collections.v 30.8 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) := `````` Robbert Krebbers committed Mar 03, 2016 268 `````` try 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 303 304 305 306 307 308 `````` 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. Global Instance set_unfold_of_list (l : list A) x : SetUnfold (x ∈ of_list l) (x ∈ l). Proof. constructor; apply elem_of_of_list. Qed. `````` Robbert Krebbers committed Jun 02, 2015 309 ``````End of_option_list. `````` Robbert Krebbers committed Jun 05, 2014 310 `````` `````` Robbert Krebbers committed Feb 24, 2016 311 ``````(** * Guard *) `````` Robbert Krebbers committed Jun 05, 2014 312 313 ``````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 314 315 316 317 318 319 320 321 322 `````` 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 323 324 325 `````` 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 326 327 328 329 330 `````` 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 331 332 333 `````` 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 334 335 `````` Lemma bind_empty {A B} (f : A → M B) X : X ≫= f ≡ ∅ ↔ X ≡ ∅ ∨ ∀ x, x ∈ X → f x ≡ ∅. `````` Robbert Krebbers committed Feb 24, 2016 336 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 27, 2015 337 ``````End collection_monad_base. `````` Robbert Krebbers committed Jun 05, 2014 338 `````` `````` Robbert Krebbers committed Feb 19, 2013 339 ``````(** * More theorems *) `````` Robbert Krebbers committed Jan 05, 2013 340 341 ``````Section collection. Context `{Collection A C}. `````` Robbert Krebbers committed Nov 18, 2015 342 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Jan 05, 2013 343 `````` `````` Robbert Krebbers committed Nov 15, 2014 344 `````` Global Instance: Lattice C. `````` Robbert Krebbers committed Feb 17, 2016 345 `````` Proof. split. apply _. firstorder auto. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 346 347 `````` Global Instance difference_proper : Proper ((≡) ==> (≡) ==> (≡)) (@difference C _). `````` Robbert Krebbers committed Feb 03, 2017 348 349 350 351 `````` 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 352 `````` Lemma non_empty_inhabited x X : x ∈ X → X ≢ ∅. `````` Robbert Krebbers committed Feb 17, 2016 353 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 354 `````` Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}. `````` Robbert Krebbers committed Feb 17, 2016 355 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 356 `````` Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y. `````` Robbert Krebbers committed Feb 17, 2016 357 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 08, 2015 358 `````` Lemma subseteq_empty_difference X Y : X ⊆ Y → X ∖ Y ≡ ∅. `````` Robbert Krebbers committed Feb 17, 2016 359 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 360 `````` Lemma difference_diag X : X ∖ X ≡ ∅. `````` Robbert Krebbers committed Feb 17, 2016 361 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 362 `````` Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z. `````` Robbert Krebbers committed Feb 17, 2016 363 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 03, 2017 364 `````` Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y). `````` Robbert Krebbers committed Feb 17, 2016 365 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 366 `````` Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z. `````` Robbert Krebbers committed Feb 17, 2016 367 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 20, 2015 368 `````` Lemma disjoint_union_difference X Y : X ∩ Y ≡ ∅ → (X ∪ Y) ∖ X ≡ Y. `````` Robbert Krebbers committed Feb 17, 2016 369 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 370 `````` `````` Robbert Krebbers committed Feb 19, 2013 371 372 373 374 375 376 `````` 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 377 378 `````` 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 379 380 381 382 `````` 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 383 384 `````` 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 385 386 387 `````` 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 388 389 `````` 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 390 391 392 `````` End leibniz. Section dec. `````` Robbert Krebbers committed Nov 17, 2015 393 `````` Context `{∀ (x : A) (X : C), Decision (x ∈ X)}. `````` Robbert Krebbers committed Feb 19, 2013 394 `````` Lemma not_elem_of_intersection x X Y : x ∉ X ∩ Y ↔ x ∉ X ∨ x ∉ Y. `````` Robbert