collections.v 31.8 KB
 Robbert Krebbers committed Feb 08, 2015 1 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) `````` Robbert Krebbers committed Aug 29, 2012 2 3 4 5 ``````(* 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 Feb 13, 2016 6 ``````From stdpp Require Export base tactics orders. `````` Robbert Krebbers committed Aug 29, 2012 7 `````` `````` Robbert Krebbers committed May 02, 2014 8 9 ``````Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y, ∀ x, x ∈ X → x ∈ Y. `````` Robbert Krebbers committed Feb 24, 2016 10 ``````Typeclasses Opaque collection_subseteq. `````` Robbert Krebbers committed May 02, 2014 11 `````` `````` Robbert Krebbers committed Feb 19, 2013 12 ``````(** * Basic theorems *) `````` Robbert Krebbers committed Nov 12, 2012 13 14 ``````Section simple_collection. Context `{SimpleCollection A C}. `````` Robbert Krebbers committed Nov 18, 2015 15 16 `````` Implicit Types x y : A. Implicit Types X Y : C. `````` Robbert Krebbers committed Jun 11, 2012 17 `````` `````` Robbert Krebbers committed Aug 29, 2012 18 `````` Lemma elem_of_empty x : x ∈ ∅ ↔ False. `````` Robbert Krebbers committed Oct 19, 2012 19 `````` Proof. split. apply not_elem_of_empty. done. Qed. `````` Robbert Krebbers committed Jun 11, 2012 20 21 22 23 `````` 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. `````` Robbert Krebbers committed Nov 15, 2014 24 25 26 `````` Global Instance: EmptySpec C. Proof. firstorder auto. Qed. Global Instance: JoinSemiLattice C. `````` Robbert Krebbers committed Oct 19, 2012 27 `````` Proof. firstorder auto. Qed. `````` Robbert Krebbers committed Feb 15, 2016 28 29 `````` Global Instance: AntiSymm (≡) (@collection_subseteq A C _). Proof. done. Qed. `````` Robbert Krebbers committed Jun 11, 2012 30 `````` Lemma elem_of_subseteq X Y : X ⊆ Y ↔ ∀ x, x ∈ X → x ∈ Y. `````` Robbert Krebbers committed Oct 19, 2012 31 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jun 11, 2012 32 33 `````` Lemma elem_of_equiv X Y : X ≡ Y ↔ ∀ x, x ∈ X ↔ x ∈ Y. Proof. firstorder. Qed. `````` Robbert Krebbers committed Aug 21, 2012 34 35 `````` Lemma elem_of_equiv_alt X Y : X ≡ Y ↔ (∀ x, x ∈ X → x ∈ Y) ∧ (∀ x, x ∈ Y → x ∈ X). `````` Robbert Krebbers committed Jun 11, 2012 36 `````` Proof. firstorder. Qed. `````` Robbert Krebbers committed Feb 19, 2013 37 38 `````` Lemma elem_of_equiv_empty X : X ≡ ∅ ↔ ∀ x, x ∉ X. Proof. firstorder. Qed. `````` Robbert Krebbers committed Jun 05, 2014 39 40 41 42 43 44 `````` 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. `````` Robbert Krebbers committed Aug 26, 2014 45 46 47 48 `````` 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. `````` Robbert Krebbers committed Oct 19, 2012 49 50 51 `````` Lemma elem_of_subseteq_singleton x X : x ∈ X ↔ {[ x ]} ⊆ X. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 52 53 `````` - intros ??. rewrite elem_of_singleton. by intros ->. - intros Ex. by apply (Ex x), elem_of_singleton. `````` Robbert Krebbers committed Oct 19, 2012 54 `````` Qed. `````` Robbert Krebbers committed Nov 18, 2015 55 `````` Global Instance singleton_proper : Proper ((=) ==> (≡)) (singleton (B:=C)). `````` Robbert Krebbers committed May 02, 2014 56 `````` Proof. by repeat intro; subst. Qed. `````` Robbert Krebbers committed Nov 18, 2015 57 58 `````` Global Instance elem_of_proper : Proper ((=) ==> (≡) ==> iff) ((∈) : A → C → Prop) | 5. `````` Robbert Krebbers committed May 02, 2014 59 `````` Proof. intros ???; subst. firstorder. Qed. `````` Robbert Krebbers committed May 07, 2013 60 `````` Lemma elem_of_union_list Xs x : x ∈ ⋃ Xs ↔ ∃ X, X ∈ Xs ∧ x ∈ X. `````` Robbert Krebbers committed Oct 19, 2012 61 62 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 63 `````` - induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|]. `````` Robbert Krebbers committed May 02, 2014 64 `````` setoid_rewrite elem_of_cons. apply elem_of_union in HXs. naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 65 `````` - intros [X []]. induction 1; simpl; [by apply elem_of_union_l |]. `````` Robbert Krebbers committed May 02, 2014 66 `````` intros. apply elem_of_union_r; auto. `````` Robbert Krebbers committed Oct 19, 2012 67 `````` Qed. `````` Robbert Krebbers committed Nov 