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 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 Mar 23, 2016 8 9 ``````Instance collection_disjoint `{ElemOf A C} : Disjoint C := λ X Y, ∀ x, x ∈ X → x ∈ Y → False. `````` Robbert Krebbers committed May 02, 2014 10 11 ``````Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y, ∀ x, x ∈ X → x ∈ Y. `````` Robbert Krebbers committed Mar 23, 2016 12 ``````Typeclasses Opaque collection_disjoint collection_subseteq. `````` Robbert Krebbers committed May 02, 2014 13 `````` `````` Robbert Krebbers committed Feb 19, 2013 14 ``````(** * Basic theorems *) `````` Robbert Krebbers committed Nov 12, 2012 15 16 ``````Section simple_collection. Context `{SimpleCollection A C}. `````` Robbert Krebbers committed Nov 18, 2015 17 18 `````` Implicit Types x y : A. Implicit Types X Y : C. `````` Robbert Krebbers committed Jun 11, 2012 19 `````` `````` Robbert Krebbers committed Aug 29, 2012 20 `````` Lemma elem_of_empty x : x ∈ ∅ ↔ False. `````` Robbert Krebbers committed Oct 19, 2012 21 `````` Proof. split. apply not_elem_of_empty. done. Qed. `````` Robbert Krebbers committed Jun 11, 2012 22 23 24 25 `````` 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 26 27 28 `````` Global Instance: EmptySpec C. Proof. firstorder auto. Qed. Global Instance: JoinSemiLattice C. `````` Robbert Krebbers committed Oct 19, 2012 29 `````` Proof. firstorder auto. Qed. `````` Robbert Krebbers committed Feb 15, 2016 30 31 `````` Global Instance: AntiSymm (≡) (@collection_subseteq A C _). Proof. done. Qed. `````` Robbert Krebbers committed Jun 11, 2012 32 `````` Lemma elem_of_subseteq X Y : X ⊆ Y ↔ ∀ x, x ∈ X → x ∈ Y. `````` Robbert Krebbers committed Oct 19, 2012 33 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jun 11, 2012 34 35 `````` Lemma elem_of_equiv X Y : X ≡ Y ↔ ∀ x, x ∈ X ↔ x ∈ Y. Proof. firstorder. Qed. `````` Robbert Krebbers committed Aug 21, 2012 36 37 `````` 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 38 `````` Proof. firstorder. Qed. `````` Robbert Krebbers committed Feb 19, 2013 39 40 `````` Lemma elem_of_equiv_empty X : X ≡ ∅ ↔ ∀ x, x ∉ X. Proof. firstorder. Qed. `````` Robbert Krebbers committed Mar 23, 2016 41 42 43 `````` Lemma elem_of_disjoint X Y : X ⊥ Y ↔ ∀ x, x ∈ X → x ∈ Y → False. Proof. done. Qed. `````` Robbert Krebbers committed Jun 05, 2014 44 45 46 47 48 49 `````` 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 50 51 52 53 `````` 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 54 55 56 `````` Lemma elem_of_subseteq_singleton x X : x ∈ X ↔ {[ x ]} ⊆ X. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 57 58 `````` - intros ??. rewrite elem_of_singleton. by intros ->. - intros Ex. by apply (Ex x), elem_of_singleton. `````` Robbert Krebbers committed Oct 19, 2012 59 `````` Qed. `````` Robbert Krebbers committed Mar 23, 2016 60 `````` `````` Robbert Krebbers committed Nov 18, 2015 61 `````` Global Instance singleton_proper : Proper ((=) ==> (≡)) (singleton (B:=C)). `````` Robbert Krebbers committed May 02, 2014 62 `````` Proof. by repeat intro; subst. Qed. `````` Robbert Krebbers committed Nov 18, 2015 63 `````` Global Instance elem_of_proper : `````` Robbert Krebbers committed Mar 23, 2016 64 `````` Proper ((=) ==> (≡) ==> iff) (@elem_of A C _) | 5. `````` Robbert Krebbers committed May 02, 2014 65 `````` Proof. intros ???; subst. firstorder. Qed. `````` Robbert Krebbers committed Mar 23, 2016 66 67 `````` Global Instance disjoint_prope : Proper ((≡) ==> (≡) ==> iff) (@disjoint C _). Proof. intros ??????. by rewrite !elem_of_disjoint; setoid_subst. Qed. `````` Robbert Krebbers committed May 07, 2013 68 `````` Lemma elem_of_union_list Xs x : x ∈ ⋃ Xs ↔ ∃ X, X ∈ Xs ∧ x ∈ X. `````` Robbert Krebbers committed Oct 19, 2012 69 70 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 71 `````` - induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|]. `````` Robbert Krebbers committed May 02, 2014 72 `````` setoid_rewrite elem_of_cons. apply elem_of_union in HXs. naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 73 `````` - intros [X []]. induction 1; simpl; [by apply elem_of_union_l |]. `````` Robbert Krebbers committed May 02, 2014 74 `````` intros. apply