collections.v 28 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 5 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file collects definitions and theorems on collections. Most importantly, it implements some tactics to automatically solve goals involving collections. *) `````` Robbert Krebbers committed Feb 13, 2016 6 ``````From prelude Require Export base tactics orders. `````` Robbert Krebbers committed Nov 11, 2015 7 8 9 10 11 12 13 `````` Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y, ∀ x, x ∈ X → x ∈ Y. (** * Basic theorems *) Section simple_collection. Context `{SimpleCollection A C}. `````` Robbert Krebbers committed Nov 18, 2015 14 15 `````` Implicit Types x y : A. Implicit Types X Y : C. `````` Robbert Krebbers committed Nov 11, 2015 16 17 18 19 20 21 22 23 24 25 26 `````` Lemma elem_of_empty x : x ∈ ∅ ↔ False. Proof. split. apply not_elem_of_empty. done. Qed. Lemma elem_of_union_l x X Y : x ∈ X → x ∈ X ∪ Y. Proof. intros. apply elem_of_union. auto. Qed. Lemma elem_of_union_r x X Y : x ∈ Y → x ∈ X ∪ Y. Proof. intros. apply elem_of_union. auto. Qed. Global Instance: EmptySpec C. Proof. firstorder auto. Qed. Global Instance: JoinSemiLattice C. Proof. firstorder auto. Qed. `````` Robbert Krebbers committed Feb 15, 2016 27 28 `````` Global Instance: AntiSymm (≡) (@collection_subseteq A C _). Proof. done. Qed. `````` Robbert Krebbers committed Nov 11, 2015 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 `````` Lemma elem_of_subseteq X Y : X ⊆ Y ↔ ∀ x, x ∈ X → x ∈ Y. Proof. done. Qed. Lemma elem_of_equiv X Y : X ≡ Y ↔ ∀ x, x ∈ X ↔ x ∈ Y. Proof. firstorder. Qed. Lemma elem_of_equiv_alt X Y : X ≡ Y ↔ (∀ x, x ∈ X → x ∈ Y) ∧ (∀ x, x ∈ Y → x ∈ X). Proof. firstorder. Qed. Lemma elem_of_equiv_empty X : X ≡ ∅ ↔ ∀ x, x ∉ X. Proof. firstorder. Qed. Lemma collection_positive_l X Y : X ∪ Y ≡ ∅ → X ≡ ∅. Proof. rewrite !elem_of_equiv_empty. setoid_rewrite elem_of_union. naive_solver. Qed. Lemma collection_positive_l_alt X Y : X ≢ ∅ → X ∪ Y ≢ ∅. Proof. eauto using collection_positive_l. Qed. Lemma elem_of_singleton_1 x y : x ∈ {[y]} → x = y. Proof. by rewrite elem_of_singleton. Qed. Lemma elem_of_singleton_2 x y : x = y → x ∈ {[y]}. Proof. by rewrite elem_of_singleton. Qed. Lemma elem_of_subseteq_singleton x X : x ∈ X ↔ {[ x ]} ⊆ X. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 51 52 `````` - intros ??. rewrite elem_of_singleton. by intros ->. - intros Ex. by apply (Ex x), elem_of_singleton. `````` Robbert Krebbers committed Nov 11, 2015 53 `````` Qed. `````` Robbert Krebbers committed Nov 18, 2015 54 `````` Global Instance singleton_proper : Proper ((=) ==> (≡)) (singleton (B:=C)). `````` Robbert Krebbers committed Nov 11, 2015 55 `````` Proof. by repeat intro; subst. Qed. `````` Robbert Krebbers committed Nov 18, 2015 56 57 `````` Global Instance elem_of_proper : Proper ((=) ==> (≡) ==> iff) ((∈) : A → C → Prop) | 5. `````` Robbert Krebbers committed Nov 11, 2015 58 59 60 61 `````` Proof. intros ???; subst. firstorder. Qed. Lemma elem_of_union_list Xs x : x ∈ ⋃ Xs ↔ ∃ X, X ∈ Xs ∧ x ∈ X. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 62 `````` - induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|]. `````` Robbert Krebbers committed Nov 11, 2015 63 `````` setoid_rewrite elem_of_cons. apply elem_of_union in HXs. naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 64 `````` - intros [X []]. induction 1; simpl; [by apply elem_of_union_l |]. `````` Robbert Krebbers committed Nov 11, 2015 65 66 `````` intros. apply elem_of_union_r; auto. Qed. `````` Robbert Krebbers committed Nov 18, 2015 67 `````` Lemma non_empty_singleton x : ({[ x ]} : C) ≢ ∅. `````` Robbert Krebbers committed Nov 11, 2015 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 `````` Proof. intros [E _]. by apply (elem_of_empty x), E, elem_of_singleton. Qed. Lemma not_elem_of_singleton x y : x ∉ {[ y ]} ↔ x ≠ y. Proof. by rewrite elem_of_singleton. Qed. Lemma not_elem_of_union x X Y : x ∉ X ∪ Y ↔ x ∉ X ∧ x ∉ Y. Proof. rewrite elem_of_union. tauto. Qed. Section leibniz. Context `{!LeibnizEquiv C}. Lemma elem_of_equiv_L X Y : X = Y ↔ ∀ x, x ∈ X ↔ x ∈ Y. Proof. unfold_leibniz. apply elem_of_equiv. Qed. Lemma elem_of_equiv_alt_L X Y : X = Y ↔ (∀ x, x ∈ X → x ∈ Y) ∧ (∀ x, x ∈ Y → x ∈ X). Proof. unfold_leibniz. apply elem_of_equiv_alt. Qed. Lemma elem_of_equiv_empty_L X : X = ∅ ↔ ∀ x, x ∉ X. Proof. unfold_leibniz. apply elem_of_equiv_empty. Qed. Lemma collection_positive_l_L X Y : X ∪ Y = ∅ → X = ∅. Proof. unfold_leibniz. apply collection_positive_l. Qed. Lemma collection_positive_l_alt_L X Y : X ≠ ∅ → X ∪ Y ≠ ∅. Proof. unfold_leibniz. apply collection_positive_l_alt. Qed. Lemma non_empty_singleton_L x : {[ x ]} ≠ ∅. Proof. unfold_leibniz. apply non_empty_singleton. Qed. End leibniz. Section dec. Context `{∀ X Y : C, Decision (X ⊆ Y)}. Global Instance elem_of_dec_slow (x : A) (X : C) : Decision (x ∈ X) | 100. Proof. refine (cast_if (decide_rel (⊆) {[ x ]} X)); by rewrite elem_of_subseteq_singleton. Defined. End dec. End simple_collection. Definition of_option `{Singleton A C, Empty C} (x : option A) : C := match x with None => ∅ | Some a => {[ a ]} end. Fixpoint of_list `{Singleton A C, Empty C, Union C} (l : list A) : C := match l with [] => ∅ | x :: l => {[ x ]} ∪ of_list l end. Section of_option_list. Context `{SimpleCollection A C}. Lemma elem_of_of_option (x : A) o : x ∈ of_option o ↔ o = Some x. Proof. destruct o; simpl; rewrite ?elem_of_empty, ?elem_of_singleton; naive_solver. Qed. Lemma elem_of_of_list (x : A) l : x ∈ of_list l ↔ x ∈ l. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 116 `````` - induction l; simpl; [by rewrite elem_of_empty|]. `````` Robbert Krebbers committed Nov 11, 2015 117 `````` rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 118 `````` - induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` Qed. End of_option_list. Global Instance collection_guard `{CollectionMonad M} : MGuard M := λ P dec A x, match dec with left H => x H | _ => ∅ end. Section collection_monad_base. Context `{CollectionMonad M}. Lemma elem_of_guard `{Decision P} {A} (x : A) (X : M A) : x ∈ guard P; X ↔ P ∧ x ∈ X. Proof. unfold mguard, collection_guard; simpl; case_match; rewrite ?elem_of_empty; naive_solver. Qed. Lemma elem_of_guard_2 `{Decision P} {A} (x : A) (X : M A) : P → x ∈ X → x ∈ guard P; X. Proof. by rewrite elem_of_guard. Qed. Lemma guard_empty `{Decision P} {A} (X : M A) : guard P; X ≡ ∅ ↔ ¬P ∨ X ≡ ∅. Proof. rewrite !elem_of_equiv_empty; setoid_rewrite elem_of_guard. destruct (decide P); naive_solver. Qed. Lemma bind_empty {A B} (f : A → M B) X : X ≫= f ≡ ∅ ↔ X ≡ ∅ ∨ ∀ x, x ∈ X → f x ≡ ∅. Proof. setoid_rewrite elem_of_equiv_empty; setoid_rewrite elem_of_bind. naive_solver. Qed. End collection_monad_base. (** * Tactics *) (** Given a hypothesis [H : _ ∈ _], the tactic [destruct_elem_of H] will recursively split [H] for [(∪)], [(∩)], [(∖)], [map], [∅], [{[_]}]. *) Tactic Notation "decompose_elem_of" hyp(H) := let rec go H := lazymatch type of H with | _ ∈ ∅ => apply elem_of_empty in H; destruct H | ?x ∈ {[ ?y ]} => apply elem_of_singleton in H; try first [subst y | subst x] | ?x ∉ {[ ?y ]} => apply not_elem_of_singleton in H | _ ∈ _ ∪ _ => apply elem_of_union in H; destruct H as [H|H]; [go H|go H] | _ ∉ _ ∪ _ => let H1 := fresh H in let H2 := fresh H in apply not_elem_of_union in H; destruct H as [H1 H2]; go H1; go H2 | _ ∈ _ ∩ _ => let H1 := fresh H in let H2 := fresh H in apply elem_of_intersection in H; destruct H as [H1 H2]; go H1; go H2 | _ ∈ _ ∖ _ => let H1 := fresh H in let H2 := fresh H in apply elem_of_difference in H; destruct H as [H1 H2]; go H1; go H2 | ?x ∈ _ <\$> _ => apply elem_of_fmap in H; destruct H as [? [? H]]; try (subst x); go H | _ ∈ _ ≫= _ => let H1 := fresh H in let H2 := fresh H in apply elem_of_bind in H; destruct H as [? [H1 H2]]; go H1; go H2 | ?x ∈ mret ?y => apply elem_of_ret in H; try first [subst y | subst x] | _ ∈ mjoin _ ≫= _ => let H1 := fresh H in let H2 := fresh H in apply elem_of_join in H; destruct H as [? [H1 H2]]; go H1; go H2 | _ ∈ guard _; _ => let H1 := fresh H in let H2 := fresh H in apply elem_of_guard in H; destruct H as [H1 H2]; go H2 | _ ∈ of_option _ => apply elem_of_of_option in H | _ ∈ of_list _ => apply elem_of_of_list in H | _ => idtac end in go H. Tactic Notation "decompose_elem_of" := repeat_on_hyps (fun H => decompose_elem_of H). Ltac decompose_empty := repeat match goal with | H : ∅ ≡ ∅ |- _ => clear H | H : ∅ = ∅ |- _ => clear H | H : ∅ ≡ _ |- _ => symmetry in H | H : ∅ = _ |- _ => symmetry in H | H : _ ∪ _ ≡ ∅ |- _ => apply empty_union in H; destruct H | H : _ ∪ _ ≢ ∅ |- _ => apply non_empty_union in H; destruct H | H : {[ _ ]} ≡ ∅ |- _ => destruct (non_empty_singleton _ H) | H : _ ∪ _ = ∅ |- _ => apply empty_union_L in H; destruct H | H : _ ∪ _ ≠ ∅ |- _ => apply non_empty_union_L in H; destruct H | H : {[ _ ]} = ∅ |- _ => destruct (non_empty_singleton_L _ H) | H : guard _ ; _ ≡ ∅ |- _ => apply guard_empty in H; destruct H end. (** The first pass of our collection tactic consists of eliminating all occurrences of [(∪)], [(∩)], [(∖)], [(<\$>)], [∅], [{[_]}], [(≡)], and [(⊆)], by rewriting these into logically equivalent propositions. For example we rewrite [A → x ∈ X ∪ ∅] into [A → x ∈ X ∨ False]. *) Ltac unfold_elem_of := repeat_on_hyps (fun H => repeat match type of H with | context [ _ ⊆ _ ] => setoid_rewrite elem_of_subseteq in H | context [ _ ⊂ _ ] => setoid_rewrite subset_spec in H | context [ _ ≡ ∅ ] => setoid_rewrite elem_of_equiv_empty in H | context [ _ ≡ _ ] => setoid_rewrite elem_of_equiv_alt in H | context [ _ = ∅ ] => setoid_rewrite elem_of_equiv_empty_L in H | context [ _ = _ ] => setoid_rewrite elem_of_equiv_alt_L in H | context [ _ ∈ ∅ ] => setoid_rewrite elem_of_empty in H | context [ _ ∈ {[ _ ]} ] => setoid_rewrite elem_of_singleton in H | context [ _ ∈ _ ∪ _ ] => setoid_rewrite elem_of_union in H | context [ _ ∈ _ ∩ _ ] => setoid_rewrite elem_of_intersection in H | context [ _ ∈ _ ∖ _ ] => setoid_rewrite elem_of_difference in H | context [ _ ∈ _ <\$> _ ] => setoid_rewrite elem_of_fmap in H | context [ _ ∈ mret _ ] => setoid_rewrite elem_of_ret in H | context [ _ ∈ _ ≫= _ ] => setoid_rewrite elem_of_bind in H | context [ _ ∈ mjoin _ ] => setoid_rewrite elem_of_join in H | context [ _ ∈ guard _; _ ] => setoid_rewrite elem_of_guard in H | context [ _ ∈ of_option _ ] => setoid_rewrite elem_of_of_option in H | context [ _ ∈ of_list _ ] => setoid_rewrite elem_of_of_list in H end); repeat match goal with | |- context [ _ ⊆ _ ] => setoid_rewrite elem_of_subseteq | |- context [ _ ⊂ _ ] => setoid_rewrite subset_spec | |- context [ _ ≡ ∅ ] => setoid_rewrite elem_of_equiv_empty | |- context [ _ ≡ _ ] => setoid_rewrite elem_of_equiv_alt | |- context [ _ = ∅ ] => setoid_rewrite elem_of_equiv_empty_L | |- context [ _ = _ ] => setoid_rewrite elem_of_equiv_alt_L | |- context [ _ ∈ ∅ ] => setoid_rewrite elem_of_empty | |- context [ _ ∈ {[ _ ]} ] => setoid_rewrite elem_of_singleton | |- context [ _ ∈ _ ∪ _ ] => setoid_rewrite elem_of_union | |- context [ _ ∈ _ ∩ _ ] => setoid_rewrite elem_of_intersection | |- context [ _ ∈ _ ∖ _ ] => setoid_rewrite elem_of_difference | |- context [ _ ∈ _ <\$> _ ] => setoid_rewrite elem_of_fmap | |- context [ _ ∈ mret _ ] => setoid_rewrite elem_of_ret | |- context [ _ ∈ _ ≫= _ ] => setoid_rewrite elem_of_bind | |- context [ _ ∈ mjoin _ ] => setoid_rewrite elem_of_join | |- context [ _ ∈ guard _; _ ] => setoid_rewrite elem_of_guard | |- context [ _ ∈ of_option _ ] => setoid_rewrite elem_of_of_option | |- context [ _ ∈ of_list _ ] => setoid_rewrite elem_of_of_list end. `````` Robbert Krebbers committed Jan 16, 2016 253 254 255 ``````(** Since [firstorder] fails or loops on very small goals generated by [solve_elem_of] already. We use the [naive_solver] tactic as a substitute. This tactic either fails or proves the goal. *) `````` Robbert Krebbers committed Nov 11, 2015 256 257 258 259 260 ``````Tactic Notation "solve_elem_of" tactic3(tac) := setoid_subst; decompose_empty; unfold_elem_of; naive_solver tac. `````` Robbert Krebbers committed Jan 16, 2016 261 262 263 264 265 266 267 268 ``````Tactic