18, 2015 68 `````` Lemma non_empty_singleton x : ({[ x ]} : C) ≢ ∅. `````` Robbert Krebbers committed Oct 19, 2012 69 70 71 72 73 74 `````` 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. `````` Robbert Krebbers committed Feb 19, 2013 75 76 77 78 79 80 81 82 83 `````` 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. `````` Robbert Krebbers committed Jun 05, 2014 84 85 86 87 `````` 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. `````` Robbert Krebbers committed Feb 19, 2013 88 89 90 91 92 93 94 95 96 97 98 99 `````` 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. `````` Robbert Krebbers committed Nov 12, 2012 100 101 ``````End simple_collection. `````` Robbert Krebbers committed Feb 24, 2016 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 ``````(** * 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. 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 268 `````` try fast_done; `````` Robbert Krebbers committed Feb 24, 2016 269 270 271 272 273 274 275 276 277 278 279 280 281 `````` 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 282 283 284 285 ``````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 286 287 288 ``````(** * 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 Jun 02, 2015 289 290 ``````Fixpoint of_list `{Singleton A C, Empty C, Union C} (l : list A) : C := match l with [] => ∅ | x :: l => {[ x ]} ∪ of_list l end. `````` Robbert Krebbers committed Jan 27, 2015 291 `````` `````` Robbert Krebbers committed Jun 02, 2015 292 293 ``````Section of_option_list. Context `{SimpleCollection A C}. `````` Robbert Krebbers committed Feb 24, 2016 294 295 `````` Lemma elem_of_of_option (x : A) mx: x ∈ of_option mx ↔ mx = Some x. Proof. destruct mx; set_solver. Qed. `````` Robbert Krebbers committed Jun 02, 2015 296 297 298 `````` Lemma elem_of_of_list (x : A) l : x ∈ of_list l ↔ x ∈ l. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 299 `````` - induction l; simpl; [by rewrite elem_of_empty|]. `````` Robbert Krebbers committed Feb 01, 2017 300 `````` rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 301 `````` - induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto. `````` Robbert Krebbers committed Jun 02, 2015 302 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 303 304 305 `````` 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 306 307 308 `````` 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 Jun 02, 2015 309 ``````End of_option_list. `````` Robbert Krebbers committed Jun 05, 2014 310 `````` `````` Robbert Krebbers committed Mar 04, 2016 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 ``````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 327 328 329 330 `````` 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 331 332 ``````End list_unfold. `````` Robbert Krebbers committed Feb 24, 2016 333 ``````(** * Guard *) `````` Robbert Krebbers committed Jun 05, 2014 334 335 ``````Global Instance collection_guard `{CollectionMonad M} : MGuard M := λ P dec A x, match dec with left H => x H | _ => ∅ end. `````` Robbert Krebbers committed Jan 27, 2015 336 337 338 339 340 341 342 343 344 `````` 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. `````` Robbert Krebbers committed Apr 16, 2015 345 346 347 `````` 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. `````` Robbert Krebbers committed Jan 27, 2015 348 349 350 351 352 `````` 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 353 354 355 `````` 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 Jan 27, 2015 356 357 `````` Lemma bind_empty {A B} (f : A → M B) X : X ≫= f ≡ ∅ ↔ X ≡ ∅ ∨ ∀ x, x ∈ X → f x ≡ ∅. `````` Robbert Krebbers committed Feb 24, 2016 358 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 27, 2015 359 ``````End collection_monad_base. `````` Robbert Krebbers committed Jun 05, 2014 360 `````` `````` Robbert Krebbers committed Feb 19, 2013 361 ``````(** * More theorems *) `````` Robbert Krebbers committed Jan 05, 2013 362 363 ``````Section collection. Context `{Collection A C}. `````` Robbert Krebbers committed Nov 18, 2015 364 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Jan 05, 2013 365 `````` `````` Robbert Krebbers committed Nov 15, 2014 366 `````` Global Instance: Lattice C. `````` Robbert Krebbers committed Feb 17, 2016 367 `````` Proof. split. apply _. firstorder auto. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 368 369 `````` Global Instance difference_proper : Proper ((≡) ==> (≡) ==> (≡)) (@difference C _). `````` Robbert Krebbers committed Feb 03, 2017 370 371 372 373 `````` 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 374 `````` Lemma non_empty_inhabited x X : x ∈ X → X ≢ ∅. `````` Robbert Krebbers committed Feb 17, 2016 375 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 376 `````` Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}. `````` Robbert Krebbers committed Feb 17, 2016 377 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 378 `````` Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y. `````` Robbert Krebbers committed Feb 17, 2016 379 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 08, 2015 380 `````` Lemma subseteq_empty_difference X Y : X ⊆ Y → X ∖ Y ≡ ∅. `````` Robbert Krebbers committed Feb 17, 2016 381 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 382 `````` Lemma difference_diag X : X ∖ X ≡ ∅. `````` Robbert Krebbers committed Feb 17, 2016 383 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 384 `````` Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z. `````` Robbert Krebbers committed Feb 17, 2016 385 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 03, 2017 386 `````` Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y). `````` Robbert Krebbers committed Feb 17, 2016 387 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 388 `````` Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z. `````` Robbert Krebbers committed Feb 17, 2016 389 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 20, 2015 390 `````` Lemma disjoint_union_difference X Y : X ∩ Y ≡ ∅ → (X ∪ Y) ∖ X ≡ Y. `````` Robbert Krebbers committed Feb 17, 2016 391 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 392 `````` `````` Robbert Krebbers committed Feb 19, 2013 393 394 395 396 397 398 `````` 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. `````` Robbert Krebbers committed Feb 08, 2015 399 400 `````` Lemma subseteq_empty_difference_L X Y : X ⊆ Y → X ∖ Y = ∅. Proof. unfold_leibniz. apply subseteq_empty_difference. Qed. `````` Robbert Krebbers committed Feb 19, 2013 401 402 403 404 `````` 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. `````` Robbert Krebbers committed Feb 03, 2017 405 406 `````` 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. `````` Robbert Krebbers committed Feb 19, 2013 407 408 409 `````` 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 Nov 20, 2015 410 411 `````` Lemma disjoint_union_difference_L X Y : X ∩ Y = ∅ → (X ∪ Y) ∖ X = Y. Proof. unfold_leibniz. apply disjoint_union_difference. Qed. `````` Robbert Krebbers committed Feb 19, 2013 412 413 414 `````` End leibniz. Section dec. `````` Robbert Krebbers committed Nov 17, 2015 415 `````` Context `{∀ (x : A) (X : C), Decision (x ∈ X)}. `````` Robbert Krebbers committed Feb 19, 2013 416 `````` Lemma not_elem_of_intersection x X Y : x ∉ X ∩ Y ↔ x ∉ X ∨ x ∉ Y. `````` Robbert Krebbers committed May 02, 2014 417 `````` Proof. rewrite elem_of_intersection. destruct (decide (x ∈ X)); tauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 418 `````` Lemma not_elem_of_difference x X Y : x ∉ X ∖ Y ↔ x ∉ X ∨ x ∈ Y. `````` Robbert Krebbers committed May 02, 2014 419 `````` Proof. rewrite elem_of_difference. destruct (decide (x ∈ Y)); tauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 420 421 `````` Lemma union_difference X Y : X ⊆ Y → Y ≡ X ∪ Y ∖ X. Proof. `````` Robbert Krebbers committed May 02, 2014 422 423 `````` split; intros x; rewrite !elem_of_union, elem_of_difference; [|intuition]. destruct (decide (x ∈ X)); intuition. `````` Robbert Krebbers committed Feb 19, 2013 424 425 `````` Qed. Lemma non_empty_difference X Y : X ⊂ Y → Y ∖ X ≢ ∅. `````` Robbert Krebbers committed Feb 24, 2016 426 `````` Proof. intros [HXY1 HXY2] Hdiff. destruct HXY2. set_solver. Qed. `````` Robbert Krebbers committed Feb 08, 2015 427 `````` Lemma empty_difference_subseteq X Y : X ∖ Y ≡ ∅ → X ⊆ Y. `````` Robbert Krebbers committed Feb 24, 2016 428 