elem_of_union_r; auto. `````` Robbert Krebbers committed Oct 19, 2012 75 `````` Qed. `````` Robbert Krebbers committed Nov 18, 2015 76 `````` Lemma non_empty_singleton x : ({[ x ]} : C) ≢ ∅. `````` Robbert Krebbers committed Oct 19, 2012 77 78 79 80 81 82 `````` 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 83 84 85 86 87 88 89 90 91 `````` 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 92 93 94 95 `````` 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 96 97 98 99 100 101 102 103 104 105 106 107 `````` 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 108 109 ``````End simple_collection. `````` Robbert Krebbers committed Feb 24, 2016 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 ``````(** * Tactics *) (** The tactic [set_unfold] transforms all occurrences of [(∪)], [(∩)], [(∖)], [(<\$>)], [∅], [{[_]}], [(≡)], and [(⊆)] into logically equivalent propositions involving just [∈]. For example, [A → x ∈ X ∪ ∅] becomes [A → x ∈ X ∨ False]. This transformation is implemented using type classes instead of [rewrite]ing to ensure that we traverse each term at most once. *) Class SetUnfold (P Q : Prop) := { set_unfold : P ↔ Q }. Arguments set_unfold _ _ {_}. Hint Mode SetUnfold + - : typeclass_instances. Class SetUnfoldSimpl (P Q : Prop) := { set_unfold_simpl : SetUnfold P Q }. Hint Extern 0 (SetUnfoldSimpl _ _) => csimpl; constructor : typeclass_instances. Instance set_unfold_fallthrough P : SetUnfold P P | 1000. done. Qed. Definition set_unfold_1 `{SetUnfold P Q} : P → Q := proj1 (set_unfold P Q). Definition set_unfold_2 `{SetUnfold P Q} : Q → P := proj2 (set_unfold P Q). Lemma set_unfold_impl P Q P' Q' : SetUnfold P P' → SetUnfold Q Q' → SetUnfold (P → Q) (P' → Q'). Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed. Lemma set_unfold_and P Q P' Q' : SetUnfold P P' → SetUnfold Q Q' → SetUnfold (P ∧ Q) (P' ∧ Q'). Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed. Lemma set_unfold_or P Q P' Q' : SetUnfold P P' → SetUnfold Q Q' → SetUnfold (P ∨ Q) (P' ∨ Q'). Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed. Lemma set_unfold_iff P Q P' Q' : SetUnfold P P' → SetUnfold Q Q' → SetUnfold (P ↔ Q) (P' ↔ Q'). Proof. constructor. by rewrite (set_unfold P P'), (set_unfold Q Q'). Qed. Lemma set_unfold_not P P' : SetUnfold P P' → SetUnfold (¬P) (¬P'). Proof. constructor. by rewrite (set_unfold P P'). Qed. Lemma set_unfold_forall {A} (P P' : A → Prop) : (∀ x, SetUnfold (P x) (P' x)) → SetUnfold (∀ x, P x) (∀ x, P' x). Proof. constructor. naive_solver. Qed. Lemma set_unfold_exist {A} (P P' : A → Prop) : (∀ x, SetUnfold (P x) (P' x)) → SetUnfold (∃ x, P x) (∃ x, P' x). Proof. constructor. naive_solver. Qed. (* Avoid too eager application of the above instances (and thus too eager unfolding of type class transparent definitions). *) Hint Extern 0 (SetUnfold (_ → _) _) => class_apply set_unfold_impl : typeclass_instances. Hint Extern 0 (SetUnfold (_ ∧ _) _) => class_apply set_unfold_and : typeclass_instances. Hint Extern 0 (SetUnfold (_ ∨ _) _) => class_apply set_unfold_or : typeclass_instances. Hint Extern 0 (SetUnfold (_ ↔ _) _) => class_apply set_unfold_iff : typeclass_instances. Hint Extern 0 (SetUnfold (¬ _) _) => class_apply set_unfold_not : typeclass_instances. Hint Extern 1 (SetUnfold (∀ _, _) _) => class_apply set_unfold_forall : typeclass_instances. Hint Extern 0 (SetUnfold (∃ _, _) _) => class_apply set_unfold_exist : typeclass_instances. Section set_unfold_simple. Context `{SimpleCollection A C}. Implicit Types x y : A. Implicit Types X Y : C. Global Instance set_unfold_empty x : SetUnfold (x ∈ ∅) False. Proof. constructor; apply elem_of_empty. Qed. Global Instance set_unfold_singleton x y : SetUnfold (x ∈ {[ y ]}) (x = y). Proof. constructor; apply elem_of_singleton. Qed. Global Instance set_unfold_union x X Y P Q : SetUnfold (x ∈ X) P → SetUnfold (x ∈ Y) Q → SetUnfold (x ∈ X ∪ Y) (P ∨ Q). Proof. intros ??; constructor. by rewrite elem_of_union, (set_unfold (x ∈ X) P), (set_unfold (x ∈ Y) Q). Qed. Global Instance set_unfold_equiv_same X : SetUnfold (X ≡ X) True | 1. Proof. done. Qed. Global Instance set_unfold_equiv_empty_l X (P : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (∅ ≡ X) (∀ x, ¬P x) | 5. Proof. intros ?; constructor. rewrite (symmetry_iff equiv), elem_of_equiv_empty; naive_solver. Qed. Global Instance set_unfold_equiv_empty_r (P : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → SetUnfold (X ≡ ∅) (∀ x, ¬P x) | 5. Proof. constructor. rewrite elem_of_equiv_empty; naive_solver. Qed. Global Instance set_unfold_equiv (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X ≡ Y) (∀ x, P x ↔ Q x) | 10. Proof. constructor. rewrite elem_of_equiv; naive_solver. Qed. Global Instance set_unfold_subseteq (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X ⊆ Y) (∀ x, P x → Q x). Proof. constructor. rewrite elem_of_subseteq; naive_solver. Qed. Global Instance set_unfold_subset (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X ⊂ Y) ((∀ x, P x → Q x) ∧ ¬∀ x, P x ↔ Q x). Proof. constructor. rewrite subset_spec, elem_of_subseteq, elem_of_equiv. repeat f_equiv; naive_solver. Qed. `````` Robbert Krebbers committed Mar 23, 2016 207 208 209 210 `````` Global Instance set_unfold_disjoint (P Q : A → Prop) : (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) → SetUnfold (X ⊥ Y) (∀ x, P x → Q x → False). Proof. constructor. rewrite elem_of_disjoint. naive_solver. Qed. `````` Robbert Krebbers committed Feb 24, 2016 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 `````` 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 280 `````` try fast_done; `````` Robbert Krebbers committed Feb 24, 2016 281 282 283 284 285 286 287 288 289 290 291 292 293 `````` 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 294 295 296 297 ``````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 298 299 300 ``````(** * 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 301 302 ``````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 303 `````` `````` Robbert Krebbers committed Jun 02, 2015 304 305 ``````Section of_option_list. Context `{SimpleCollection A C}. `````` Robbert Krebbers committed Feb 24, 2016 306 307 `````` 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 308 309 310 `````` Lemma elem_of_of_list (x : A) l : x ∈ of_list l ↔ x ∈ l. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 311 `````` - induction l; simpl; [by rewrite elem_of_empty|]. `````` Robbert Krebbers committed Feb 01, 2017 312 `````` rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 313 `````` - induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto. `````` Robbert Krebbers committed Jun 02, 2015 314 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 315 316 317 `````` 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 318 319 320 `````` 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 321 ``````End of_option_list. `````` Robbert Krebbers committed Jun 05, 2014 322 `````` `````` Robbert Krebbers committed Mar 04, 2016 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 ``````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 339 340 341 342 `````` 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 343 344 ``````End list_unfold. `````` Robbert Krebbers committed Feb 24, 2016 345 ``````(** * Guard *) `````` Robbert Krebbers committed Jun 05, 2014 346 347 ``````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 348 349 350 351 352 353 354 355 356 `````` 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 357 358 359 `````` 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 360 361 362 363 364 `````` 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 365 366 367 `````` 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 368 369 `````` Lemma bind_empty {A B} (f : A → M B) X : X ≫= f ≡ ∅ ↔ X ≡ ∅ ∨ ∀ x, x ∈ X → f x ≡ ∅. `````` Robbert Krebbers committed Feb 24, 2016 370 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 27, 2015 371 ``````End collection_monad_base. `````` Robbert Krebbers committed Jun 05, 2014 372 `````` `````` Robbert Krebbers committed Feb 19, 2013 373 ``````(** * More theorems *) `````` Robbert Krebbers committed Jan 05, 2013 374 375 ``````Section collection. Context `{Collection A C}. `````` Robbert Krebbers committed Nov 18, 2015 376 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Jan 05, 2013 377 `````` `````` Robbert Krebbers committed Nov 15, 2014 378 `````` Global Instance: Lattice C. `````` Robbert Krebbers committed Feb 17, 2016 379 `````` Proof. split. apply _. firstorder auto. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 380 381 `````` Global Instance difference_proper : Proper ((≡) ==> (≡) ==> (≡)) (@difference C _). `````` Robbert Krebbers committed Feb 03, 2017 382 383 384 385 `````` 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 386 `````` Lemma non_empty_inhabited x X : x ∈ X → X ≢ ∅. `````` Robbert Krebbers committed Feb 17, 2016 387 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 388 `````` Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}. `````` Robbert Krebbers committed Feb 17, 2016 389 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 390 `````` Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y. `````` Robbert Krebbers committed Feb 17, 2016 391 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 08, 2015 392 `````` Lemma subseteq_empty_difference X Y : X ⊆ Y → X ∖ Y ≡ ∅. `````` Robbert Krebbers committed Feb 17, 2016 393 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 394 `````` Lemma difference_diag X : X ∖ X ≡ ∅. `````` Robbert Krebbers committed Feb 17, 2016 395 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 396 `````` Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z. `````` Robbert Krebbers committed Feb 17, 2016 397 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 03, 2017 398 `````` Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y). `````` Robbert Krebbers committed Feb 17, 2016 399 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 400 `````` Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z. `````` Robbert Krebbers committed Feb 17, 2016 401 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Mar 23, 2016 402 `````` Lemma disjoint_union_difference X Y : X ⊥ Y → (X ∪ Y) ∖ X ≡ Y. `````` Robbert Krebbers committed Feb 17, 2016 403 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 05, 2013 404 `````` `````` Robbert Krebbers committed Feb 19, 2013 405 406 407 408 409 410 `````` 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 411 412 `````` 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 413 414 415 416 `````` 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 417 418 `````` 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 419 420 421 `````` Lemma difference_intersection_distr_l_L X Y Z : (X ∩ Y) ∖ Z = X ∖ Z ∩ Y ∖ Z. Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed. `````` Robbert Krebbers committed Mar 23, 2016 422 `````` Lemma disjoint_union_difference_L X Y : X ⊥ Y → (X ∪ Y) ∖ X = Y. `````` Robbert Krebbers committed Nov 20, 2015 423 `````` Proof. unfold_leibniz. apply disjoint_union_difference. Qed. `````` Robbert Krebbers committed Feb 19, 2013 424 425 426 `````` End leibniz. Section dec. `````` Robbert Krebbers committed Nov 17, 2015 427 `````` Context `{∀ (x : A) (X : C), Decision (x ∈ X)}. `````` Robbert Krebbers committed Feb 19, 2013 428 `````` Lemma not_elem_of_intersection x X Y : x ∉ X ∩ Y ↔ x ∉ X ∨ x ∉ Y. `````` Robbert Krebbers committed May 02, 2014 429 `````` Proof. rewrite elem_of_intersection. destruct (decide (x ∈ X)); tauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 430 `````` Lemma not_elem_of_difference x X Y : x ∉ X ∖ Y ↔ x ∉ X ∨ x ∈ Y. `````` Robbert Krebbers committed May 02, 2014 431 `````` Proof. rewrite elem_of_difference. destruct (decide (x ∈ Y)); tauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 432 433 `````` Lemma union_difference X Y : X ⊆ Y → Y ≡ X ∪ Y ∖ X. Proof. `````` Robbert Krebbers committed May 02, 2014 434 435 `````` split; intros x; rewrite !elem_of_union, elem_of_difference; [|intuition]. destruct (decide (x ∈ X)); intuition. `````` Robbert Krebbers committed Feb 19, 2013 436 437 `````` Qed. Lemma non_empty_difference X Y : X ⊂ Y → Y ∖ X ≢ ∅. `````` Robbert Krebbers committed Feb 24, 2016 438 `````` Proof. intros [HXY1 HXY2] Hdiff. destruct HXY2. set_solver. Qed. `````` Robbert Krebbers committed Feb 08, 2015 439 `````` Lemma empty_difference_subseteq X Y : X ∖ Y ≡ ∅ → X ⊆ Y. `````` Robbert Krebbers committed Feb 24, 2016 440 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Feb 19, 2013 441 442 443 444 445 `````` 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 446 447 `````` 