orders.v 24.3 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file collects common properties of pre-orders and semi lattices. This theory will mainly be used for the theory on collections and finite maps. *) `````` Robbert Krebbers committed Feb 13, 2016 5 ``````From Coq Require Export Sorted. `````` Robbert Krebbers committed Mar 03, 2016 6 ``````From prelude Require Export tactics list. `````` Robbert Krebbers committed Nov 11, 2015 7 8 9 10 11 12 13 14 15 16 17 18 19 `````` (** * Arbitrary pre-, parial and total orders *) (** Properties about arbitrary pre-, partial, and total orders. We do not use the relation [⊆] because we often have multiple orders on the same structure *) Section orders. Context {A} {R : relation A}. Implicit Types X Y : A. Infix "⊆" := R. Notation "X ⊈ Y" := (¬X ⊆ Y). Infix "⊂" := (strict R). Lemma reflexive_eq `{!Reflexive R} X Y : X = Y → X ⊆ Y. Proof. by intros <-. Qed. `````` Robbert Krebbers committed Feb 11, 2016 20 21 `````` Lemma anti_symm_iff `{!PartialOrder R} X Y : X = Y ↔ R X Y ∧ R Y X. Proof. split. by intros ->. by intros [??]; apply (anti_symm _). Qed. `````` Robbert Krebbers committed Nov 11, 2015 22 23 24 25 26 27 28 29 30 31 `````` Lemma strict_spec X Y : X ⊂ Y ↔ X ⊆ Y ∧ Y ⊈ X. Proof. done. Qed. Lemma strict_include X Y : X ⊂ Y → X ⊆ Y. Proof. by intros [? _]. Qed. Lemma strict_ne X Y : X ⊂ Y → X ≠ Y. Proof. by intros [??] <-. Qed. Lemma strict_ne_sym X Y : X ⊂ Y → Y ≠ X. Proof. by intros [??] <-. Qed. Lemma strict_transitive_l `{!Transitive R} X Y Z : X ⊂ Y → Y ⊆ Z → X ⊂ Z. Proof. `````` Ralf Jung committed Feb 20, 2016 32 33 `````` intros [? HXY] ?. split; [by trans Y|]. contradict HXY. by trans Z. `````` Robbert Krebbers committed Nov 11, 2015 34 35 36 `````` Qed. Lemma strict_transitive_r `{!Transitive R} X Y Z : X ⊆ Y → Y ⊂ Z → X ⊂ Z. Proof. `````` Ralf Jung committed Feb 20, 2016 37 38 `````` intros ? [? HYZ]. split; [by trans Y|]. contradict HYZ. by trans X. `````` Robbert Krebbers committed Nov 11, 2015 39 40 41 42 43 44 45 46 47 48 `````` Qed. Global Instance: Irreflexive (strict R). Proof. firstorder. Qed. Global Instance: Transitive R → StrictOrder (strict R). Proof. split; try apply _. eauto using strict_transitive_r, strict_include. Qed. Global Instance preorder_subset_dec_slow `{∀ X Y, Decision (X ⊆ Y)} (X Y : A) : Decision (X ⊂ Y) | 100 := _. `````` Robbert Krebbers committed Feb 11, 2016 49 `````` Lemma strict_spec_alt `{!AntiSymm (=) R} X Y : X ⊂ Y ↔ X ⊆ Y ∧ X ≠ Y. `````` Robbert Krebbers committed Nov 11, 2015 50 51 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 52 53 `````` - intros [? HYX]. split. done. by intros <-. - intros [? HXY]. split. done. by contradict HXY; apply (anti_symm R). `````` Robbert Krebbers committed Nov 11, 2015 54 55 56 57 58 `````` Qed. Lemma po_eq_dec `{!PartialOrder R, ∀ X Y, Decision (X ⊆ Y)} (X Y : A) : Decision (X = Y). Proof. refine (cast_if_and (decide (X ⊆ Y)) (decide (Y ⊆ X))); `````` Robbert Krebbers committed Feb 11, 2016 59 `````` abstract (rewrite anti_symm_iff; tauto). `````` Robbert Krebbers committed Nov 11, 2015 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 `````` Defined. Lemma total_not `{!Total R} X Y : X ⊈ Y → Y ⊆ X. Proof. intros. destruct (total R X Y); tauto. Qed. Lemma total_not_strict `{!Total R} X Y : X ⊈ Y → Y ⊂ X. Proof. red; auto using total_not. Qed. Global Instance trichotomy_total `{!Trichotomy (strict R), !Reflexive R} : Total R. Proof. intros X Y. destruct (trichotomy (strict R) X Y) as [[??]