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 19, 2013 429 430 431 432 433 `````` 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. `````` Robbert Krebbers committed Feb 08, 2015 434 435 `````` Lemma empty_difference_subseteq_L X Y : X ∖ Y = ∅ → X ⊆ Y. Proof. unfold_leibniz. apply empty_difference_subseteq. Qed. `````` Robbert Krebbers committed Feb 19, 2013 436 437 438 439 440 441 `````` End dec. End collection. Section collection_ops. Context `{CollectionOps A C}. `````` Robbert Krebbers committed Jan 05, 2013 442 443 444 445 446 `````` 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 447 `````` - revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|]. `````` Robbert Krebbers committed May 02, 2014 448 449 `````` 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 450 `````` eexists (x1 :: xs), y. intuition (simplify_option_eq; auto). `````` Robbert Krebbers committed Feb 17, 2016 451 `````` - intros (xs & y & Hxs & ? & Hx). revert x Hx. `````` Robbert Krebbers committed Feb 17, 2016 452 `````` induction Hxs; intros; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Jan 05, 2013 453 454 455 456 457 458 459 460 461 `````` 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 462 `````` intros HY HXs Hf. induction Xs; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Jan 05, 2013 463 464 465 `````` intros x Hx. rewrite elem_of_intersection_with in Hx. decompose_Forall. destruct Hx as (? & ? & ? & ? & ?). eauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 466 ``````End collection_ops. `````` Robbert Krebbers committed Jan 05, 2013 467 `````` `````` Robbert Krebbers committed Aug 29, 2012 468 ``````(** * Sets without duplicates up to an equivalence *) `````` Robbert Krebbers committed May 07, 2013 469 ``````Section NoDup. `````` Robbert Krebbers committed Nov 12, 2012 470 `````` Context `{SimpleCollection A B} (R : relation A) `{!Equivalence R}. `````` Robbert Krebbers committed Jun 11, 2012 471 472 `````` Definition elem_of_upto (x : A) (X : B) := ∃ y, y ∈ X ∧ R x y. `````` Robbert Krebbers committed May 07, 2013 473 `````` Definition set_NoDup (X : B) := ∀ x y, x ∈ X → y ∈ X → R x y → x = y. `````` Robbert Krebbers committed Jun 11, 2012 474 475 `````` Global Instance: Proper ((≡) ==> iff) (elem_of_upto x). `````` Robbert Krebbers committed Jan 05, 2013 476 `````` Proof. intros ??? E. unfold elem_of_upto. by setoid_rewrite E. Qed. `````` Robbert Krebbers committed Jun 11, 2012 477 478 479 `````` Global Instance: Proper (R ==> (≡) ==> iff) elem_of_upto. Proof. intros ?? E1 ?? E2. split; intros [z [??]]; exists z. `````` Robbert Krebbers committed Feb 17, 2016 480 481 `````` - rewrite <-E1, <-E2; intuition. - rewrite E1, E2; intuition. `````` Robbert Krebbers committed Jun 11, 2012 482 `````` Qed. `````` Robbert Krebbers committed May 07, 2013 483 `````` Global Instance: Proper ((≡) ==> iff) set_NoDup. `````` Robbert Krebbers committed Jun 11, 2012 484 485 486 `````` Proof. firstorder. Qed. Lemma elem_of_upto_elem_of x X : x ∈ X → elem_of_upto x X. `````` Robbert Krebbers committed Feb 17, 2016 487 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 488 `````` Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅. `````` Robbert Krebbers committed Feb 17, 2016 489 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Aug 21, 2012 490 `````` Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y. `````` Robbert Krebbers committed Feb 17, 2016 491 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 492 `````` `````` Robbert Krebbers committed Aug 21, 2012 493 494 `````` 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 495 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 496 `````` Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y. `````` Robbert Krebbers committed Feb 17, 2016 497 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 498 `````` `````` Robbert Krebbers committed May 07, 2013 499 `````` Lemma set_NoDup_empty: set_NoDup ∅. `````` Robbert Krebbers committed Feb 17, 2016 500 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 501 502 `````` Lemma set_NoDup_add x X : ¬elem_of_upto x X → set_NoDup X → set_NoDup ({[ x ]} ∪ X). `````` Robbert Krebbers committed Feb 17, 2016 503 `````` Proof. unfold set_NoDup, elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 504 505 `````` Lemma set_NoDup_inv_add x X : x ∉ X → set_NoDup ({[ x ]} ∪ X) → ¬elem_of_upto x X. `````` Robbert Krebbers committed Aug 21, 2012 506 507 `````` Proof. intros Hin Hnodup [y [??]]