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 448 449 450 451 452 453 `````` End dec. End collection. Section collection_ops. Context `{CollectionOps A C}. `````` Robbert Krebbers committed Jan 05, 2013 454 455 456 457 458 `````` 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 459 `````` - revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|]. `````` Robbert Krebbers committed May 02, 2014 460 461 `````` 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 462 `````` eexists (x1 :: xs), y. intuition (simplify_option_eq; auto). `````` Robbert Krebbers committed Feb 17, 2016 463 `````` - intros (xs & y & Hxs & ? & Hx). revert x Hx. `````` Robbert Krebbers committed Feb 17, 2016 464 `````` induction Hxs; intros; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Jan 05, 2013 465 466 467 468 469 470 471 472 473 `````` 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 474 `````` intros HY HXs Hf. induction Xs; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Jan 05, 2013 475 476 477 `````` intros x Hx. rewrite elem_of_intersection_with in Hx. decompose_Forall. destruct Hx as (? & ? & ? & ? & ?). eauto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 478 ``````End collection_ops. `````` Robbert Krebbers committed Jan 05, 2013 479 `````` `````` Robbert Krebbers committed Aug 29, 2012 480 ``````(** * Sets without duplicates up to an equivalence *) `````` Robbert Krebbers committed May 07, 2013 481 ``````Section NoDup. `````` Robbert Krebbers committed Nov 12, 2012 482 `````` Context `{SimpleCollection A B} (R : relation A) `{!Equivalence R}. `````` Robbert Krebbers committed Jun 11, 2012 483 484 `````` Definition elem_of_upto (x : A) (X : B) := ∃ y, y ∈ X ∧ R x y. `````` Robbert Krebbers committed May 07, 2013 485 `````` Definition set_NoDup (X : B) := ∀ x y, x ∈ X → y ∈ X → R x y → x = y. `````` Robbert Krebbers committed Jun 11, 2012 486 487 `````` Global Instance: Proper ((≡) ==> iff) (elem_of_upto x). `````` Robbert Krebbers committed Jan 05, 2013 488 `````` Proof. intros ??? E. unfold elem_of_upto. by setoid_rewrite E. Qed. `````` Robbert Krebbers committed Jun 11, 2012 489 490 491 `````` Global Instance: Proper (R ==> (≡) ==> iff) elem_of_upto. Proof. intros ?? E1 ?? E2. split; intros [z [??]]; exists z. `````` Robbert Krebbers committed Feb 17, 2016 492 493 `````` - rewrite <-E1, <-E2; intuition. - rewrite E1, E2; intuition. `````` Robbert Krebbers committed Jun 11, 2012 494 `````` Qed. `````` Robbert Krebbers committed May 07, 2013 495 `````` Global Instance: Proper ((≡) ==> iff) set_NoDup. `````` Robbert Krebbers committed Jun 11, 2012 496 497 498 `````` Proof. firstorder. Qed. Lemma elem_of_upto_elem_of x X : x ∈ X → elem_of_upto x X. `````` Robbert Krebbers committed Feb 17, 2016 499 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 500 `````` Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅. `````` Robbert Krebbers committed Feb 17, 2016 501 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Aug 21, 2012 502 `````` Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y. `````` Robbert Krebbers committed Feb 17, 2016 503 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 504 `````` `````` Robbert Krebbers committed Aug 21, 2012 505 506 `````` 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 507 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 508 `````` Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y. `````` Robbert Krebbers committed Feb 17, 2016 509 `````` Proof. unfold elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 510 `````` `````` Robbert Krebbers committed May 07, 2013 511 `````` Lemma set_NoDup_empty: set_NoDup ∅. `````` Robbert Krebbers committed Feb 17, 2016 512 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 513 514 `````` Lemma set_NoDup_add x X : ¬elem_of_upto x X → set_NoDup X → set_NoDup ({[ x ]} ∪ X). `````` Robbert Krebbers committed Feb 17, 2016 515 `````` Proof. unfold set_NoDup, elem_of_upto. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 516 517 `````` Lemma set_NoDup_inv_add x X : x ∉ X → set_NoDup ({[ x ]} ∪ X) → ¬elem_of_upto x X. `````` Robbert Krebbers committed Aug 21, 2012 518 519 `````` Proof. intros Hin Hnodup [y [??]]