_list(Hs) "by" tactic3(tac) := clear Hs; set_solver by tac. Tactic Notation "set_solver" "+" hyp_list(Hs) "by" tactic3(tac) := clear -Hs; set_solver by tac. Tactic Notation "set_solver" := set_solver by idtac. Tactic Notation "set_solver" "-" hyp_list(Hs) := clear Hs; set_solver. Tactic Notation "set_solver" "+" hyp_list(Hs) := clear -Hs; set_solver. `````` Robbert Krebbers committed Mar 03, 2016 313 314 315 316 ``````Hint Extern 1000 (_ ∉ _) => set_solver : set_solver. Hint Extern 1000 (_ ∈ _) => set_solver : set_solver. Hint Extern 1000 (_ ⊆ _) => set_solver : set_solver. `````` Robbert Krebbers committed Feb 24, 2016 317 318 319 ``````(** * Conversion of option and list *) Definition of_option `{Singleton A C, Empty C} (mx : option A) : C := match mx with None => ∅ | Some x => {[ x ]} end. `````` Robbert Krebbers committed Nov 11, 2015 320 321 322 323 324 ``````Fixpoint of_list `{Singleton A C, Empty C, Union C} (l : list A) : C := match l with [] => ∅ | x :: l => {[ x ]} ∪ of_list l end. Section of_option_list. Context `{SimpleCollection A C}. `````` Robbert Krebbers committed Feb 24, 2016 325 326 `````` Lemma elem_of_of_option (x : A) mx: x ∈ of_option mx ↔ mx = Some x. Proof. destruct mx; set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 327 328 329 `````` Lemma elem_of_of_list (x : A) l : x ∈ of_list l ↔ x ∈ l. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 330 `````` - induction l; simpl; [by rewrite elem_of_empty|]. `````` Robbert Krebbers committed Nov 11, 2015 331 `````` rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 332 `````` - induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto. `````` Robbert Krebbers committed Nov 11, 2015 333 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 334 335 336 `````` Global Instance set_unfold_of_option (mx : option A) x : SetUnfold (x ∈ of_option mx) (mx = Some x). Proof. constructor; apply elem_of_of_option. Qed. `````` Robbert Krebbers committed Mar 04, 2016 337 338 339 `````` Global Instance set_unfold_of_list (l : list A) x P : SetUnfold (x ∈ l) P → SetUnfold (x ∈ of_list l) P. Proof. constructor. by rewrite elem_of_of_list, (set_unfold (x ∈ l) P). Qed. `````` Robbert Krebbers committed Nov 11, 2015 340 341 ``````End of_option_list. `````` Robbert Krebbers committed Mar 04, 2016 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 ``````Section list_unfold. Context {A : Type}. Implicit Types x : A. Implicit Types l : list A. Global Instance set_unfold_nil x : SetUnfold (x ∈ []) False. Proof. constructor; apply elem_of_nil. Qed. Global Instance set_unfold_cons x y l P : SetUnfold (x ∈ l) P → SetUnfold (x ∈ y :: l) (x = y ∨ P). Proof. constructor. by rewrite elem_of_cons, (set_unfold (x ∈ l) P). Qed. Global Instance set_unfold_app x l k P Q : SetUnfold (x ∈ l) P → SetUnfold (x ∈ k) Q → SetUnfold (x ∈ l ++ k) (P ∨ Q). Proof. intros ??; constructor. by rewrite elem_of_app, (set_unfold (x ∈ l) P), (set_unfold (x ∈ k) Q). Qed. `````` Robbert Krebbers committed Mar 04, 2016 358 359 360 361 `````` Global Instance set_unfold_included l k (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ l) (P x)) → (∀ x, SetUnfold (x ∈ k) (Q x)) → SetUnfold (l `included` k) (∀ x, P x → Q x). Proof. by constructor; unfold included; set_unfold. Qed. `````` Robbert Krebbers committed Mar 04, 2016 362 363 ``````End list_unfold. `````` Robbert Krebbers committed