Krebbers committed May 02, 2014 395 `````` Proof. rewrite elem_of_intersection. destruct (decide (x ∈ X)); tauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 396 `````` Lemma not_elem_of_difference x X Y : x ∉ X ∖ Y ↔ x ∉ X ∨ x ∈ Y. `````` Robbert Krebbers committed May 02, 2014 397 `````` Proof. rewrite elem_of_difference. destruct (decide (x ∈ Y)); tauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 398 399 `````` Lemma union_difference X Y : X ⊆ Y → Y ≡ X ∪ Y ∖ X. Proof. `````` Robbert Krebbers committed May 02, 2014 400 401 `````` split; intros x; rewrite !elem_of_union, elem_of_difference; [|intuition]. destruct (decide (x ∈ X)); intuition. `````` Robbert Krebbers committed Feb 19, 2013 402 403 `````` Qed. Lemma non_empty_difference X Y : X ⊂ Y → Y ∖ X ≢ ∅. `````` Robbert Krebbers committed Feb 24, 2016 404 `````` Proof. intros [HXY1 HXY2] Hdiff. destruct HXY2. set_solver. Qed. `````` Robbert Krebbers committed Feb 08, 2015 405 `````` Lemma empty_difference_subseteq X Y : X ∖ Y ≡ ∅ → X ⊆ Y. `````` Robbert Krebbers committed Feb 24, 2016 406 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 19, 2013 407 408 409 410 411 `````` 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 412 413 `````` 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 414 415 416 417 418 419 `````` End dec. End collection. Section collection_ops. Context `{CollectionOps A C}. `````` Robbert Krebbers committed Jan 05, 2013 420 421 422 423 424 `````` 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 425 `````` - revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|]. `````` Robbert Krebbers committed May 02, 2014 426 427 `````` 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 428 `````` eexists (x1 :: xs), y. intuition (simplify_option_eq; auto). `````` Robbert Krebbers committed Feb 17, 2016 429 `````` - intros (xs & y & Hxs & ? & Hx). revert x Hx. `````` Robbert Krebbers committed Feb 17, 2016 430 `````` induction Hxs; intros; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Jan 05, 2013 431 432 433 434 435 436 437 438 439 `````` 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 440 `````` intros HY HXs Hf. induction Xs; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Jan 05, 2013 441 442 443 `````` intros x Hx. rewrite elem_of_intersection_with in Hx. decompose_Forall. destruct Hx as (? & ? & ? & ? & ?). eauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 444 ``````End collection_ops. `````` Robbert Krebbers committed Jan 05, 2013 445 `````` `````` Robbert Krebbers committed Aug 29, 2012 446 ``````(** * Sets without duplicates up to an equivalence *) `````` Robbert Krebbers committed May 07, 2013 447 ``````Section NoDup. `````` Robbert Krebbers committed Nov 12, 2012 448 `````` Context `{SimpleCollection A B} (R : relation A) `{!Equivalence R}. `````` Robbert Krebbers committed Jun 11, 2012 449 450 `````` Definition elem_of_upto (x : A) (X : B) := ∃ y, y ∈ X ∧ R x y. `````` Robbert Krebbers committed May 07, 2013 451 `````` Definition set_NoDup (X : B) := ∀ x y, x ∈ X → y ∈ X → R x y → x = y. `````` Robbert Krebbers committed Jun 11, 2012 452 453 `````` Global Instance: Proper ((≡) ==> iff) (elem_of_upto x). `````` Robbert Krebbers committed Jan 05, 2013 454 `````` Proof. intros ??? E. unfold elem_of_upto. by setoid_rewrite E. Qed. `````` Robbert Krebbers committed Jun 11, 2012 455 456 457 `````` Global Instance: Proper (R ==> (≡) ==> iff) elem_of_upto. Proof. intros ?? E1 ?? E2. split; intros [z [??]]; exists z. `````` Robbert Krebbers committed Feb 17, 2016 458 459 `````` - rewrite <-E1, <-E2; intuition. - rewrite E1, E2; intuition. `````` Robbert Krebbers committed Jun 11, 2012 460 `````` Qed. `````` Robbert Krebbers committed May 07, 2013 461 `````` Global Instance: Proper ((≡) ==> iff) set_NoDup. `````` Robbert Krebbers committed Jun 11, 2012 462 463 464 `````` Proof. firstorder. Qed. Lemma elem_of_upto_elem_of x X : x ∈ X → elem_of_upto x X. `````` Robbert Krebbers committed Feb 17, 2016 465 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 466 `````` Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅. `````` Robbert Krebbers committed Feb 17, 2016 467 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Aug 21, 2012 468 `````` Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y. `````` Robbert Krebbers committed Feb 17, 2016 469 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 470 `````` `````` Robbert Krebbers committed Aug 21, 2012 471 472 `````` 