. `````` Robbert Krebbers committed Feb 17, 2016 520 `````` rewrite (Hnodup x y) in Hin; set_solver. `````` Robbert Krebbers committed Aug 21, 2012 521 `````` Qed. `````` Robbert Krebbers committed May 07, 2013 522 `````` Lemma set_NoDup_inv_union_l X Y : set_NoDup (X ∪ Y) → set_NoDup X. `````` Robbert Krebbers committed Feb 17, 2016 523 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 524 `````` Lemma set_NoDup_inv_union_r X Y : set_NoDup (X ∪ Y) → set_NoDup Y. `````` Robbert Krebbers committed Feb 17, 2016 525 `````` Proof. unfold set_NoDup. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 526 ``````End NoDup. `````` Robbert Krebbers committed Jun 11, 2012 527 `````` `````` Robbert Krebbers committed Aug 29, 2012 528 ``````(** * Quantifiers *) `````` Robbert Krebbers committed Jun 11, 2012 529 ``````Section quantifiers. `````` Robbert Krebbers committed Nov 12, 2012 530 `````` Context `{SimpleCollection A B} (P : A → Prop). `````` Robbert Krebbers committed Jun 11, 2012 531 `````` `````` Robbert Krebbers committed May 07, 2013 532 533 534 535 `````` 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 536 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 537 `````` Lemma set_Forall_singleton x : set_Forall {[ x ]} ↔ P x. `````` Robbert Krebbers committed Feb 17, 2016 538 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 539 `````` Lemma set_Forall_union X Y : set_Forall X → set_Forall Y → set_Forall (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 540 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 541 `````` Lemma set_Forall_union_inv_1 X Y : set_Forall (X ∪ Y) → set_Forall X. `````` Robbert Krebbers committed Feb 17, 2016 542 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 543 `````` Lemma set_Forall_union_inv_2 X Y : set_Forall (X ∪ Y) → set_Forall Y. `````` Robbert Krebbers committed Feb 17, 2016 544 `````` Proof. unfold set_Forall. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 545 546 `````` Lemma set_Exists_empty : ¬set_Exists ∅. `````` Robbert Krebbers committed Feb 17, 2016 547 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 548 `````` Lemma set_Exists_singleton x : set_Exists {[ x ]} ↔ P x. `````` Robbert Krebbers committed Feb 17, 2016 549 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 550 `````` Lemma set_Exists_union_1 X Y : set_Exists X → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 551 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 552 `````` Lemma set_Exists_union_2 X Y : set_Exists Y → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Feb 17, 2016 553 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed May 07, 2013 554 555 `````` Lemma set_Exists_union_inv X Y : set_Exists (X ∪ Y) → set_Exists X ∨ set_Exists Y. `````` Robbert Krebbers committed Feb 17, 2016 556 `````` Proof. unfold set_Exists. set_solver. Qed. `````` Robbert Krebbers committed Jun 11, 2012 557 558 ``````End quantifiers. `````` Robbert Krebbers committed Aug 21, 2012 559 ``````Section more_quantifiers. `````` Robbert Krebbers committed Jun 05, 2014 560 `````` Context `{SimpleCollection A B}. `````` Robbert Krebbers committed Aug 29, 2012 561 `````` `````` Robbert Krebbers committed May 07, 2013 562 563 564 565 566 567 `````` 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 568 569 ``````End more_quantifiers. `````` Robbert Krebbers committed Aug 29, 2012 570 571 572 ``````(** * 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 573 574 575 576 577 578 579 580 581 582 ``````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 583 `````` `````` Robbert Krebbers committed Mar 02, 2015 584 585 ``````Section fresh. Context `{FreshSpec A C}. `````` Robbert Krebbers committed Nov 18, 2015 586 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Aug 21, 2012 587 `````` `````` Robbert Krebbers committed Nov 18, 2015 588 `````` Global Instance fresh_proper: Proper ((≡) ==> (=)) (fresh (C:=C)). `````` Robbert Krebbers committed Feb 08, 2015 589 `````` Proof. intros ???. by apply fresh_proper_alt, elem_of_equiv. Qed. `````` Robbert Krebbers committed Nov 18, 2015 590 591 `````` Global Instance fresh_list_proper: Proper ((=) ==> (≡) ==> (=)) (fresh_list (C:=C)). `````` Robbert Krebbers committed Aug 29, 2012 592 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 