. `````` Robbert Krebbers committed Feb 17, 2016 508 `````` rewrite (Hnodup x y) in Hin; set_solver. `````` Robbert Krebbers committed Aug 21, 2012 509 `````` Qed. `````` Robbert Krebbers committed May 07, 2013 510 `````` Lemma set_NoDup_inv_union_l X Y : set_NoDup (X ∪ Y) → set_NoDup X. `````` Robbert Krebbers committed Feb 17, 2016 511 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 512 `````` Lemma set_NoDup_inv_union_r X Y : set_NoDup (X ∪ Y) → set_NoDup Y. `````` Robbert Krebbers committed Feb 17, 2016 513 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 514 ``````End NoDup. `````` Robbert Krebbers committed Jun 11, 2012 515 `````` `````` Robbert Krebbers committed Aug 29, 2012 516 ``````(** * Quantifiers *) `````` Robbert Krebbers committed Jun 11, 2012 517 ``````Section quantifiers. `````` Robbert Krebbers committed Nov 12, 2012 518 `````` Context `{SimpleCollection A B} (P : A → Prop). `````` Robbert Krebbers committed Jun 11, 2012 519 `````` `````` Robbert Krebbers committed May 07, 2013 520 521 522 523 `````` 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 524 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 525 `````` Lemma set_Forall_singleton x : set_Forall {[ x ]} ↔ P x. `````` Robbert Krebbers committed Feb 17, 2016 526 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 527 `````` Lemma set_Forall_union X Y : set_Forall X → set_Forall Y → set_Forall (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 528 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 529 `````` Lemma set_Forall_union_inv_1 X Y : set_Forall (X ∪ Y) → set_Forall X. `````` Robbert Krebbers committed Feb 17, 2016 530 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 531 `````` Lemma set_Forall_union_inv_2 X Y : set_Forall (X ∪ Y) → set_Forall Y. `````` Robbert Krebbers committed Feb 17, 2016 532 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 533 534 `````` Lemma set_Exists_empty : ¬set_Exists ∅. `````` Robbert Krebbers committed Feb 17, 2016 535 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 536 `````` Lemma set_Exists_singleton x : set_Exists {[ x ]} ↔ P x. `````` Robbert Krebbers committed Feb 17, 2016 537 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 538 `````` Lemma set_Exists_union_1 X Y : set_Exists X → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 539 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 540 `````` Lemma set_Exists_union_2 X Y : set_Exists Y → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 541 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 542 543 `````` Lemma set_Exists_union_inv X Y : set_Exists (X ∪ Y) → set_Exists X ∨ set_Exists Y. `````` Robbert Krebbers committed Feb 17, 2016 544 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 545 546 ``````End quantifiers. `````` Robbert Krebbers committed Aug 21, 2012 547 ``````Section more_quantifiers. `````` Robbert Krebbers committed Jun 05, 2014 548 `````` Context `{SimpleCollection A B}. `````` Robbert Krebbers committed Aug 29, 2012 549 `````` `````` Robbert Krebbers committed May 07, 2013 550 551 552 553 554 555 `````` 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. `````` Robbert Krebbers committed Aug 21, 2012 556 557 ``````End more_quantifiers. `````` Robbert Krebbers committed Aug 29, 2012 558 559 560 ``````(** * Fresh elements *) (** We collect some properties on the [fresh] operation. In particular we generalize [fresh] to generate lists of fresh elements. *) `````` Robbert Krebbers committed Mar 02, 2015 561 562 563 564 565 566 567 568 569 570 ``````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). `````` Robbert Krebbers committed Aug 21, 2012 571 `````` `````` Robbert Krebbers committed Mar 02, 2015 572 573 ``````Section fresh. Context `{FreshSpec A C}. `````` Robbert Krebbers committed Nov 18, 2015 574 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Aug 21, 2012 575 `````` `````` Robbert Krebbers committed