Notation "solve_elem_of" "-" hyp_list(Hs) "/" tactic3(tac) := clear Hs; solve_elem_of tac. Tactic Notation "solve_elem_of" "+" hyp_list(Hs) "/" tactic3(tac) := revert Hs; clear; solve_elem_of tac. Tactic Notation "solve_elem_of" := solve_elem_of eauto. Tactic Notation "solve_elem_of" "-" hyp_list(Hs) := clear Hs; solve_elem_of. Tactic Notation "solve_elem_of" "+" hyp_list(Hs) := revert Hs; clear; solve_elem_of. `````` Robbert Krebbers committed Nov 18, 2015 269 `````` `````` Robbert Krebbers committed Nov 11, 2015 270 271 272 ``````(** * More theorems *) Section collection. Context `{Collection A C}. `````` Robbert Krebbers committed Nov 18, 2015 273 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Nov 11, 2015 274 275 276 `````` Global Instance: Lattice C. Proof. split. apply _. firstorder auto. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 18, 2015 277 278 `````` Global Instance difference_proper : Proper ((≡) ==> (≡) ==> (≡)) (@difference C _). `````` Robbert Krebbers committed Nov 11, 2015 279 280 281 282 `````` 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 283 284 `````` Lemma non_empty_inhabited x X : x ∈ X → X ≢ ∅. Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 18, 2015 285 `````` Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}. `````` Robbert Krebbers committed Jan 16, 2016 286 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 287 `````` Lemma difference_twice X Y : (X ∖ Y) ∖ Y ≡ X ∖ Y. `````` Robbert Krebbers committed Jan 16, 2016 288 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 289 `````` Lemma subseteq_empty_difference X Y : X ⊆ Y → X ∖ Y ≡ ∅. `````` Robbert Krebbers committed Jan 16, 2016 290 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 291 `````` Lemma difference_diag X : X ∖ X ≡ ∅. `````` Robbert Krebbers committed Jan 16, 2016 292 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 293 `````` Lemma difference_union_distr_l X Y Z : (X ∪ Y) ∖ Z ≡ X ∖ Z ∪ Y ∖ Z. `````` Robbert Krebbers committed Jan 16, 2016 294 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 295 `````` Lemma difference_union_distr_r X Y Z : Z ∖ (X ∪ Y) ≡ (Z ∖ X) ∩ (Z ∖ Y). `````` Robbert Krebbers committed Jan 16, 2016 296 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 297 `````` Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z. `````` Robbert Krebbers committed Jan 16, 2016 298 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 20, 2015 299 `````` Lemma disjoint_union_difference X Y : X ∩ Y ≡ ∅ → (X ∪ Y) ∖ X ≡ Y. `````` Robbert Krebbers committed Jan 16, 2016 300 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 `````` Section leibniz. Context `{!LeibnizEquiv C}. Lemma intersection_singletons_L x : {[x]} ∩ {[x]} = {[x]}. Proof. unfold_leibniz. apply intersection_singletons. Qed. Lemma difference_twice_L X Y : (X ∖ Y) ∖ Y = X ∖ Y. Proof. unfold_leibniz. apply difference_twice. Qed. Lemma subseteq_empty_difference_L X Y : X ⊆ Y → X ∖ Y = ∅. Proof. unfold_leibniz. apply subseteq_empty_difference. Qed. Lemma difference_diag_L X : X ∖ X = ∅. Proof. unfold_leibniz. apply difference_diag. Qed. Lemma difference_union_distr_l_L X Y Z : (X ∪ Y) ∖ Z = X ∖ Z ∪ Y ∖ Z. Proof. unfold_leibniz. apply difference_union_distr_l. Qed. Lemma difference_union_distr_r_L X Y Z : Z ∖ (X ∪ Y) = (Z ∖ X) ∩ (Z ∖ Y). Proof. unfold_leibniz. apply difference_union_distr_r. Qed. Lemma difference_intersection_distr_l_L X Y Z : (X ∩ Y) ∖ Z = X ∖ Z ∩ Y ∖ Z. Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed. `````` Robbert Krebbers committed Nov 20, 2015 319 320 `````` Lemma disjoint_union_difference_L X Y : X ∩ Y = ∅ → (X ∪ Y) ∖ X = Y. Proof. unfold_leibniz. apply disjoint_union_difference. Qed. `````` Robbert Krebbers committed Nov 11, 2015 321 322 323 `````` End leibniz. Section dec. `````` Robbert Krebbers committed Nov 17, 2015 324 `````` Context `{∀ (x : A) (X : C), Decision (x ∈ X)}. `````` Robbert Krebbers committed Nov 11, 2015 325 326 327 328 329 330 331 332 333 334 335 336 `````` Lemma not_elem_of_intersection x X Y : x ∉ X ∩ Y ↔ x ∉ X ∨ x ∉ Y. Proof. rewrite elem_of_intersection. destruct (decide (x ∈ X)); tauto. Qed. Lemma not_elem_of_difference x X Y : x ∉ X ∖ Y ↔ x ∉ X ∨ x ∈ Y. Proof. rewrite elem_of_difference. destruct (decide (x ∈ Y)); tauto. Qed. Lemma union_difference X Y : X ⊆ Y → Y ≡ X ∪ Y ∖ X. Proof. split; intros x; rewrite !elem_of_union, elem_of_difference; [|intuition]. destruct (decide (x ∈ X)); intuition. Qed. Lemma non_empty_difference X Y : X ⊂ Y → Y ∖ X ≢ ∅. Proof. intros [HXY1 HXY2] Hdiff. destruct HXY2. intros x. `````` Robbert Krebbers committed Jan 16, 2016 337 `````` destruct (decide (x ∈ X)); solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 338 339 `````` Qed. Lemma empty_difference_subseteq X Y : X ∖ Y ≡ ∅ → X ⊆ Y. `````` Robbert Krebbers committed Jan 16, 2016 340 `````` Proof. intros ? x ?; apply dec_stable; solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 `````` Context `{!LeibnizEquiv C}. Lemma union_difference_L X Y : X ⊆ Y → Y = X ∪ Y ∖ X. Proof. unfold_leibniz. apply union_difference. Qed. Lemma non_empty_difference_L X Y : X ⊂ Y → Y ∖ X ≠ ∅. Proof. unfold_leibniz. apply non_empty_difference. Qed. Lemma empty_difference_subseteq_L X Y : X ∖ Y = ∅ → X ⊆ Y. Proof. unfold_leibniz. apply empty_difference_subseteq. Qed. End dec. End collection. Section collection_ops. Context `{CollectionOps A C}. Lemma elem_of_intersection_with_list (f : A → A → option A) Xs Y x : x ∈ intersection_with_list f Y Xs ↔ ∃ xs y, Forall2 (∈) xs Xs ∧ y ∈ Y ∧ foldr (λ x, (≫= f x)) (Some y) xs = Some x. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 359 `````` - revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|]. `````` Robbert Krebbers committed Nov 11, 2015 360 361 `````` 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 362 `````` eexists (x1 :: xs), y. intuition (simplify_option_eq; auto). `````` Robbert Krebbers committed Feb 17, 2016 363 `````` - intros (xs & y & Hxs & ? & Hx). revert x Hx. `````` Robbert Krebbers committed Feb 17, 2016 364 `````` induction Hxs; intros; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Nov 11, 2015 365 366 367 368 369 370 371 372 373 `````` 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 374 `````` intros HY HXs Hf. induction Xs; simplify_option_eq; [done |]. `````` Robbert Krebbers committed Nov 11, 2015 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 `````` intros x Hx. rewrite elem_of_intersection_with in Hx. decompose_Forall. destruct Hx as (? & ? & ? & ? & ?). eauto. Qed. End collection_ops. (** * Sets without duplicates up to an equivalence *) Section NoDup. Context `{SimpleCollection A B} (R : relation A) `{!Equivalence R}. Definition elem_of_upto (x : A) (X : B) := ∃ y, y ∈ X ∧ R x y. Definition set_NoDup (X : B) := ∀ x y, x ∈ X → y ∈ X → R x y → x = y. Global Instance: Proper ((≡) ==> iff) (elem_of_upto x). Proof. intros ??? E. unfold elem_of_upto. by setoid_rewrite E. Qed. Global Instance: Proper (R ==> (≡) ==> iff) elem_of_upto. Proof. intros ?? E1 ?? E2. split; intros [z [??]]; exists z. `````` Robbert Krebbers committed Feb 17, 2016 392 393 `````` - rewrite <-E1, <-E2; intuition. - rewrite E1, E2; intuition. `````` Robbert Krebbers committed Nov 11, 2015 394 395 396 397 398 `````` Qed. Global Instance: Proper ((≡) ==> iff) set_NoDup. Proof. firstorder. Qed. Lemma elem_of_upto_elem_of x X : x ∈ X → elem_of_upto x X. `````` Robbert Krebbers committed Jan 16, 2016 399 `````` Proof. unfold elem_of_upto. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 400 `````` Lemma elem_of_upto_empty x : ¬elem_of_upto x ∅. `````` Robbert Krebbers committed Jan 16, 2016 401 `````` Proof. unfold elem_of_upto. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 402 `````` Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]} ↔ R x y. `````` Robbert Krebbers committed Jan 16, 2016 403 `````` Proof. unfold elem_of_upto. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 404 405 406 `````` 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 Jan 16, 2016 407 `````` Proof. unfold elem_of_upto. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 408 `````` Lemma not_elem_of_upto x X : ¬elem_of_upto x X → ∀ y, y ∈ X → ¬R x y. `````` Robbert Krebbers committed Jan 16, 2016 409 `````` Proof. unfold elem_of_upto. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 410 411 412 413 414 `````` Lemma set_NoDup_empty: set_NoDup ∅. Proof. unfold set_NoDup. solve_elem_of. Qed. Lemma set_NoDup_add x X : ¬elem_of_upto x X → set_NoDup X → set_NoDup ({[ x ]} ∪ X). `````` Robbert Krebbers committed Jan 16, 2016 415 `````` Proof. unfold set_NoDup, elem_of_upto. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 `````` Lemma set_NoDup_inv_add x X : x ∉ X → set_NoDup ({[ x ]} ∪ X) → ¬elem_of_upto x X. Proof. intros Hin Hnodup [y [??]]. rewrite (Hnodup x y) in Hin; solve_elem_of. Qed. Lemma set_NoDup_inv_union_l X Y : set_NoDup (X ∪ Y) → set_NoDup X. Proof. unfold set_NoDup. solve_elem_of. Qed. Lemma set_NoDup_inv_union_r X Y : set_NoDup (X ∪ Y) → set_NoDup Y. Proof. unfold set_NoDup. solve_elem_of. Qed. End NoDup. (** * Quantifiers *) Section quantifiers. Context `{SimpleCollection A B} (P : A → Prop). Definition set_Forall X := ∀ x, x ∈ X → P x. Definition set_Exists X := ∃ x, x ∈ X ∧ P x. Lemma set_Forall_empty : set_Forall ∅. Proof. unfold set_Forall. solve_elem_of. Qed. Lemma set_Forall_singleton x : set_Forall {[ x ]} ↔ P x. Proof. unfold set_Forall. solve_elem_of. Qed. Lemma set_Forall_union X Y : set_Forall X → set_Forall Y → set_Forall (X ∪ Y). Proof. unfold set_Forall. solve_elem_of. Qed. Lemma set_Forall_union_inv_1 X Y : set_Forall (X ∪ Y) → set_Forall X. Proof. unfold set_Forall. solve_elem_of. Qed. Lemma set_Forall_union_inv_2 X Y : set_Forall (X ∪ Y) → set_Forall Y. Proof. unfold set_Forall. solve_elem_of. Qed. Lemma set_Exists_empty : ¬set_Exists ∅. `````` Robbert Krebbers committed Jan 16, 2016 447 `````` Proof. unfold set_Exists. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 448 `````` Lemma set_Exists_singleton x : set_Exists {[ x ]} ↔ P x. `````` Robbert Krebbers committed Jan 16, 2016 449 `````` Proof. unfold set_Exists. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 450 `````` Lemma set_Exists_union_1 X Y : set_Exists X → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Jan 16, 2016 451 `````` Proof. unfold set_Exists. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 452 `````` Lemma set_Exists_union_2 X Y : set_Exists Y → set_Exists (X ∪ Y). `````` Robbert Krebbers committed Jan 16, 2016 453 `````` Proof. unfold set_Exists. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 454 455 `````` Lemma set_Exists_union_inv X Y : set_Exists (X ∪ Y) → set_Exists X ∨ set_Exists Y. `````` Robbert Krebbers committed Jan 16, 2016 456 `````` Proof. unfold set_Exists. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 ``````End quantifiers. Section more_quantifiers. Context `{SimpleCollection A B}. Lemma set_Forall_weaken (P Q : A → Prop) (Hweaken : ∀ x, P x → Q x) X : set_Forall P X → set_Forall Q X. Proof. unfold set_Forall. naive_solver. Qed. Lemma set_Exists_weaken (P Q : A → Prop) (Hweaken : ∀ x, P x → Q x) X : set_Exists P X → set_Exists Q X. Proof. unfold set_Exists. naive_solver. Qed. End more_quantifiers. (** * Fresh elements *) (** We collect some properties on the [fresh] operation. In particular we generalize [fresh] to generate lists of fresh elements. *) Fixpoint fresh_list `{Fresh A C, Union C, Singleton A C} (n : nat) (X : C) : list A := match n with | 0 => [] | S n => let x := fresh X in x :: fresh_list n ({[ x ]} ∪ X) end. Inductive Forall_fresh `{ElemOf A C} (X : C) : list A → Prop := | Forall_fresh_nil : Forall_fresh X [] | Forall_fresh_cons x xs : x ∉ xs → x ∉ X → Forall_fresh X xs → Forall_fresh X (x :: xs). Section fresh. Context `{FreshSpec A C}. `````` Robbert Krebbers committed Nov 18, 2015 486 `````` Implicit Types X Y : C. `````` Robbert Krebbers committed Nov 11, 2015 487 `````` `````` Robbert Krebbers committed Nov 18, 2015 488 `````` Global Instance fresh_proper: Proper ((≡) ==> (=)) (fresh (C:=C)). `````` Robbert Krebbers