593 `````` intros ? n ->. induction n as [|n IH]; intros ?? E; f_equal/=; [by rewrite E|]. `````` Robbert Krebbers committed May 02, 2014 594 `````` apply IH. by rewrite E. `````` Robbert Krebbers committed Aug 29, 2012 595 `````` Qed. `````` Robbert Krebbers committed Mar 02, 2015 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 `````` 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 611 612 `````` Lemma Forall_fresh_subseteq X Y xs : Forall_fresh X xs → Y ⊆ X → Forall_fresh Y xs. `````` Robbert Krebbers committed Feb 17, 2016 613 `````` Proof. rewrite !Forall_fresh_alt; set_solver. Qed. `````` Robbert Krebbers committed Mar 02, 2015 614 `````` `````` Robbert Krebbers committed Aug 21, 2012 615 616 `````` 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 617 `````` Lemma fresh_list_is_fresh n X x : x ∈ fresh_list n X → x ∉ X. `````` Robbert Krebbers committed Aug 21, 2012 618 `````` Proof. `````` Robbert Krebbers committed Mar 02, 2015 619 `````` revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|]. `````` Robbert Krebbers committed May 02, 2014 620 `````` rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|]. `````` Robbert Krebbers committed Feb 17, 2016 621 `````` apply IH in Hin; set_solver. `````` Robbert Krebbers committed Aug 21, 2012 622 `````` Qed. `````` Robbert Krebbers committed Mar 02, 2015 623 `````` Lemma NoDup_fresh_list n X : NoDup (fresh_list n X). `````` Robbert Krebbers committed Aug 21, 2012 624 `````` Proof. `````` Robbert Krebbers committed May 07, 2013 625 `````` revert X. induction n; simpl; constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 626 `````` intros Hin; apply fresh_list_is_fresh in Hin; set_solver. `````` Robbert Krebbers committed Mar 02, 2015 627 628 629 630 `````` 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 631 632 `````` Qed. End fresh. `````` Robbert Krebbers committed Nov 12, 2012 633 `````` `````` Robbert Krebbers committed Feb 19, 2013 634 ``````(** * Properties of implementations of collections that form a monad *) `````` Robbert Krebbers committed Nov 12, 2012 635 636 637 ``````Section collection_monad. Context `{CollectionMonad M}. `````` Ralf Jung committed Feb 15, 2016 638 639 `````` Global Instance collection_fmap_mono {A B} : Proper (pointwise_relation _ (=) ==> (⊆) ==> (⊆)) (@fmap M _ A B). `````` Robbert Krebbers committed Feb 24, 2016 640 `````` Proof. intros f g ? X Y ?; set_solver by eauto. Qed. `````` Robbert Krebbers committed Nov 16, 2015 641 642 `````` Global Instance collection_fmap_proper {A B} : Proper (pointwise_relation _ (=) ==> (≡) ==> (≡)) (@fmap M _ A B). `````` Robbert Krebbers committed Feb 24, 2016 643 `````` Proof. intros f g ? X Y [??]; split; set_solver by eauto. Qed. `````` Ralf Jung committed Feb 15, 2016 644 645 `````` Global Instance collection_bind_mono {A B} : Proper (((=) ==> (⊆)) ==> (⊆) ==> (⊆)) (@mbind M _ A B). `````` Robbert Krebbers committed Feb 17, 2016 646 `````` Proof. unfold respectful; intros f g Hfg X Y ?; set_solver. Qed. `````` Robbert Krebbers committed Nov 16, 2015 647 648 `````` Global Instance collection_bind_proper {A B} : Proper (((=) ==> (≡)) ==> (≡) ==> (≡)) (@mbind M _ A B). `````` Robbert Krebbers committed Feb 17, 2016 649 `````` Proof. unfold respectful; intros f g Hfg X Y [??]; split; set_solver. Qed. `````` Ralf Jung committed Feb 15, 2016 650 651 `````` Global Instance collection_join_mono {A} : Proper ((⊆) ==> (⊆)) (@mjoin M _ A). `````` Robbert Krebbers committed Feb 17, 2016 652 `````` Proof. intros X Y ?; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 653 654 `````` Global Instance collection_join_proper {A} : Proper ((≡) ==> (≡)) (@mjoin M _ A). `````` Robbert Krebbers committed Feb 17, 2016 655 `````` Proof. intros X Y [??]; split; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 656 `````` `````` Robbert Krebbers committed Jan 27, 2015 657 `````` Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x. `````` Robbert Krebbers committed Feb 17, 2016 658 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Jan 27, 2015 659 `````` Lemma collection_guard_True {A} `{Decision P} (X : M A) : P → guard P; X ≡ X. `````` Robbert Krebbers committed Feb 17, 2016 660 