Feb 24, 2016 364 ``````(** * Guard *) `````` Robbert Krebbers committed Nov 11, 2015 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 ``````Global Instance collection_guard `{CollectionMonad M} : MGuard M := λ P dec A x, match dec with left H => x H | _ => ∅ end. Section collection_monad_base. Context `{CollectionMonad M}. Lemma elem_of_guard `{Decision P} {A} (x : A) (X : M A) : x ∈ guard P; X ↔ P ∧ x ∈ X. Proof. unfold mguard, collection_guard; simpl; case_match; rewrite ?elem_of_empty; naive_solver. Qed. Lemma elem_of_guard_2 `{Decision P} {A} (x : A) (X : M A) : P → x ∈ X → x ∈ guard P; X. Proof. by rewrite elem_of_guard. Qed. Lemma guard_empty `{Decision P} {A} (X : M A) : guard P; X ≡ ∅ ↔ ¬P ∨ X ≡ ∅. Proof. rewrite !elem_of_equiv_empty; setoid_rewrite elem_of_guard. destruct (decide P); naive_solver. Qed. `````` Robbert Krebbers committed Feb 24, 2016 384 385 386 `````` Global Instance set_unfold_guard `{Decision P} {A} (x : A) X Q : SetUnfold (x ∈ X) Q → SetUnfold (x ∈ guard P; X) (P ∧ Q). Proof. constructor. by rewrite elem_of_guard, (set_unfold (x ∈ X) Q). Qed. `````` Robbert Krebbers committed Nov 11, 2015 387 388 `````` Lemma bind_empty {A B} (f : A → M B) X : X ≫= f ≡ ∅ ↔ X ≡ ∅ ∨ ∀ x, x ∈ X → f x ≡ ∅. `````` Robbert Krebbers committed Feb 24, 2016 389 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 390 391 392 393 394 ``````End collection_monad_base. (** * More theorems *) Section collection. Context `{Collection A C}. `````` Robbert Krebbers committed Nov 18, 2015 395 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Nov 11, 2015 396 397 `````` Global Instance: Lattice C. `````` Robbert Krebbers committed Feb 17, 2016 398 `````` Proof. split. apply _. firstorder auto. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 399 400 `````` Global Instance difference_proper : Proper ((≡) ==> (≡) ==> (≡)) (@difference C _). `````` Robbert Krebbers committed Nov 11, 2015 401 402 403 404 `````` Proof. intros X1 X2 HX Y1 Y2 HY; apply elem_of_equiv; intros x. by rewrite !elem_of_difference, HX, HY. Qed. `````` Ralf Jung committed Feb 16, 2016 405 `````` Lemma non_empty_inhabited x X : x ∈ X → X ≢ ∅. `````` Robbert Krebbers committed Feb 17, 2016 406 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 407 `````` Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}. `````` Robbert Krebbers committed Feb 17, 2016 408 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 409 `````` Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y. `````` Robbert Krebbers committed Feb 17, 2016 410 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 411 `````` Lemma subseteq_empty_difference X Y : X ⊆ Y → X ∖ Y ≡ ∅. `````` Robbert Krebbers committed Feb 17, 2016 412 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 413 `````` Lemma difference_diag X : X ∖ X ≡ ∅. `````` Robbert Krebbers committed Feb 17, 2016 414 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 415 `````` Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z. `````` Robbert Krebbers committed Feb 17, 2016 416 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 417 `````` Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y). `````` Robbert Krebbers committed Feb 17, 2016 418 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 419 `````` Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z. `````` Robbert Krebbers committed Feb 17, 