committed Nov 11, 2015 489 `````` Proof. intros ???. by apply fresh_proper_alt, elem_of_equiv. Qed. `````` Robbert Krebbers committed Nov 18, 2015 490 491 `````` Global Instance fresh_list_proper: Proper ((=) ==> (≡) ==> (=)) (fresh_list (C:=C)). `````` Robbert Krebbers committed Nov 11, 2015 492 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 493 `````` intros ? n ->. induction n as [|n IH]; intros ?? E; f_equal/=; [by rewrite E|]. `````` Robbert Krebbers committed Nov 11, 2015 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 `````` apply IH. by rewrite E. Qed. Lemma Forall_fresh_NoDup X xs : Forall_fresh X xs → NoDup xs. Proof. induction 1; by constructor. Qed. Lemma Forall_fresh_elem_of X xs x : Forall_fresh X xs → x ∈ xs → x ∉ X. Proof. intros HX; revert x; rewrite <-Forall_forall. by induction HX; constructor. Qed. Lemma Forall_fresh_alt X xs : Forall_fresh X xs ↔ NoDup xs ∧ ∀ x, x ∈ xs → x ∉ X. Proof. split; eauto using Forall_fresh_NoDup, Forall_fresh_elem_of. rewrite <-Forall_forall. intros [Hxs Hxs']. induction Hxs; decompose_Forall_hyps; constructor; auto. Qed. Lemma Forall_fresh_subseteq X Y xs : Forall_fresh X xs → Y ⊆ X → Forall_fresh Y xs. `````` Robbert Krebbers committed Jan 16, 2016 513 `````` Proof. rewrite !Forall_fresh_alt; solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 `````` Lemma fresh_list_length n X : length (fresh_list n X) = n. Proof. revert X. induction n; simpl; auto. Qed. Lemma fresh_list_is_fresh n X x : x ∈ fresh_list n X → x ∉ X. Proof. revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|]. rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|]. apply IH in Hin; solve_elem_of. Qed. Lemma NoDup_fresh_list n X : NoDup (fresh_list n X). Proof. revert X. induction n; simpl; constructor; auto. intros Hin; apply fresh_list_is_fresh in Hin; solve_elem_of. Qed. Lemma Forall_fresh_list X n : Forall_fresh X (fresh_list n X). Proof. rewrite Forall_fresh_alt; eauto using NoDup_fresh_list, fresh_list_is_fresh. Qed. End fresh. (** * Properties of implementations of collections that form a monad *) Section collection_monad. Context `{CollectionMonad M}. `````` Ralf Jung committed Feb 15, 2016 538 539 540 `````` Global Instance collection_fmap_mono {A B} : Proper (pointwise_relation _ (=) ==> (⊆) ==> (⊆)) (@fmap M _ A B). Proof. intros f g ? X Y ?; solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 16, 2015 541 542 `````` Global Instance collection_fmap_proper {A B} : Proper (pointwise_relation _ (=) ==> (≡) ==> (≡)) (@fmap M _ A B). `````` Robbert Krebbers committed Jan 16, 2016 543 `````` Proof. intros f g ? X Y [??]; split; solve_elem_of. Qed. `````` Ralf Jung committed Feb 15, 2016 544 545 546 `````` Global Instance collection_bind_mono {A B} : Proper (((=) ==> (⊆)) ==> (⊆) ==> (⊆)) (@mbind M _ A B). Proof. unfold respectful; intros f g Hfg X Y ?; solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 16, 2015 547 548 `````` Global Instance collection_bind_proper {A B} : Proper (((=) ==> (≡)) ==> (≡) ==> (≡)) (@mbind M _ A B). `````` Robbert Krebbers committed Jan 16, 2016 549 `````` Proof. unfold respectful; intros f g Hfg X Y [??]; split; solve_elem_of. Qed. `````` Ralf Jung committed Feb 15, 2016 550 551 552 `````` Global Instance collection_join_mono {A} : Proper ((⊆) ==> (⊆)) (@mjoin M _ A). Proof. intros X Y ?; solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 553 554 `````` Global Instance collection_join_proper {A} : Proper ((≡) ==> (≡)) (@mjoin M _ A). `````` Robbert Krebbers committed Jan 16, 2016 555 `````` Proof. intros X Y [??]; split; solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 556 557 `````` Lemma collection_bind_singleton {A B} (f : A → M B) x : {[ x ]} ≫= f ≡ f x. `````` Robbert Krebbers committed Jan 16, 2016 558 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 559 `````` Lemma collection_guard_True {A} `{Decision P} (X : M A) : P → guard P; X ≡ X. `````` Robbert Krebbers committed Jan 16, 2016 560 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 18, 2015 561 `````` Lemma