|[<-|[??]]]; intuition. Qed. End orders. Section strict_orders. Context {A} {R : relation A}. Implicit Types X Y : A. Infix "⊂" := R. Lemma irreflexive_eq `{!Irreflexive R} X Y : X = Y → ¬X ⊂ Y. Proof. intros ->. apply (irreflexivity R). Qed. `````` Robbert Krebbers committed Feb 11, 2016 80 `````` Lemma strict_anti_symm `{!StrictOrder R} X Y : `````` Robbert Krebbers committed Nov 11, 2015 81 `````` X ⊂ Y → Y ⊂ X → False. `````` Ralf Jung committed Feb 20, 2016 82 `````` Proof. intros. apply (irreflexivity R X). by trans Y. Qed. `````` Robbert Krebbers committed Nov 11, 2015 83 84 85 86 87 `````` Global Instance trichotomyT_dec `{!TrichotomyT R, !StrictOrder R} X Y : Decision (X ⊂ Y) := match trichotomyT R X Y with | inleft (left H) => left H | inleft (right H) => right (irreflexive_eq _ _ H) `````` Robbert Krebbers committed Feb 11, 2016 88 `````` | inright H => right (strict_anti_symm _ _ H) `````` Robbert Krebbers committed Nov 11, 2015 89 90 91 92 93 94 95 `````` end. Global Instance trichotomyT_trichotomy `{!TrichotomyT R} : Trichotomy R. Proof. intros X Y. destruct (trichotomyT R X Y) as [[|]|]; tauto. Qed. End strict_orders. Ltac simplify_order := repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 96 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Nov 11, 2015 97 98 99 100 `````` | H : ?R ?x ?x |- _ => by destruct (irreflexivity _ _ H) | H1 : ?R ?x ?y |- _ => match goal with | H2 : R y x |- _ => `````` Robbert Krebbers committed Feb 11, 2016 101 `````` assert (x = y) by (by apply (anti_symm R)); clear H1 H2 `````` Robbert Krebbers committed Nov 11, 2015 102 103 `````` | H2 : R y ?z |- _ => unless (R x z) by done; `````` Ralf Jung committed Feb 20, 2016 104 `````` assert (R x z) by (by trans y) `````` Robbert Krebbers committed Nov 11, 2015 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 `````` end end. (** * Sorting *) (** Merge sort. Adapted from the implementation of Hugo Herbelin in the Coq standard library, but without using the module system. *) Section merge_sort. Context {A} (R : relation A) `{∀ x y, Decision (R x y)}. Fixpoint list_merge (l1 : list A) : list A → list A := fix list_merge_aux l2 := match l1, l2 with | [], _ => l2 | _, [] => l1 | x1 :: l1, x2 :: l2 => if decide_rel R x1 x2 then x1 :: list_merge l1 (x2 :: l2) else x2 :: list_merge_aux l2 end. Global Arguments list_merge !_ !_ /. Local Notation stack := (list (option (list A))). Fixpoint merge_list_to_stack (st : stack) (l : list A) : stack := match st with | [] => [Some l] | None :: st => Some l :: st | Some l' :: st => None :: merge_list_to_stack st (list_merge l' l) end. Fixpoint merge_stack (st : stack) : list A := match st with | [] => [] | None :: st => merge_stack st | Some l :: st => list_merge l (merge_stack st) end. Fixpoint merge_sort_aux (st : stack) (l : list A) : list A := match l with | [] => merge_stack st | x :: l => merge_sort_aux (merge_list_to_stack st [x]) l end. Definition merge_sort : list A → list A := merge_sort_aux []. End merge_sort. (** ** Properties of the [Sorted] and [StronglySorted] predicate *) Section sorted. Context {A} (R : relation A). Lemma Sorted_StronglySorted `{!Transitive R} l : Sorted R l → StronglySorted R l. Proof. by apply Sorted.Sorted_StronglySorted. Qed. `````` Robbert Krebbers committed Feb 11, 2016 153 `````` Lemma StronglySorted_unique `{!AntiSymm (=) R} l1 l2 : `````` Robbert Krebbers committed Nov 11, 2015 154 155 156 157 158 159 160 161 162 163 `````` StronglySorted R l1 → StronglySorted R l2 → l1 ≡ₚ l2 → l1 = l2. Proof. intros Hl1; revert l2. induction Hl1 as [|x1 l1 ? IH Hx1]; intros l2 Hl2 E. { symmetry. by apply Permutation_nil. } destruct Hl2 as [|x2 l2 ? Hx2]. { by apply Permutation_nil in E. } assert (x1 = x2); subst. { rewrite Forall_forall in Hx1, Hx2. assert (x2 ∈ x1 :: l1) as Hx2' by (by rewrite E; left). assert (x1 ∈ x2 :: l2) as Hx1' by (by rewrite <-E; left). `````` Robbert Krebbers