2016 420 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Mar 23, 2016 421 `````` Lemma disjoint_union_difference X Y : X ⊥ Y → (X ∪ Y) ∖ X ≡ Y. `````` Robbert Krebbers committed Feb 17, 2016 422 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 `````` Section leibniz. Context `{!LeibnizEquiv C}. Lemma intersection_singletons_L x : {[x]} ∩ {[x]} = {[x]}. Proof. unfold_leibniz. apply intersection_singletons. Qed. Lemma difference_twice_L X Y : (X ∖ Y) ∖ Y = X ∖ Y. Proof. unfold_leibniz. apply difference_twice. Qed. Lemma subseteq_empty_difference_L X Y : X ⊆ Y → X ∖ Y = ∅. Proof. unfold_leibniz. apply subseteq_empty_difference. Qed. Lemma difference_diag_L X : X ∖ X = ∅. Proof. unfold_leibniz. apply difference_diag. Qed. Lemma difference_union_distr_l_L X Y Z : (X ∪ Y) ∖ Z = X ∖ Z ∪ Y ∖ Z. Proof. unfold_leibniz. apply difference_union_distr_l. Qed. Lemma difference_union_distr_r_L X Y Z : Z ∖ (X ∪ Y) = (Z ∖ X) ∩ (Z ∖ Y). Proof. unfold_leibniz. apply difference_union_distr_r. Qed. Lemma difference_intersection_distr_l_L X Y Z : (X ∩ Y) ∖ Z = X ∖ Z ∩ Y ∖ Z. Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed. `````` Robbert Krebbers committed Mar 23, 2016 441 `````` Lemma disjoint_union_difference_L X Y : X ⊥ Y → (X ∪ Y) ∖ X = Y. `````` Robbert Krebbers committed Nov 20, 2015 442 `````` Proof. unfold_leibniz. apply disjoint_union_difference. Qed. `````` Robbert Krebbers committed Nov 11, 2015 443 444 445 `````` End leibniz. Section dec. `````` Robbert Krebbers committed Nov 17, 2015 446 `````` Context `{∀ (x : A) (X : C), Decision (x ∈ X)}. `````` Robbert Krebbers committed Nov 11, 2015 447 448 449 450 451 452 453 454 455 456 `````` Lemma not_elem_of_intersection x X Y : x ∉ X ∩ Y ↔ x ∉ X ∨ x ∉ Y. Proof. rewrite elem_of_intersection. destruct (decide (x ∈ X)); tauto. Qed. Lemma not_elem_of_difference x X Y : x ∉ X ∖ Y ↔ x ∉ X ∨ x ∈ Y. Proof. rewrite elem_of_difference. destruct (decide (x ∈ Y)); tauto. Qed. Lemma union_difference X Y : X ⊆ Y → Y ≡ X ∪ Y ∖ X. Proof. split; intros x; rewrite !elem_of_union, elem_of_difference; [|intuition]. destruct (decide (x ∈ X)); intuition. Qed. Lemma non_empty_difference X Y : X ⊂ Y → Y ∖ X ≢ ∅. `````` Robbert Krebbers committed Feb 24, 2016 457 `````` Proof. intros [HXY1 HXY2] Hdiff. destruct HXY2. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 458 `````` Lemma empty_difference_subseteq X Y : X ∖ Y ≡ ∅ → X ⊆ Y. `````` Robbert Krebbers committed Feb 24, 2016 459 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 `````` Context `{!LeibnizEquiv C}. Lemma union_difference_L X Y : X ⊆ Y → Y = X ∪ Y ∖ X. Proof. unfold_leibniz. apply union_difference. Qed. Lemma non_empty_difference_L X Y : X ⊂ Y → Y ∖ X ≠ ∅. Proof. unfold_leibniz. apply non_empty_difference. Qed. Lemma empty_difference_subseteq_L X Y : X ∖ Y = ∅ → X ⊆ Y. Proof. unfold_leibniz. apply empty_difference_subseteq. Qed. End dec. End collection. Section collection_ops. Context `{CollectionOps A C}. Lemma elem_of_intersection_with_list (f : A → A → option A) Xs Y x : x ∈ intersection_with_list f Y Xs ↔ ∃ xs y, Forall2 (∈) xs Xs ∧ y ∈ Y ∧ foldr (λ x, (≫= f x)) (Some y) xs = Some x. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 478 `````` - revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|]. `````` Robbert Krebbers committed Nov 11, 2015 479 480 `````` rewrite elem_of_intersection_with in HXs; destruct HXs as (x1&x2&?