collection_fmap_compose {A B C} (f : A → B) (g : B → C) (X : M A) : `````` Robbert Krebbers committed Nov 11, 2015 562 `````` g ∘ f <\$> X ≡ g <\$> (f <\$> X). `````` Robbert Krebbers committed Jan 16, 2016 563 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 564 565 `````` 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 Jan 16, 2016 566 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 567 568 `````` Lemma elem_of_fmap_2 {A B} (f : A → B) (X : M A) (x : A) : x ∈ X → f x ∈ f <\$> X. `````` Robbert Krebbers committed Jan 16, 2016 569 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 570 571 `````` 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 Jan 16, 2016 572 `````` Proof. solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 573 574 575 576 577 `````` 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 17, 2016 578 579 `````` - revert l. induction k; solve_elem_of. - induction 1; solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 580 581 582 `````` Qed. Lemma collection_mapM_length {A B} (f : A → M B) l k : l ∈ mapM f k → length l = length k. `````` Robbert Krebbers committed Jan 16, 2016 583 `````` Proof. revert l; induction k; solve_elem_of. Qed. `````` Robbert Krebbers committed Nov 11, 2015 584 585 586 587 `````` Lemma elem_of_mapM_fmap {A B} (f : A → B) (g : B → M A) l k : Forall (λ x, ∀ y, y ∈ g x → f y = x) l → k ∈ mapM g l → fmap f k = l. Proof. intros Hl. revert k. induction Hl; simpl; intros; `````` Robbert Krebbers committed Feb 17, 2016 588 `````` decompose_elem_of; f_equal/=; auto. `````` Robbert Krebbers committed Nov 11, 2015 589 590 591 592 593 594 595 596 597 598 599 600 `````` Qed. Lemma elem_of_mapM_Forall {A B} (f : A → M B) (P : B → Prop) l k : l ∈ mapM f k → Forall (λ x, ∀ y, y ∈ f x → P y) k → Forall P l. Proof. rewrite elem_of_mapM. apply Forall2_Forall_l. Qed. Lemma elem_of_mapM_Forall2_l {A B C} (f : A → M B) (P: B → C → Prop) l1 l2 k : l1 ∈ mapM f k → Forall2 (λ x y, ∀ z, z ∈ f x → P z y) k l2 → Forall2 P l1 l2. Proof. rewrite elem_of_mapM. intros Hl1. revert l2. induction Hl1; inversion_clear 1; constructor; auto. Qed. End collection_monad. `````` Robbert Krebbers committed Dec 11, 2015 601 602 603 604 605 606 `````` (** 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 607 608 609 610 611 `````` Global Instance set_finite_subseteq : Proper (flip (⊆) ==> impl) (@set_finite A B _). Proof. intros X Y HX [l Hl]; exists l; solve_elem_of. Qed. 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 612 613 614 `````` 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 615 `````` Proof. exists [x]; intros y ->%elem_of_singleton; left. Qed. `````` Robbert Krebbers committed Dec 11, 2015 616 617 618 619 620 621 `````` 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 Jan 16, 2016 622 `````` Proof. intros [l ?]; exists l; solve_elem_of. Qed. `````` Robbert Krebbers committed Dec 11, 2015 623 `````` Lemma union_finite_inv_r X Y : set_finite (X ∪ Y) → set_finite Y. `````` Robbert Krebbers committed Jan 16, 2016 624 `````` Proof. intros [l ?]; exists l; solve_elem_of. Qed. `````` Robbert Krebbers committed Dec 11, 2015 625 626 627 628 629 ``````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 630 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. `````` Robbert Krebbers committed Dec 11, 2015 631 `````` Lemma intersection_finite_r X Y : set_finite Y → set_finite (X ∩ Y). `````` Ralf Jung committed Jan 04, 2016 632 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed. `````` Robbert Krebbers committed Dec 11, 2015 633 `````` Lemma difference_finite X Y : set_finite X → set_finite (X ∖ Y). `````` Ralf Jung committed Jan 04, 2016 634 `````` Proof. intros [l ?]; exists l; intros x [??]%elem_of_difference; auto. Qed. `````` Robbert Krebbers committed Jan 16, 2016 635 636 637 638 639 640 `````` 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). intros x ?; destruct (decide (x ∈ Y)); rewrite elem_of_app; solve_elem_of. Qed. `````` Robbert Krebbers committed Dec 11, 2015 641 ``End more_finite.``