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 473 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 474 `````` Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y. `````` Robbert Krebbers committed Feb 17, 2016 475 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 476 `````` `````` Robbert Krebbers committed May 07, 2013 477 `````` Lemma set_NoDup_empty: set_NoDup ∅. `````` Robbert Krebbers committed Feb 17, 2016 478 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 479 480 `````` Lemma set_NoDup_add x X : ¬elem_of_upto x X → set_NoDup X → set_NoDup ({[ x ]} ∪ X). `````` Robbert Krebbers committed Feb 17, 2016 481 `````` Proof. unfold set_NoDup, elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 482 483 `````` Lemma set_NoDup_inv_add x X : x ∉ X → set_NoDup ({[ x ]} ∪ X) → ¬elem_of_upto x X. `````` Robbert Krebbers committed Aug 21, 2012 484 485 `````` Proof. intros Hin Hnodup [y [??]]. `````` Robbert Krebbers committed Feb 17, 2016 486 `````` rewrite (Hnodup x y) in Hin; set_solver. `````` Robbert Krebbers committed Aug 21, 2012 487 `````` Qed. `````` Robbert Krebbers committed May 07, 2013 488 `````` Lemma set_NoDup_inv_union_l X Y : set_NoDup (X ∪ Y) → set_NoDup X. `````` Robbert Krebbers committed Feb 17, 2016 489 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 490 `````` Lemma set_NoDup_inv_union_r X Y : set_NoDup (X ∪ Y) → set_NoDup Y. `````` Robbert Krebbers committed Feb 17, 2016 491 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 492 ``````End NoDup. `````` Robbert Krebbers committed Jun 11, 2012 493 `````` `````` Robbert Krebbers committed Aug 29, 2012 494 ``````(** * Quantifiers *) `````` Robbert Krebbers committed Jun 11, 2012 495 ``````Section quantifiers. `````` Robbert Krebbers committed Nov 12, 2012 496 `````` Context `{SimpleCollection A B} (P : A → Prop). `````` Robbert Krebbers committed Jun 11, 2012 497 `````` `````` Robbert Krebbers committed May 07, 2013 498 499 500 501 `````` 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 502 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 503 `````` Lemma set_Forall_singleton x : set_Forall {[ x ]} ↔ P x. `````` Robbert Krebbers committed Feb 17, 2016 504 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 505 `````` Lemma set_Forall_union X Y : set_Forall X → set_Forall Y → set_Forall (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 506 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 507 `````` Lemma set_Forall_union_inv_1 X Y : set_Forall (X ∪ Y) → set_Forall X. `````` Robbert Krebbers committed Feb 17, 2016 508 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 509 `````` Lemma set_Forall_union_inv_2 X Y : set_Forall (X ∪ Y) → set_Forall Y. `````` Robbert Krebbers committed Feb 17, 2016 510 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 511 512 `````` Lemma set_Exists_empty : ¬set_Exists ∅. `````` Robbert Krebbers committed Feb 17, 2016 513 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 514 `````` Lemma set_Exists_singleton x : set_Exists {[ x ]} ↔ P x. `````` Robbert Krebbers committed Feb 17, 2016 515 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 516 `````` Lemma set_Exists_union_1 X Y : set_Exists X → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 517 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 518 `````` Lemma set_Exists_union_2 X Y : set_Exists Y → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 519 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 520 521 `````` Lemma set_Exists_union_inv X Y : set_Exists (X ∪ Y) → set_Exists X ∨ set_Exists Y. `````` Robbert Krebbers committed Feb 17, 2016 522 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 523 524 ``````End quantifiers. `````` Robbert Krebbers committed Aug 21, 2012 525 ``````Section more_quantifiers. `````` Robbert Krebbers committed Jun 05, 2014 526 `````` Context `{SimpleCollection A B}. `````` Robbert Krebbers committed Aug 29, 2012 527 `````` `````` Robbert Krebbers committed May 07, 2013 528 529 530 531 532 533 `````` 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 534 535 ``````End more_quantifiers. `````` Robbert Krebbers committed Aug 29, 2012 536 537 538 ``````(** * 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 539 540 541 542 543 544 545 546 547 548 ``````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 549 `````` `````` Robbert Krebbers committed Mar 02, 2015 550 551 ``````Section fresh. Context `{FreshSpec A C}. `````` Robbert Krebbers committed Nov 18, 2015 