&?&?). destruct (IHXs x2) as (xs & y & hy & ? & ?); trivial. `````` Robbert Krebbers committed Feb 17, 2016 481 `````` eexists (x1 :: xs), y. intuition (simplify_option_eq; auto). `````` Robbert Krebbers committed Feb 17, 2016 482 `````` - intros (xs & y & Hxs & ? & Hx). revert x Hx. `````` Robbert Krebbers committed Feb 17, 2016 483 `````` induction Hxs; intros; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Nov 11, 2015 484 485 486 487 488 489 490 491 492 `````` rewrite elem_of_intersection_with. naive_solver. Qed. Lemma intersection_with_list_ind (P Q : A → Prop) f Xs Y : (∀ y, y ∈ Y → P y) → Forall (λ X, ∀ x, x ∈ X → Q x) Xs → (∀ x y z, Q x → P y → f x y = Some z → P z) → ∀ x, x ∈ intersection_with_list f Y Xs → P x. Proof. `````` Robbert Krebbers committed Feb 17, 2016 493 `````` intros HY HXs Hf. induction Xs; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Nov 11, 2015 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 `````` intros x Hx. rewrite elem_of_intersection_with in Hx. decompose_Forall. destruct Hx as (? & ? & ? & ? & ?). eauto. Qed. End collection_ops. (** * Sets without duplicates up to an equivalence *) Section NoDup. Context `{SimpleCollection A B} (R : relation A) `{!Equivalence R}. Definition elem_of_upto (x : A) (X : B) := ∃ y, y ∈ X ∧ R x y. Definition set_NoDup (X : B) := ∀ x y, x ∈ X → y ∈ X → R x y → x = y. Global Instance: Proper ((≡) ==> iff) (elem_of_upto x). Proof. intros ??? E. unfold elem_of_upto. by setoid_rewrite E. Qed. Global Instance: Proper (R ==> (≡) ==> iff) elem_of_upto. Proof. intros ?? E1 ?? E2. split; intros [z [??]]; exists z. `````` Robbert Krebbers committed Feb 17, 2016 511 512 `````` - rewrite <-E1, <-E2; intuition. - rewrite E1, E2; intuition. `````` Robbert Krebbers committed Nov 11, 2015 513 514 515 516 517 `````` Qed. Global Instance: Proper ((≡) ==> iff) set_NoDup. Proof. firstorder. Qed. Lemma elem_of_upto_elem_of x X : x ∈ X → elem_of_upto x X. `````` Robbert Krebbers committed Feb 17, 2016 518 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 519 `````` Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅. `````` Robbert Krebbers committed Feb 17, 2016 520 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 521 `````` Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y. `````` Robbert Krebbers committed Feb 17, 2016 522 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 523 524 525 `````` Lemma elem_of_upto_union X Y x : elem_of_upto x (X ∪ Y) ↔ elem_of_upto x X ∨ elem_of_upto x Y. `````` Robbert Krebbers committed Feb 17, 2016 526 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 527 `````` Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y. `````` Robbert Krebbers committed Feb 17, 2016 528 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 529 530 `````` Lemma set_NoDup_empty: set_NoDup ∅. `````` Robbert Krebbers committed Feb 17, 2016 531 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 532 533 `````` Lemma set_NoDup_add x X : ¬elem_of_upto x X → set_NoDup X → set_NoDup ({[ x ]} ∪ X). `````` Robbert Krebbers committed Feb 17, 2016 534 `````` Proof. unfold set_NoDup, elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 535 536 537 538 `````` Lemma set_NoDup_inv_add x X : x ∉ X → set_NoDup ({[ x ]} ∪ X) → ¬elem_of_upto x X. Proof. intros Hin Hnodup [y [??]]