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 18, 2015 661 `````` Lemma collection_fmap_compose {A B C} (f : A → B) (g : B → C) (X : M A) : `````` Robbert Krebbers committed Nov 12, 2012 662 `````` g ∘ f <\$> X ≡ g <\$> (f <\$> X). `````` Robbert Krebbers committed Feb 17, 2016 663 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 664 665 `````` 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 666 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 667 668 `````` 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 669 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 670 671 `````` 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 672 `````` Proof. set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 673 674 675 676 677 `````` 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 678 `````` - revert l. induction k; set_solver by eauto. `````` Robbert Krebbers committed Feb 17, 2016 679 `````` - induction 1; set_solver. `````` Robbert Krebbers committed Nov 12, 2012 680 `````` Qed. `````` Robbert Krebbers committed Jun 17, 2013 681 `````` Lemma collection_mapM_length {A B} (f : A → M B) l k : `````` Robbert Krebbers committed Nov 12, 2012 682 `````` l ∈ mapM f k → length l = length k. `````` Robbert Krebbers committed Feb 24, 2016 683 `````` Proof. revert l; induction k; set_solver by eauto. Qed. `````` Robbert Krebbers committed Nov 12, 2012 684 `````` Lemma elem_of_mapM_fmap {A B} (f : A → B) (g : B → M A) l k : `````` Robbert Krebbers committed May 07, 2013 685 `````` 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 686 `````` Proof. intros Hl. revert k. induction Hl; set_solver. Qed. `````` Robbert Krebbers committed Nov 12, 2012 687 `````` Lemma elem_of_mapM_Forall {A B} (f : A → M B) (P : B → Prop) l k : `````` Robbert Krebbers committed May 07, 2013 688 `````` l ∈ mapM f k → Forall (λ x, ∀ y, y ∈ f x → P y) k → Forall P l. `````` Robbert Krebbers committed Jan 05, 2013 689 `````` Proof. rewrite elem_of_mapM. apply Forall2_Forall_l. Qed. `````` Robbert Krebbers committed May 07, 2013 690 691 `````` 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 692 693 694 695 696 `````` 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 697 ``````End collection_monad. `````` Robbert Krebbers committed Dec 11, 2015 698 699 700 701 702 703 `````` (** 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 704 705 `````` Global Instance set_finite_subseteq : Proper (flip (⊆) ==> impl) (@set_finite A B _). `````` Robbert Krebbers committed Feb 17, 2016 706 `````` Proof. intros X Y HX [l Hl]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Jan 16, 2016 707 708 `````` 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 709 710 711 `````` 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 712 `````` Proof. exists [x]; intros y ->%elem_of_singleton; left. Qed. `````` Robbert Krebbers committed Dec 11, 2015 713 714 715 716 717 718 `````` 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 719 `````` Proof. intros [l ?]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Dec 11, 2015 720 `````` Lemma union_finite_inv_r X Y : set_finite (X ∪ Y) → set_finite Y. `````` Robbert Krebbers committed Feb 17, 2016 721 `````` Proof. intros [l ?]; exists l; set_solver. Qed. `````` Robbert Krebbers committed Dec 11, 2015 722 723 724 725 726 ``````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 727 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. `````` Robbert Krebbers committed Dec 11, 2015 728 `````` Lemma intersection_finite_r X Y : set_finite Y → set_finite (X ∩ Y). `````` Ralf Jung committed Jan 04, 2016 729 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. `````` Robbert Krebbers committed Dec 11, 2015 730 `````` Lemma difference_finite X Y : set_finite X → set_finite (X ∖ Y). `````` Ralf Jung committed Jan 04, 2016 731 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_difference; auto. Qed. `````` Robbert Krebbers committed Jan 16, 2016 732 733 734 735 `````` 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 736 `````` intros x ?; destruct (decide (x ∈ Y)); rewrite elem_of_app; set_solver. `````` Robbert Krebbers committed Jan 16, 2016 737 `````` Qed. `````` Robbert Krebbers committed Dec 11, 2015 738 ``End more_finite.``