Nov 18, 2015 576 `````` Global Instance fresh_proper: Proper ((≡) ==> (=)) (fresh (C:=C)). `````` Robbert Krebbers committed Feb 08, 2015 577 `````` Proof. intros ???. by apply fresh_proper_alt, elem_of_equiv. Qed. `````` Robbert Krebbers committed Nov 18, 2015 578 579 `````` Global Instance fresh_list_proper: Proper ((=) ==> (≡) ==> (=)) (fresh_list (C:=C)). `````` Robbert Krebbers committed Aug 29, 2012 580 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 581 `````` intros ? n ->. induction n as [|n IH]; intros ?? E; f_equal/=; [by rewrite E|]. `````` Robbert Krebbers committed May 02, 2014 582 `````` apply IH. by rewrite E. `````` Robbert Krebbers committed Aug 29, 2012 583 `````` Qed. `````` Robbert Krebbers committed Mar 02, 2015 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 `````` 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. `````` Robbert Krebbers committed Jun 05, 2015 599 600 `````` Lemma Forall_fresh_subseteq X Y xs : Forall_fresh X xs → Y ⊆ X → Forall_fresh Y xs. `````` Robbert Krebbers committed Feb 17, 2016 601 `````` Proof. rewrite !Forall_fresh_alt; set_solver. Qed. `````` Robbert Krebbers committed Mar 02, 2015 602 `````` `````` Robbert Krebbers committed Aug 21, 2012 603 604 `````` Lemma fresh_list_length n X : length (fresh_list n X) = n. Proof. revert X. induction n; simpl; auto. Qed. `````` Robbert Krebbers committed Nov 12, 2012 605 `````` Lemma fresh_list_is_fresh n X x : x ∈ fresh_list n X → x ∉ X. `````` Robbert Krebbers committed Aug 21, 2012 606 `````` Proof. `````` Robbert Krebbers committed Mar 02, 2015 607 `````` revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|]. `````` Robbert Krebbers committed May 02, 2014 608 `````` rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|]. `````` Robbert Krebbers committed Feb 17, 2016 609 `````` apply IH in Hin; set_solver. `````` Robbert Krebbers committed Aug 21, 2012 610 `````` Qed. `````` Robbert Krebbers committed Mar 02, 2015 611 `````` Lemma NoDup_fresh_list n X : NoDup (fresh_list n X). `````` Robbert Krebbers committed Aug 21, 2012 612 `````` Proof. `````` Robbert Krebbers committed May 07, 2013 613 `````` revert X. induction n; simpl; constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 614 `````` intros Hin; apply fresh_list_is_fresh in Hin; set_solver. `````` Robbert Krebbers committed Mar 02, 2015 615 616 617 618 `````` 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. `````` Robbert Krebbers committed Aug 21, 2012 619 620 `````` Qed. End fresh. `````` Robbert Krebbers committed Nov 12, 2012 621 `````` `````` Robbert Krebbers committed Feb 19, 2013 622 ``````(** * Properties of implementations of collections that form a monad *) `````` Robbert Krebbers committed Nov 12, 2012 623 624 625 ``````Section collection_monad. Context `{CollectionMonad M}. `````` Ralf Jung committed Feb 15, 2016 626 627 `````` Global Instance collection_fmap_mono {A B} : Proper (pointwise_relation _ (=) ==> (⊆) ==> (⊆)) (@fmap M _ A B). `````` Robbert Krebbers committed Feb 24, 2016 628 `````` Proof. intros f g ? X Y ?; set_solver by eauto. Qed. `````` Robbert Krebbers committed Nov 16, 2015 629 630 `````` Global Instance collection_fmap_proper {A B} : Proper (pointwise_relation _ (=) ==> (≡) ==> (≡)) (@fmap M _ A B). `````` Robbert Krebbers committed Feb 24, 2016 631 `````` Proof. intros f g ? X Y [??]; split; set_solver by eauto. Qed. `````` Ralf Jung committed Feb 15, 2016 632 633 `````` Global Instance collection_bind_mono {A B} : Proper (((=) ==> (⊆)) ==> (⊆) ==> (⊆)) (@mbind M _ A B). `````` Robbert Krebbers committed Feb 17, 2016 634 `````` Proof. unfold respectful; intros f g Hfg X Y ?; set_solver. Qed. `````` Robbert Krebbers committed Nov 16, 2015 635 636 `````` Global Instance collection_bind_proper {A B} : Proper (((=) ==> (≡)) ==> (≡) ==> (≡)) (@mbind M _ A B). `````` Robbert Krebbers committed Feb 17, 2016 637 `````` Proof. unfold respectful; intros f g Hfg X Y [??]; split; set_solver. Qed. `````` Ralf Jung committed Feb 15, 2016 638 639 `````` Global Instance collection_join_mono {A} : Proper ((⊆) ==> (⊆)) (@mjoin M _ A). `````` Robbert Krebbers committed Feb 17, 2016 640 `````` Proof. intros X Y ?; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 