. `````` Robbert Krebbers committed Feb 17, 2016 539 `````` rewrite (Hnodup x y) in Hin; set_solver. `````` Robbert Krebbers committed Nov 11, 2015 540 541 `````` Qed. Lemma set_NoDup_inv_union_l X Y : set_NoDup (X ∪ Y) → set_NoDup X. `````` Robbert Krebbers committed Feb 17, 2016 542 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 543 `````` Lemma set_NoDup_inv_union_r X Y : set_NoDup (X ∪ Y) → set_NoDup Y. `````` Robbert Krebbers committed Feb 17, 2016 544 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 545 546 547 548 549 550 551 552 553 554 ``````End NoDup. (** * Quantifiers *) Section quantifiers. Context `{SimpleCollection A B} (P : A → Prop). Definition set_Forall X := ∀ x, x ∈ X → P x. Definition set_Exists X := ∃ x, x ∈ X ∧ P x. Lemma set_Forall_empty : set_Forall ∅. `````` Robbert Krebbers committed Feb 17, 2016 555 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 556 `````` Lemma set_Forall_singleton x : set_Forall {[ x ]} ↔ P x. `````` Robbert Krebbers committed Feb 17, 2016 557 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 558 `````` Lemma set_Forall_union X Y : set_Forall X → set_Forall Y → set_Forall (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 559 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 560 `````` Lemma set_Forall_union_inv_1 X Y : set_Forall (X ∪ Y) → set_Forall X. `````` Robbert Krebbers committed Feb 17, 2016 561 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 562 `````` Lemma set_Forall_union_inv_2 X Y : set_Forall (X ∪ Y) → set_Forall Y. `````` Robbert Krebbers committed Feb 17, 2016 563 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 564 565 `````` Lemma set_Exists_empty : ¬set_Exists ∅. `````` Robbert Krebbers committed Feb 17, 2016 566 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 567 `````` Lemma set_Exists_singleton x : set_Exists {[ x ]} ↔ P x. `````` Robbert Krebbers committed Feb 17, 2016 568 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 569 `````` Lemma set_Exists_union_1 X Y : set_Exists X → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 570 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 571 `````` Lemma set_Exists_union_2 X Y : set_Exists Y → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 572 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 573 574 `````` Lemma set_Exists_union_inv X Y : set_Exists (X ∪ Y) → set_Exists X ∨ set_Exists Y. `````` Robbert Krebbers committed Feb 17, 2016 575 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 ``````End quantifiers. Section more_quantifiers. Context `{SimpleCollection A B}. Lemma set_Forall_weaken (P Q : A → Prop) (Hweaken : ∀ x, P x → Q x) X : set_Forall P X → set_Forall Q X. Proof. unfold set_Forall. naive_solver. Qed. Lemma set_Exists_weaken (P Q : A → Prop) (Hweaken : ∀ x, P x → Q x) X : set_Exists P X → set_Exists Q X. Proof. unfold set_Exists. naive_solver. Qed. End more_quantifiers. (** * Fresh elements *) (** We collect some properties on the [fresh] operation. In particular we generalize [fresh] to generate lists of fresh elements. *) Fixpoint fresh_list `{Fresh A C, Union C, Singleton A C} (n : nat) (X : C) : list A := match n with | 0 => [] | S n => let x := fresh X in x :: fresh_list n ({[ x ]} ∪ X) end. Inductive Forall_fresh `{ElemOf A C} (X : C) : list A → Prop := | Forall_fresh_nil : Forall_fresh X [] | Forall_fresh_cons x xs : x ∉ xs → x ∉ X → Forall_fresh X xs → Forall_fresh X (x :: xs). Section fresh. Context `{FreshSpec A C}. `````` Robbert Krebbers committed Nov 18, 2015 605 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Nov 11, 2015 606 `````` `````` Robbert Krebbers committed Nov 18, 2015 607 `````` Global Instance fresh_proper: Proper ((≡) ==> (=)) (fresh (C:=C)). `````` Robbert Krebbers committed Nov 11, 2015 608 `````` Proof. intros ???. by apply fresh_proper_alt, elem_of_equiv. Qed. `````` Robbert Krebbers committed Nov 18, 2015 609 610 `````` Global Instance fresh_list_proper: Proper ((=) ==> (≡) ==> (=)) (fresh_list (C:=C)). `````` Robbert Krebbers committed Nov 11, 2015 611 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 612 `````` intros ? n ->. induction n as [|n IH]; intros ?? E; f_equal/=; [by rewrite E|]. `````` Robbert Krebbers committed Nov 11, 2015 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 `````` apply IH. by rewrite E. Qed. Lemma Forall_fresh_NoDup X xs : Forall_fresh X xs → NoDup xs. Proof. induction 1; by constructor. Qed. Lemma Forall_fresh_elem_of X xs x : Forall_fresh X xs → x ∈ xs → x ∉ X. Proof. intros HX; revert x; rewrite <-Forall_forall. by induction HX; constructor. Qed. Lemma Forall_fresh_alt X xs : Forall_fresh X xs ↔ NoDup xs ∧ ∀ x, x ∈ xs → x ∉ X. Proof. split; eauto using Forall_fresh_NoDup, Forall_fresh_elem_of. rewrite <-Forall_forall. intros [Hxs Hxs']. induction Hxs; decompose_Forall_hyps; constructor; auto. Qed. Lemma Forall_fresh_subseteq X Y xs : Forall_fresh X xs → Y ⊆ X → Forall_fresh Y xs. `````` Robbert Krebbers committed Feb 17, 2016 632 `````` Proof. rewrite !Forall_fresh_alt; set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 633 634 635 636 637 638 639 `````` Lemma fresh_list_length n X : length (fresh_list n X) = n. Proof. revert X. induction n; simpl; auto. Qed. Lemma fresh_list_is_fresh n X x : x ∈ fresh_list n X → x ∉ X. Proof. revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|]. rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|]. `````` Robbert Krebbers committed Feb 17, 2016 640 `````` apply IH in Hin; set_solver. `````` Robbert Krebbers committed Nov 11, 2015 641 642 643 644 `````` Qed. Lemma NoDup_fresh_list n X : NoDup (fresh_list n X). Proof. revert X. induction n; simpl; constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 645 `````` intros Hin; apply fresh_list_is_fresh in Hin; set_solver. `````` Robbert Krebbers committed Nov 11, 2015 646 647 648 649 650 651 652 653 654 655 656 `````` Qed. Lemma Forall_fresh_list X n : Forall_fresh X (fresh_list n X). Proof. rewrite Forall_fresh_alt; eauto using NoDup_fresh_list, fresh_list_is_fresh. Qed. End fresh. (** * Properties of implementations of collections that form a monad *) Section collection_monad. Context `{CollectionMonad M}. `````` Ralf Jung committed Feb 15, 2016 657 658 `````` Global Instance collection_fmap_mono {A B} : Proper (pointwise_relation _ (=) ==> (⊆) ==> (⊆)) (@fmap M _ A B). `````` Robbert Krebbers committed Feb 24, 2016 659 `````` Proof. intros f g ? X Y ?; set_solver by eauto. Qed. `````` Robbert Krebbers committed Nov 16, 2015 660 661 `````` Global Instance collection_fmap_proper {A B} : Proper (pointwise_relation _ (=) ==> (≡) ==> (≡)) (@fmap M _ A B). `````` Robbert Krebbers committed Feb 24, 2016 662 `````` Proof. intros f g ? X Y [??]; split; set_solver by eauto. Qed. `````` Ralf Jung committed