552 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Aug 21, 2012 553 `````` `````` Robbert Krebbers committed Nov 18, 2015 554 `````` Global Instance fresh_proper: Proper ((≡) ==> (=)) (fresh (C:=C)). `````` Robbert Krebbers committed Feb 08, 2015 555 `````` Proof. intros ???. by apply fresh_proper_alt, elem_of_equiv. Qed. `````` Robbert Krebbers committed Nov 18, 2015 556 557 `````` Global Instance fresh_list_proper: Proper ((=) ==> (≡) ==> (=)) (fresh_list (C:=C)). `````` Robbert Krebbers committed Aug 29, 2012 558 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 559 `````` intros ? n ->. induction n as [|n IH]; intros ?? E; f_equal/=; [by rewrite E|]. `````` Robbert Krebbers committed May 02, 2014 560 `````` apply IH. by rewrite E. `````` Robbert Krebbers committed Aug 29, 2012 561 `````` Qed. `````` Robbert Krebbers committed Mar 02, 2015 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 `````` 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 577 578 `````` Lemma Forall_fresh_subseteq X Y xs : Forall_fresh X xs → Y ⊆ X → Forall_fresh Y xs. `````` Robbert Krebbers committed Feb 17, 2016 579 `````` Proof. rewrite !Forall_fresh_alt; set_solver. Qed. `````` Robbert Krebbers committed Mar 02, 2015 580 `````` `````` Robbert Krebbers committed Aug 21, 2012 581 582 `````` 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 583 `````` Lemma fresh_list_is_fresh n X x : x ∈ fresh_list n X → x ∉ X. `````` Robbert Krebbers committed Aug 21, 2012 584 `````` Proof. `````` Robbert Krebbers committed Mar 02, 2015 585 `````` revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|]. `````` Robbert Krebbers committed May 02, 2014 586 `````` rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|]. `````` Robbert Krebbers committed Feb 17, 2016 587 `````` apply IH in Hin; set_solver. `````` Robbert Krebbers committed Aug 21, 2012 588 `````` Qed. `````` Robbert Krebbers committed Mar 02, 2015 589 `````` Lemma NoDup_fresh_list n X : NoDup (fresh_list n X). `````` Robbert Krebbers committed Aug 21, 2012 590 `````` Proof. `````` Robbert Krebbers committed May 07, 2013 591 `````` revert X. induction n; simpl; constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 592 `````` intros Hin; apply fresh_list_is_fresh in Hin; set_solver. `````` Robbert Krebbers committed Mar 02, 2015 593 594 595 596 `````` 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 597 598 `````` Qed. End fresh. `````` Robbert Krebbers committed Nov 12, 2012 599 `````` `````` Robbert Krebbers committed Feb 19, 2013 600 ``````(** * Properties of implementations of collections that form a monad *) `````` Robbert Krebbers committed Nov 12, 2012 601 602 603 ``````Section collection_monad. Context `{CollectionMonad M}. `````` Ralf Jung committed Feb 15, 2016 604 605 `````` Global Instance collection_fmap_mono {A B} : Proper (pointwise_relation _ (=) ==> (⊆) ==> (⊆)) (@fmap M _ A B). `````` Robbert Krebbers committed Feb 24, 2016 606 `````` Proof. intros f g ? X Y ?; set_solver by eauto. Qed. `````` Robbert Krebbers committed Nov 16, 2015 607 608 `````` Global Instance collection_fmap_proper {A B} : Proper (pointwise_relation _ (=) ==> (≡) ==> (≡)) (@fmap M _ A B). `````` Robbert Krebbers committed Feb 24, 2016 609 `````` Proof. intros f g ? X Y [??]; split; set_solver by eauto. Qed. `````` Ralf Jung committed Feb 15, 2016 610 611 `````` Global Instance collection_bind_mono {A B} : Proper (((=) ==> (⊆)) ==> (⊆) ==> (⊆)) (@mbind M _ A B). `````` Robbert Krebbers committed Feb 17, 2016 612 `````` Proof. unfold respectful; intros f g Hfg X Y ?; set_solver. Qed. `````` Robbert Krebbers committed Nov 16, 2015 613 614 `````` Global Instance collection_bind_proper {A B} : Proper (((=) ==> (≡)) ==> (≡) ==> (≡)) (@mbind M _ A B). `````` Robbert Krebbers committed Feb 17, 2016 615 `````` Proof. unfold respectful; intros f g Hfg X Y [??]; split; set_solver. Qed. `````` Ralf Jung committed Feb 15, 2016 616 617 `````` Global Instance collection_join_mono {A} : Proper ((⊆) ==> (⊆)) (@mjoin M _ A). `````` Robbert Krebbers committed Feb 17, 2016 618 `````` Proof. intros X Y ?; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 619 620 `````` Global Instance collection_join_proper {A} : Proper ((≡) ==> (≡)) (@mjoin M _ A). `````` Robbert Krebbers committed Feb 17, 2016 621 `````` Proof. intros X Y [??]; split; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 622 `````` `````` Robbert Krebbers committed Jan 27, 2015 623 `````` Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x. `````` Robbert Krebbers committed Feb 17, 2016 624 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 27, 2015 625 `````` Lemma collection_guard_True {A} `{Decision P} (X : M A) : P → guard P; X ≡ X. `````` Robbert Krebbers committed Feb 17, 2016 626 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 627 `````` Lemma collection_fmap_compose {A B C} (f : A → B) (g : B → C) (X : M A) : `````` Robbert Krebbers committed Nov 12, 2012 628 `````` g ∘ f <\$> X ≡ g <\$> (f <\$> X). `````` Robbert Krebbers committed Feb 17, 2016 629 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 630 631 `````` 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 632 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 633 634 `````` 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 635 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 636 637 `````` 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 638 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 639 640 641 642 643 `````` 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 644 `````` - revert l. induction k; set_solver by eauto. `````` Robbert Krebbers committed Feb 17, 2016 645 `````` - induction 1; set_solver. `````` Robbert Krebbers committed Nov 12, 2012 646 `````` Qed. `````` Robbert Krebbers committed Jun 17, 2013 647 `````` Lemma collection_mapM_length {A B} (f : A → M B) l k : `````` Robbert Krebbers committed Nov 12, 2012 648 `````` l ∈ mapM f k → length l = length k. `````` Robbert Krebbers committed Feb 24, 2016 649 `````` Proof. revert l; induction k; set_solver by eauto. Qed. `````` Robbert Krebbers committed Nov 12, 2012 650 `````` Lemma elem_of_mapM_fmap {A B} (f : A → B) (g : B → M A) l k : `````` Robbert Krebbers committed May 07, 2013 651 `````` 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 652 `````` Proof. intros Hl. revert k. induction Hl; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 653 `````` Lemma elem_of_mapM_Forall {A B} (f : A → M B) (P : B → Prop) l k : `````` Robbert Krebbers committed May 07, 2013 654 `````` l ∈ mapM f k → Forall (λ x, ∀ y, y ∈ f x → P y) k → Forall P l. `````` Robbert Krebbers committed Jan 05, 2013 655 `````` Proof. rewrite elem_of_mapM. apply Forall2_Forall_l. Qed. `````` Robbert Krebbers committed May 07, 2013 656 657 `````` 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 658 659 660 661 662 `````` 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 663 ``````End collection_monad. `````` Robbert Krebbers committed Dec 11, 2015 664 665 666 667 668 669 `````` (** 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 670 671 `````` Global Instance set_finite_subseteq : Proper (flip (⊆) ==> impl) (@set_finite A B _). `````` Robbert Krebbers committed Feb 17, 2016 672 `````` Proof. intros X Y HX [l Hl]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Jan 16, 2016 673 674 `````` 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 675 676 677 `````` 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 678 `````` Proof. exists [x]; intros y ->%elem_of_singleton; left. Qed. `````` Robbert Krebbers committed Dec 11, 2015 679 680 681 682 683 684 `````` 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 685 `````` Proof. intros [l ?]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Dec 11, 2015 686 `````` Lemma union_finite_inv_r X Y : set_finite (X ∪ Y) → set_finite Y. `````` Robbert Krebbers committed Feb 17, 2016 687 `````` Proof. intros [l ?]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Dec 11, 2015 688 689 690 691 692 ``````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 693 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. `````` Robbert Krebbers committed Dec 11, 2015 694 `````` Lemma intersection_finite_r X Y : set_finite Y → set_finite (X ∩ Y). `````` Ralf Jung committed Jan 04, 2016 695 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. `````` Robbert Krebbers committed Dec 11, 2015 696 `````` Lemma difference_finite X Y : set_finite X → set_finite (X ∖ Y). `````` Ralf Jung committed Jan 04, 2016 697 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_difference; auto. Qed. `````` Robbert Krebbers committed Jan 16, 2016 698 699 700 701 `````` 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 702 `````` intros x ?; destruct (decide (x ∈ Y)); rewrite elem_of_app; set_solver. `````` Robbert Krebbers committed Jan 16, 2016 703 `````` Qed. `````` Robbert Krebbers committed Dec 11, 2015 704 ``End more_finite.``