641 642 `````` Global Instance collection_join_proper {A} : Proper ((≡) ==> (≡)) (@mjoin M _ A). `````` Robbert Krebbers committed Feb 17, 2016 643 `````` Proof. intros X Y [??]; split; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 644 `````` `````` Robbert Krebbers committed Jan 27, 2015 645 `````` Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x. `````` Robbert Krebbers committed Feb 17, 2016 646 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 27, 2015 647 `````` Lemma collection_guard_True {A} `{Decision P} (X : M A) : P → guard P; X ≡ X. `````` Robbert Krebbers committed Feb 17, 2016 648 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 649 `````` Lemma collection_fmap_compose {A B C} (f : A → B) (g : B → C) (X : M A) : `````` Robbert Krebbers committed Nov 12, 2012 650 `````` g ∘ f <\$> X ≡ g <\$> (f <\$> X). `````` Robbert Krebbers committed Feb 17, 2016 651 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 652 653 `````` 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 654 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 655 656 `````` 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 657 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 658 659 `````` 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 660 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 661 662 663 664 665 `````` 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 666 `````` - revert l. induction k; set_solver by eauto. `````` Robbert Krebbers committed Feb 17, 2016 667 `````` - induction 1; set_solver. `````` Robbert Krebbers committed Nov 12, 2012 668 `````` Qed. `````` Robbert Krebbers committed Jun 17, 2013 669 `````` Lemma collection_mapM_length {A B} (f : A → M B) l k : `````` Robbert Krebbers committed Nov 12, 2012 670 `````` l ∈ mapM f k → length l = length k. `````` Robbert Krebbers committed Feb 24, 2016 671 `````` Proof. revert l; induction k; set_solver by eauto. Qed. `````` Robbert Krebbers committed Nov 12, 2012 672 `````` Lemma elem_of_mapM_fmap {A B} (f : A → B) (g : B → M A) l k : `````` Robbert Krebbers committed May 07, 2013 673 `````` 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 674 `````` Proof. intros Hl. revert k. induction Hl; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 675 `````` Lemma elem_of_mapM_Forall {A B} (f : A → M B) (P : B → Prop) l k : `````` Robbert Krebbers committed May 07, 2013 676 `````` l ∈ mapM f k → Forall (λ x, ∀ y, y ∈ f x → P y) k → Forall P l. `````` Robbert Krebbers committed Jan 05, 2013 677 `````` Proof. rewrite elem_of_mapM. apply Forall2_Forall_l. Qed. `````` Robbert Krebbers committed May 07, 2013 678 679 `````` 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 → `````` Robbert Krebbers committed Jan 05, 2013 680 681 682 683 684 `````` Forall2 P l1 l2. Proof. rewrite elem_of_mapM. intros Hl1. revert l2. induction Hl1; inversion_clear 1; constructor; auto. Qed. `````` Robbert Krebbers committed Nov 12, 2012 685 ``````End collection_monad. `````` Robbert Krebbers committed Dec 11, 2015 686 687 688 689 690 691 `````` (** 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 692 693 `````` Global Instance set_finite_subseteq : Proper (flip (⊆) ==> impl) (@set_finite A B _). `````` Robbert Krebbers committed Feb 17, 2016 694 `````` Proof. intros X Y HX [l Hl]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Jan 16, 2016 695 696 `````` 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 697 698 699 `````` 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 700 `````` Proof. exists [x]; intros y ->%elem_of_singleton; left. Qed. `````` Robbert Krebbers committed Dec 11, 2015 701 702 703 704 705 706 `````` 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 707 `````` Proof. intros [l ?]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Dec 11, 2015 708 `````` Lemma union_finite_inv_r X Y : set_finite (X ∪ Y) → set_finite Y. `````` Robbert Krebbers committed Feb 17, 2016 709 `````` Proof. intros [l ?]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Dec 11, 2015 710 711 712 713 714 ``````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 715 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. ``````