Feb 15, 2016 663 664 `````` Global Instance collection_bind_mono {A B} : Proper (((=) ==> (⊆)) ==> (⊆) ==> (⊆)) (@mbind M _ A B). `````` Robbert Krebbers committed Feb 17, 2016 665 `````` Proof. unfold respectful; intros f g Hfg X Y ?; set_solver. Qed. `````` Robbert Krebbers committed Nov 16, 2015 666 667 `````` Global Instance collection_bind_proper {A B} : Proper (((=) ==> (≡)) ==> (≡) ==> (≡)) (@mbind M _ A B). `````` Robbert Krebbers committed Feb 17, 2016 668 `````` Proof. unfold respectful; intros f g Hfg X Y [??]; split; set_solver. Qed. `````` Ralf Jung committed Feb 15, 2016 669 670 `````` Global Instance collection_join_mono {A} : Proper ((⊆) ==> (⊆)) (@mjoin M _ A). `````` Robbert Krebbers committed Feb 17, 2016 671 `````` Proof. intros X Y ?; set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 672 673 `````` Global Instance collection_join_proper {A} : Proper ((≡) ==> (≡)) (@mjoin M _ A). `````` Robbert Krebbers committed Feb 17, 2016 674 `````` Proof. intros X Y [??]; split; set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 675 676 `````` Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x. `````` Robbert Krebbers committed Feb 17, 2016 677 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 678 `````` Lemma collection_guard_True {A} `{Decision P} (X : M A) : P → guard P; X ≡ X. `````` Robbert Krebbers committed Feb 17, 2016 679 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 680 `````` Lemma collection_fmap_compose {A B C} (f : A → B) (g : B → C) (X : M A) : `````` Robbert Krebbers committed Nov 11, 2015 681 `````` g ∘ f <\$> X ≡ g <\$> (f <\$> X). `````` Robbert Krebbers committed Feb 17, 2016 682 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 683 684 `````` Lemma elem_of_fmap_1 {A B} (f : A → B) (X : M A) (y : B) : y ∈ f <\$> X → ∃ x, y = f x ∧ x ∈ X. `````` Robbert Krebbers committed Feb 17, 2016 685 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 686 687 `````` Lemma elem_of_fmap_2 {A B} (f : A → B) (X : M A) (x : A) : x ∈ X → f x ∈ f <\$> X. `````` Robbert Krebbers committed Feb 17, 2016 688 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 689 690 `````` Lemma elem_of_fmap_2_alt {A B} (f : A → B) (X : M A) (x : A) (y : B) : x ∈ X → y = f x → y ∈ f <\$> X. `````` Robbert Krebbers committed Feb 17, 2016 691 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 692 693 694 695 696 `````` Lemma elem_of_mapM {A B} (f : A → M B) l k : l ∈ mapM f k ↔ Forall2 (λ x y, x ∈ f y) l k. Proof. split. `````` Robbert Krebbers committed Feb 24, 2016 697 `````` - revert l. induction k; set_solver by eauto. `````` Robbert Krebbers committed Feb 17, 2016 698 `````` - induction 1; set_solver. `````` Robbert Krebbers committed Nov 11, 2015 699 700 701 `````` Qed. Lemma collection_mapM_length {A B} (f : A → M B) l k : l ∈ mapM f k → length l = length k. `````` Robbert Krebbers committed Feb 24, 2016 702 `````` Proof. revert l; induction k; set_solver by eauto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 703 704 `````` Lemma elem_of_mapM_fmap {A B} (f : A → B) (g : B → M A) l k : Forall (λ x, ∀ y, y ∈ g x → f y = x) l → k ∈ mapM g l → fmap f k = l. `````` Robbert Krebbers committed Feb 24, 2016 705 `````` Proof. intros Hl. revert k. induction Hl; set_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 706 707 708 709 710 711 712 713 714 715 716 `````` Lemma elem_of_mapM_Forall {A B} (f : A → M B) (P : B → Prop) l k : l ∈ mapM f k → Forall (λ x, ∀ y, y ∈ f x → P y) k → Forall P l. Proof. rewrite elem_of_mapM. apply Forall2_Forall_l. Qed. Lemma elem_of_mapM_Forall2_l {A B C} (f : A → M B) (P: B → C → Prop) l1 l2 k : l1 ∈ mapM f k → Forall2 (λ x y, ∀ z, z ∈ f x → P z y) k l2 → Forall2 P l1 l2. Proof. rewrite elem_of_mapM. intros Hl1. revert l2. induction Hl1; inversion_clear 1; constructor; auto. Qed. End collection_monad. `````` Robbert Krebbers committed Dec 11, 2015 717 718 719 720 721 722 `````` (** Finite collections *) Definition set_finite `{ElemOf A B} (X : B) := ∃ l : list A, ∀ x, x ∈ X → x ∈ l. Section finite. Context `{SimpleCollection A B}. `````` Robbert Krebbers committed Jan 16, 2016 723 724 `````` Global Instance set_finite_subseteq : Proper (flip (⊆) ==> impl) (@set_finite A B _). `````` Robbert Krebbers committed Feb 17, 2016 725 `````` Proof. intros X Y HX [l Hl]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Jan 16, 2016 726 727 `````` Global Instance set_finite_proper : Proper ((≡) ==> iff) (@set_finite A B _). Proof. by intros X Y [??]; split; apply set_finite_subseteq. Qed. `````` Robbert Krebbers committed Dec 11, 2015 728 729 730 `````` Lemma empty_finite : set_finite ∅. Proof. by exists []; intros ?; rewrite elem_of_empty. Qed. Lemma singleton_finite (x : A) : set_finite {[ x ]}. `````` Ralf Jung committed Jan 04, 2016 731 `````` Proof. exists [x]; intros y ->%elem_of_singleton; left. Qed. `````` Robbert Krebbers committed Dec 11, 2015 732 733 734 735 736 737 `````` Lemma union_finite X Y : set_finite X → set_finite Y → set_finite (X ∪ Y). Proof. intros [lX ?] [lY ?]; exists (lX ++ lY); intros x. rewrite elem_of_union, elem_of_app; naive_solver. Qed. Lemma union_finite_inv_l X Y : set_finite (X ∪ Y) → set_finite X. `````` Robbert Krebbers committed Feb 17, 2016 738 `````` Proof. intros [l ?]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Dec 11, 2015 739 `````` Lemma union_finite_inv_r X Y : set_finite (X ∪ Y) → set_finite Y. `````` Robbert Krebbers committed Feb 17, 2016 740 `````` Proof. intros [l ?]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Dec 11, 2015 741 742 743 744 745 ``````End finite. Section more_finite. Context `{Collection A B}. Lemma intersection_finite_l X Y : set_finite X → set_finite (X ∩ Y). `````` Ralf Jung committed Jan 04, 2016 746 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. `````` Robbert Krebbers committed Dec 11, 2015 747 `````` Lemma intersection_finite_r X Y : set_finite Y → set_finite (X ∩ Y). `````` Ralf Jung committed Jan 04, 2016 748 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. `````` Robbert Krebbers committed Dec 11, 2015 749 `````` Lemma difference_finite X Y : set_finite X → set_finite (X ∖ Y). `````` Ralf Jung committed Jan 04, 2016 750 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_difference; auto. Qed. `````` Robbert Krebbers committed Jan 16, 2016 751 752 753 754 `````` Lemma difference_finite_inv X Y `{∀ x, Decision (x ∈ Y)} : set_finite Y → set_finite (X ∖ Y) → set_finite X. Proof. intros [l ?] [k ?]; exists (l ++ k). `````` Robbert Krebbers committed Feb 17, 2016 755 `````` intros x ?; destruct (decide (x ∈ Y)); rewrite elem_of_app; set_solver. `````` Robbert Krebbers committed Jan 16, 2016 756 `````` Qed. `````` Robbert Krebbers committed Dec 11, 2015 757 ``End more_finite.``