committed Feb 17, 2016 164 `````` inversion Hx1'; inversion Hx2'; simplify_eq; auto. } `````` Robbert Krebbers committed Feb 11, 2016 165 `````` f_equal. by apply IH, (inj (x2 ::)). `````` Robbert Krebbers committed Nov 11, 2015 166 `````` Qed. `````` Robbert Krebbers committed Feb 11, 2016 167 `````` Lemma Sorted_unique `{!Transitive R, !AntiSymm (=) R} l1 l2 : `````` Robbert Krebbers committed Nov 11, 2015 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 `````` Sorted R l1 → Sorted R l2 → l1 ≡ₚ l2 → l1 = l2. Proof. auto using StronglySorted_unique, Sorted_StronglySorted. Qed. Global Instance HdRel_dec x `{∀ y, Decision (R x y)} l : Decision (HdRel R x l). Proof. refine match l with | [] => left _ | y :: l => cast_if (decide (R x y)) end; abstract first [by constructor | by inversion 1]. Defined. Global Instance Sorted_dec `{∀ x y, Decision (R x y)} : ∀ l, Decision (Sorted R l). Proof. refine (fix go l := match l return Decision (Sorted R l) with | [] => left _ | x :: l => cast_if_and (decide (HdRel R x l)) (go l) end); clear go; abstract first [by constructor | by inversion 1]. Defined. Global Instance StronglySorted_dec `{∀ x y, Decision (R x y)} : ∀ l, Decision (StronglySorted R l). Proof. refine (fix go l := match l return Decision (StronglySorted R l) with | [] => left _ | x :: l => cast_if_and (decide (Forall (R x) l)) (go l) end); clear go; abstract first [by constructor | by inversion 1]. Defined. Context {B} (f : A → B). Lemma HdRel_fmap (R1 : relation A) (R2 : relation B) x l : (∀ y, R1 x y → R2 (f x) (f y)) → HdRel R1 x l → HdRel R2 (f x) (f <\$> l). Proof. destruct 2; constructor; auto. Qed. Lemma Sorted_fmap (R1 : relation A) (R2 : relation B) l : (∀ x y, R1 x y → R2 (f x) (f y)) → Sorted R1 l → Sorted R2 (f <\$> l). Proof. induction 2; simpl; constructor; eauto using HdRel_fmap. Qed. Lemma StronglySorted_fmap (R1 : relation A) (R2 : relation B) l : (∀ x y, R1 x y → R2 (f x) (f y)) → StronglySorted R1 l → StronglySorted R2 (f <\$> l). Proof. induction 2; csimpl; constructor; rewrite ?Forall_fmap; eauto using Forall_impl. Qed. End sorted. (** ** Correctness of merge sort *) Section merge_sort_correct. Context {A} (R : relation A) `{∀ x y, Decision (R x y)} `{!Total R}. Lemma list_merge_cons x1 x2 l1 l2 : list_merge R (x1 :: l1) (x2 :: l2) = if decide (R x1 x2) then x1 :: list_merge R l1 (x2 :: l2) else x2 :: list_merge R (x1 :: l1) l2. Proof. done. Qed. Lemma HdRel_list_merge x l1 l2 : HdRel R x l1 → HdRel R x l2 → HdRel R x (list_merge R l1 l2). Proof. destruct 1 as [|x1 l1 IH1], 1 as [|x2 l2 IH2]; rewrite ?list_merge_cons; simpl; repeat case_decide; auto. Qed. Lemma Sorted_list_merge l1 l2 : Sorted R l1 → Sorted R l2 → Sorted R (list_merge R l1 l2). Proof. intros Hl1. revert l2. induction Hl1 as [|x1 l1 IH1]; induction 1 as [|x2 l2 IH2]; rewrite ?list_merge_cons; simpl; repeat case_decide; constructor; eauto using HdRel_list_merge, HdRel_cons, total_not. Qed. Lemma merge_Permutation l1 l2 : list_merge R l1 l2 ≡ₚ l1 ++ l2. Proof. revert l2. induction l1 as [|x1 l1 IH1]; intros l2; induction l2 as [|x2 l2 IH2]; rewrite ?list_merge_cons; simpl; repeat case_decide; auto. `````` Robbert Krebbers committed Feb 17, 2016 245 246 `````` - by rewrite (right_id_L [] (++)). - by rewrite IH2, Permutation_middle. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` Qed. Local Notation stack := (list (option (list A))). Inductive merge_stack_Sorted : stack → Prop := | merge_stack_Sorted_nil : merge_stack_Sorted [] | merge_stack_Sorted_cons_None st : merge_stack_Sorted st → merge_stack_Sorted (None :: st) | merge_stack_Sorted_cons_Some l st : Sorted R l → merge_stack_Sorted st → merge_stack_Sorted (Some l :: st). Fixpoint merge_stack_flatten (st : stack) : list A := match st with | [] => [] | None :: st => merge_stack_flatten st | Some l :: st => l ++ merge_stack_flatten st end. Lemma Sorted_merge_list_to_stack st l : merge_stack_Sorted st → Sorted R l → merge_stack_Sorted (merge_list_to_stack R st l). Proof. intros Hst. revert l. induction Hst; repeat constructor; naive_solver auto using Sorted_list_merge. Qed. Lemma merge_list_to_stack_Permutation st l : merge_stack_flatten (merge_list_to_stack R st l) ≡ₚ l ++ merge_stack_flatten st. Proof. revert l. induction st as [|[l'|] st IH]; intros l; simpl; auto. `````` Robbert Krebbers committed Feb 11, 2016 275 `````` by rewrite IH, merge_Permutation, (assoc_L _), (comm (++) l). `````` Robbert Krebbers committed Nov 11, 2015 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 `````` Qed. Lemma Sorted_merge_stack st : merge_stack_Sorted st → Sorted R (merge_stack R st). Proof. induction 1; simpl; auto using Sorted_list_merge. Qed. Lemma merge_stack_Permutation st : merge_stack R st ≡ₚ merge_stack_flatten st. Proof. induction st as [|[] ? IH]; intros; simpl; auto. by rewrite merge_Permutation, IH. Qed. Lemma Sorted_merge_sort_aux st l : merge_stack_Sorted st → Sorted R (merge_sort_aux R st l). Proof. revert st. induction l; simpl; auto using Sorted_merge_stack, Sorted_merge_list_to_stack. Qed. Lemma merge_sort_aux_Permutation st l : merge_sort_aux R st l ≡ₚ merge_stack_flatten st ++ l. Proof. revert st. induction l as [|?? IH]; simpl; intros. `````` Robbert Krebbers committed Feb 17, 2016 295 296 `````` - by rewrite (right_id_L [] (++)), merge_stack_Permutation. - rewrite IH, merge_list_to_stack_Permutation; simpl. `````` Robbert Krebbers committed Nov 11, 2015 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 `````` by rewrite Permutation_middle. Qed. Lemma Sorted_merge_sort l : Sorted R (merge_sort R l). Proof. apply Sorted_merge_sort_aux. by constructor. Qed. Lemma merge_sort_Permutation l : merge_sort R l ≡ₚ l. Proof. unfold merge_sort. by rewrite merge_sort_aux_Permutation. Qed. Lemma StronglySorted_merge_sort `{!Transitive R} l : StronglySorted R (merge_sort R l). Proof. auto using Sorted_StronglySorted, Sorted_merge_sort. Qed. End merge_sort_correct. (** * Canonical pre and partial orders *) (** We extend the canonical pre-order [⊆] to a partial order by defining setoid equality as [λ X Y, X ⊆ Y ∧ Y ⊆ X]. We prove that this indeed gives rise to a setoid. *) `````` Robbert Krebbers committed Nov 18, 2015 312 ``````Instance preorder_equiv `{SubsetEq A} : Equiv A | 20 := λ X Y, X ⊆ Y ∧ Y ⊆ X. `````` Robbert Krebbers committed Nov 11, 2015 313 314 315 316 317 318 319 `````` Section preorder. Context `{SubsetEq A, !PreOrder (@subseteq A _)}. Instance preorder_equivalence: @Equivalence A (≡). Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 320 321 `````` - done. - by intros ?? [??]. `````` Ralf Jung committed Feb 20, 2016 322 `````` - by intros X Y Z [??] [??]; split; trans Y. `````` Robbert Krebbers committed Nov 11, 2015 323 `````` Qed. `````` Robbert Krebbers committed Nov 18, 2015 324 `````` Global Instance: Proper ((≡) ==> (≡) ==> iff) ((⊆) : relation A). `````` Robbert Krebbers committed Nov 11, 2015 325 326 `````` Proof. unfold equiv, preorder_equiv. intros X1 Y1 ? X2 Y2 ?. split; intro. `````` Ralf Jung committed Feb 20, 2016 327 328 `````` - trans X1. tauto. trans X2; tauto. - trans Y1. tauto. trans Y2; tauto. `````` Robbert Krebbers committed Nov 11, 2015 329 `````` Qed. `````` Robbert Krebbers committed Nov 18, 2015 330 `````` Lemma subset_spec (X Y : A) : X ⊂ Y ↔ X ⊆ Y ∧ X ≢ Y. `````` Robbert Krebbers committed Nov 11, 2015 331 332 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 333 334 `````` - intros [? HYX]. split. done. contradict HYX. by rewrite <-HYX. - intros [? HXY]. split. done. by contradict HXY. `````` Robbert Krebbers committed Nov 11, 2015 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 `````` Qed. Section dec. Context `{∀ X Y : A, Decision (X ⊆ Y)}. Global Instance preorder_equiv_dec_slow (X Y : A) : Decision (X ≡ Y) | 100 := _. Lemma subseteq_inv X Y : X ⊆ Y → X ⊂ Y ∨ X ≡ Y. Proof. rewrite subset_spec. destruct (decide (X ≡ Y)); tauto. Qed. Lemma not_subset_inv X Y : X ⊄ Y → X ⊈ Y ∨ X ≡ Y. Proof. rewrite subset_spec. destruct (decide (X ≡ Y)); tauto. Qed. End dec. Section leibniz. Context `{!LeibnizEquiv A}. Lemma subset_spec_L X Y : X ⊂ Y ↔ X ⊆ Y ∧ X ≠ Y. Proof. unfold_leibniz. apply subset_spec. Qed. Context `{∀ X Y : A, Decision (X ⊆ Y)}. Lemma subseteq_inv_L X Y : X ⊆ Y → X ⊂ Y ∨ X = Y. Proof. unfold_leibniz. apply subseteq_inv. Qed. Lemma not_subset_inv_L X Y : X ⊄ Y → X ⊈ Y ∨ X = Y. Proof. unfold_leibniz. apply not_subset_inv. Qed. End leibniz. End preorder. Typeclasses Opaque preorder_equiv. Hint Extern 0 (@Equivalence _ (≡)) => class_apply preorder_equivalence : typeclass_instances. (** * Partial orders *) Section partial_order. Context `{SubsetEq A, !PartialOrder (@subseteq A _)}. Global Instance: LeibnizEquiv A. `````` Robbert Krebbers committed Feb 11, 2016 367 `````` Proof. intros ?? [??]; by apply (anti_symm (⊆)). Qed. `````` Robbert Krebbers committed Nov 11, 2015 368 369 370 371 372 373 374 375 376 377 378 ``````End partial_order. (** * Join semi lattices *) (** General purpose theorems on join semi lattices. *) Section join_semi_lattice. Context `{Empty A, JoinSemiLattice A, !EmptySpec A}. Implicit Types X Y : A. Implicit Types Xs Ys : list A. Hint Resolve subseteq_empty union_subseteq_l union_subseteq_r union_least. Lemma union_subseteq_l_transitive X1 X2 Y : X1 ⊆ X2 → X1 ⊆ X2 ∪ Y. `````` Ralf Jung committed Feb 20, 2016 379 `````` Proof. intros. trans X2; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 380 `````` Lemma union_subseteq_r_transitive X1 X2 Y : X1 ⊆ X2 → X1 ⊆ Y ∪ X2. `````` Ralf Jung committed Feb 20, 2016 381 `````` Proof. intros. trans X2; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 382 383 384 385 386 387 388 389 390 `````` Hint Resolve union_subseteq_l_transitive union_subseteq_r_transitive. Lemma union_preserving_l X Y1 Y2 : Y1 ⊆ Y2 → X ∪ Y1 ⊆ X ∪ Y2. Proof. auto. Qed. Lemma union_preserving_r X1 X2 Y : X1 ⊆ X2 → X1 ∪ Y ⊆ X2 ∪ Y. Proof. auto. Qed. Lemma union_preserving X1 X2 Y1 Y2 : X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ∪ Y1 ⊆ X2 ∪ Y2. Proof. auto. Qed. Lemma union_empty X : X ∪ ∅ ⊆ X. Proof. by apply union_least. Qed. `````` Robbert Krebbers committed Nov 18, 2015 391 `````` Global Instance union_proper : Proper ((≡) ==> (≡) ==> (≡)) (@union A _). `````` Robbert Krebbers committed Nov 11, 2015 392 393 394 395 `````` Proof. unfold equiv, preorder_equiv. split; apply union_preserving; simpl in *; tauto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 396 `````` Global Instance: IdemP ((≡) : relation A) (∪). `````` Robbert Krebbers committed Nov 11, 2015 397 `````` Proof. split; eauto. Qed. `````` Robbert Krebbers committed Nov 18, 2015 398 `````` Global Instance: LeftId ((≡) : relation A) ∅ (∪). `````` Robbert Krebbers committed Nov 11, 2015 399 `````` Proof. split; eauto. Qed. `````` Robbert Krebbers committed Nov 18, 2015 400 `````` Global Instance: RightId ((≡) : relation A) ∅ (∪). `````` Robbert Krebbers committed Nov 11, 2015 401 `````` Proof. split; eauto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 402 `````` Global Instance: Comm ((≡) : relation A) (∪). `````` Robbert Krebbers committed Nov 11, 2015 403 `````` Proof. split; auto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 404 `````` Global Instance: Assoc ((≡) : relation A) (∪). `````` Robbert Krebbers committed Nov 11, 2015 405 406 407 408 409 410 411 412 413 `````` Proof. split; auto. Qed. Lemma subseteq_union X Y : X ⊆ Y ↔ X ∪ Y ≡ Y. Proof. repeat split; eauto. intros HXY. rewrite <-HXY. auto. Qed. Lemma subseteq_union_1 X Y : X ⊆ Y → X ∪ Y ≡ Y. Proof. apply subseteq_union. Qed. Lemma subseteq_union_2 X Y : X ∪ Y ≡ Y → X ⊆ Y. Proof. apply subseteq_union. Qed. Lemma equiv_empty X : X ⊆ ∅ → X ≡ ∅. Proof. split; eauto. Qed. `````` Robbert Krebbers committed Nov 18, 2015 414 415 `````` Global Instance union_list_proper: Proper ((≡) ==> (≡)) (union_list (A:=A)). Proof. by induction 1; simpl; try apply union_proper. Qed. `````` Robbert Krebbers committed Nov 11, 2015 416 417 418 419 420 421 422 423 424 `````` Lemma union_list_nil : ⋃ @nil A = ∅. Proof. done. Qed. Lemma union_list_cons X Xs : ⋃ (X :: Xs) = X ∪ ⋃ Xs. Proof. done. Qed. Lemma union_list_singleton X : ⋃ [X] ≡ X. Proof. simpl. by rewrite (right_id ∅ _). Qed. Lemma union_list_app Xs1 Xs2 : ⋃ (Xs1 ++ Xs2) ≡ ⋃ Xs1 ∪ ⋃ Xs2. Proof. induction Xs1 as [|X Xs1 IH]; simpl; [by rewrite (left_id ∅ _)|]. `````` Robbert Krebbers committed Feb 11, 2016 425 `````` by rewrite IH, (assoc _). `````` Robbert Krebbers committed Nov 11, 2015 426 427 428 429 430 `````` Qed. Lemma union_list_reverse Xs : ⋃ (reverse Xs) ≡ ⋃ Xs. Proof. induction Xs as [|X Xs IH]; simpl; [done |]. by rewrite reverse_cons, union_list_app, `````` Robbert Krebbers committed Feb 11, 2016 431 `````` union_list_singleton, (comm _), IH. `````` Robbert Krebbers committed Nov 11, 2015 432 433 434 435 436 437 `````` Qed. Lemma union_list_preserving Xs Ys : Xs ⊆* Ys → ⋃ Xs ⊆ ⋃ Ys. Proof. induction 1; simpl; auto using union_preserving. Qed. Lemma empty_union X Y : X ∪ Y ≡ ∅ ↔ X ≡ ∅ ∧ Y ≡ ∅. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 438 `````` - intros HXY. split; apply equiv_empty; `````` Ralf Jung committed Feb 20, 2016 439 `````` by trans (X ∪ Y); [auto | rewrite HXY]. `````` Robbert Krebbers committed Feb 17, 2016 440 `````` - intros [HX HY]. by rewrite HX, HY, (left_id _ _). `````` Robbert Krebbers committed Nov 11, 2015 441 442 443 444 `````` Qed. Lemma empty_union_list Xs : ⋃ Xs ≡ ∅ ↔ Forall (≡ ∅) Xs. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 445 446 `````` - induction Xs; simpl; rewrite ?empty_union; intuition. - induction 1 as [|?? E1 ? E2]; simpl. done. by apply empty_union. `````` Robbert Krebbers committed Nov 11, 2015 447 448 449 450 `````` Qed. Section leibniz. Context `{!LeibnizEquiv A}. `````` Robbert Krebbers committed Feb 11, 2016 451 452 `````` Global Instance: IdemP (=) (∪). Proof. intros ?. unfold_leibniz. apply (idemp _). Qed. `````` Robbert Krebbers committed Nov 11, 2015 453 454 455 456 `````` Global Instance: LeftId (=) ∅ (∪). Proof. intros ?. unfold_leibniz. apply (left_id _ _). Qed. Global Instance: RightId (=) ∅ (∪). Proof. intros ?. unfold_leibniz. apply (right_id _ _). Qed. `````` Robbert Krebbers committed Feb 11, 2016 457 458 459 460 `````` Global Instance: Comm (=) (∪). Proof. intros ??. unfold_leibniz. apply (comm _). Qed. Global Instance: Assoc (=) (∪). Proof. intros ???. unfold_leibniz. apply (assoc _). Qed. `````` Robbert Krebbers committed Nov 11, 2015 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 `````` Lemma subseteq_union_L X Y : X ⊆ Y ↔ X ∪ Y = Y. Proof. unfold_leibniz. apply subseteq_union. Qed. Lemma subseteq_union_1_L X Y : X ⊆ Y → X ∪ Y = Y. Proof. unfold_leibniz. apply subseteq_union_1. Qed. Lemma subseteq_union_2_L X Y : X ∪ Y = Y → X ⊆ Y. Proof. unfold_leibniz. apply subseteq_union_2. Qed. Lemma equiv_empty_L X : X ⊆ ∅ → X = ∅. Proof. unfold_leibniz. apply equiv_empty. Qed. Lemma union_list_singleton_L (X : A) : ⋃ [X] = X. Proof. unfold_leibniz. apply union_list_singleton. Qed. Lemma union_list_app_L (Xs1 Xs2 : list A) : ⋃ (Xs1 ++ Xs2) = ⋃ Xs1 ∪ ⋃ Xs2. Proof. unfold_leibniz. apply union_list_app. Qed. Lemma union_list_reverse_L (Xs : list A) : ⋃ (reverse Xs) = ⋃ Xs. Proof. unfold_leibniz. apply union_list_reverse. Qed. Lemma empty_union_L X Y : X ∪ Y = ∅ ↔ X = ∅ ∧ Y = ∅. Proof. unfold_leibniz. apply empty_union. Qed. Lemma empty_union_list_L Xs : ⋃ Xs = ∅ ↔ Forall (= ∅) Xs. Proof. unfold_leibniz. by rewrite empty_union_list. Qed. End leibniz. Section dec. Context `{∀ X Y : A, Decision (X ⊆ Y)}. Lemma non_empty_union X Y : X ∪ Y ≢ ∅ ↔ X ≢ ∅ ∨ Y ≢ ∅. Proof. rewrite empty_union. destruct (decide (X ≡ ∅)); intuition. Qed. Lemma non_empty_union_list Xs : ⋃ Xs ≢ ∅ → Exists (≢ ∅) Xs. Proof. rewrite empty_union_list. apply (not_Forall_Exists _). Qed. Context `{!LeibnizEquiv A}. Lemma non_empty_union_L X Y : X ∪ Y ≠ ∅ ↔ X ≠ ∅ ∨ Y ≠ ∅. Proof. unfold_leibniz. apply non_empty_union. Qed. Lemma non_empty_union_list_L Xs : ⋃ Xs ≠ ∅ → Exists (≠ ∅) Xs. Proof. unfold_leibniz. apply non_empty_union_list. Qed. End dec. End join_semi_lattice. (** * Meet semi lattices *) (** The dual of the above section, but now for meet semi lattices. *) Section meet_semi_lattice. Context `{MeetSemiLattice A}. Implicit Types X Y : A. Implicit Types Xs Ys : list A. Hint Resolve intersection_subseteq_l intersection_subseteq_r intersection_greatest. Lemma intersection_subseteq_l_transitive X1 X2 Y : X1 ⊆ X2 → X1 ∩ Y ⊆ X2. `````` Ralf Jung committed Feb 20, 2016 505 `````` Proof. intros. trans X1; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 506 `````` Lemma intersection_subseteq_r_transitive X1 X2 Y : X1 ⊆ X2 → Y ∩ X1 ⊆ X2. `````` Ralf Jung committed Feb 20, 2016 507 `````` Proof. intros. trans X1; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 508 509 510 511 512 513 514 515 516 `````` Hint Resolve intersection_subseteq_l_transitive intersection_subseteq_r_transitive. Lemma intersection_preserving_l X Y1 Y2 : Y1 ⊆ Y2 → X ∩ Y1 ⊆ X ∩ Y2. Proof. auto. Qed. Lemma intersection_preserving_r X1 X2 Y : X1 ⊆ X2 → X1 ∩ Y ⊆ X2 ∩ Y. Proof. auto. Qed. Lemma intersection_preserving X1 X2 Y1 Y2 : X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ∩ Y1 ⊆ X2 ∩ Y2. Proof. auto. Qed. `````` Robbert Krebbers committed Nov 18, 2015 517 `````` Global Instance: Proper ((≡) ==> (≡) ==> (≡)) (@intersection A _). `````` Robbert Krebbers committed Nov 11, 2015 518 519 520 521 `````` Proof. unfold equiv, preorder_equiv. split; apply intersection_preserving; simpl in *; tauto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 522 `````` Global Instance: IdemP ((≡) : relation A) (∩). `````` Robbert Krebbers committed Nov 11, 2015 523 `````` Proof. split; eauto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 524 `````` Global Instance: Comm ((≡) : relation A) (∩). `````` Robbert Krebbers committed Nov 11, 2015 525 `````` Proof. split; auto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 526 `````` Global Instance: Assoc ((≡) : relation A) (∩). `````` Robbert Krebbers committed Nov 11, 2015 527 528 529 530 531 532 533 534 535 536 `````` Proof. split; auto. Qed. Lemma subseteq_intersection X Y : X ⊆ Y ↔ X ∩ Y ≡ X. Proof. repeat split; eauto. intros HXY. rewrite <-HXY. auto. Qed. Lemma subseteq_intersection_1 X Y : X ⊆ Y → X ∩ Y ≡ X. Proof. apply subseteq_intersection. Qed. Lemma subseteq_intersection_2 X Y : X ∩ Y ≡ X → X ⊆ Y. Proof. apply subseteq_intersection. Qed. Section leibniz. Context `{!LeibnizEquiv A}. `````` Robbert Krebbers committed Feb 11, 2016 537 538 539 540 541 542 `````` Global Instance: IdemP (=) (∩). Proof. intros ?. unfold_leibniz. apply (idemp _). Qed. Global Instance: Comm (=) (∩). Proof. intros ??. unfold_leibniz. apply (comm _). Qed. Global Instance: Assoc (=) (∩). Proof. intros ???. unfold_leibniz. apply (assoc _). Qed. `````` Robbert Krebbers committed Nov 11, 2015 543 544 545 546 547 548 549 550 551 552 553 554 555 `````` Lemma subseteq_intersection_L X Y : X ⊆ Y ↔ X ∩ Y = X. Proof. unfold_leibniz. apply subseteq_intersection. Qed. Lemma subseteq_intersection_1_L X Y : X ⊆ Y → X ∩ Y = X. Proof. unfold_leibniz. apply subseteq_intersection_1. Qed. Lemma subseteq_intersection_2_L X Y : X ∩ Y = X → X ⊆ Y. Proof. unfold_leibniz. apply subseteq_intersection_2. Qed. End leibniz. End meet_semi_lattice. (** * Lower bounded lattices *) Section lattice. Context `{Empty A, Lattice A, !EmptySpec A}. `````` Robbert Krebbers committed Nov 18, 2015 556 `````` Global Instance: LeftAbsorb ((≡) : relation A) ∅ (∩). `````` Robbert Krebbers committed Nov 11, 2015 557 `````` Proof. split. by apply intersection_subseteq_l. by apply subseteq_empty. Qed. `````` Robbert Krebbers committed Nov 18, 2015 558 `````` Global Instance: RightAbsorb ((≡) : relation A) ∅ (∩). `````` Robbert Krebbers committed Feb 11, 2016 559 `````` Proof. intros ?. by rewrite (comm _), (left_absorb _ _). Qed. `````` Robbert Krebbers committed Jan 16, 2016 560 `````` Lemma union_intersection_l (X Y Z : A) : X ∪ (Y ∩ Z) ≡ (X ∪ Y) ∩ (X ∪ Z). `````` Robbert Krebbers committed Nov 11, 2015 561 `````` Proof. `````` Robbert Krebbers committed Jan 16, 2016 562 `````` split; [apply union_least|apply lattice_distr]. `````` Robbert Krebbers committed Nov 11, 2015 563 564 `````` { apply intersection_greatest; auto using union_subseteq_l. } apply intersection_greatest. `````` Robbert Krebbers committed Feb 17, 2016 565 566 `````` - apply union_subseteq_r_transitive, intersection_subseteq_l. - apply union_subseteq_r_transitive, intersection_subseteq_r. `````` Robbert Krebbers committed Nov 11, 2015 567 `````` Qed. `````` Robbert Krebbers committed Jan 16, 2016 568 `````` Lemma union_intersection_r (X Y Z : A) : (X ∩ Y) ∪ Z ≡ (X ∪ Z) ∩ (Y ∪ Z). `````` Robbert Krebbers committed Feb 11, 2016 569 `````` Proof. by rewrite !(comm _ _ Z), union_intersection_l. Qed. `````` Robbert Krebbers committed Jan 16, 2016 570 `````` Lemma intersection_union_l (X Y Z : A) : X ∩ (Y ∪ Z) ≡ (X ∩ Y) ∪ (X ∩ Z). `````` Robbert Krebbers committed Nov 11, 2015 571 `````` Proof. `````` Robbert Krebbers committed Jan 16, 2016 572 `````` split. `````` Robbert Krebbers committed Feb 17, 2016 573 `````` - rewrite union_intersection_l. `````` Robbert Krebbers committed Nov 11, 2015 574 575 `````` apply intersection_greatest. { apply union_subseteq_r_transitive, intersection_subseteq_l. } `````` Robbert Krebbers committed Jan 16, 2016 576 `````` rewrite union_intersection_r. `````` Robbert Krebbers committed Nov 11, 2015 577 `````` apply intersection_preserving; auto using union_subseteq_l. `````` Robbert Krebbers committed Feb 17, 2016 578 `````` - apply intersection_greatest. `````` Robbert Krebbers committed Nov 11, 2015 579 580 581 582 583 `````` { apply union_least; auto using intersection_subseteq_l. } apply union_least. + apply intersection_subseteq_r_transitive, union_subseteq_l. + apply intersection_subseteq_r_transitive, union_subseteq_r. Qed. `````` Robbert Krebbers committed Jan 16, 2016 584 `````` Lemma intersection_union_r (X Y Z : A) : (X ∪ Y) ∩ Z ≡ (X ∩ Z) ∪ (Y ∩ Z). `````` Robbert Krebbers committed Feb 11, 2016 585 `````` Proof. by rewrite !(comm _ _ Z), intersection_union_l. Qed. `````` Robbert Krebbers committed Nov 11, 2015 586 587 588 589 590 591 592 `````` Section leibniz. Context `{!LeibnizEquiv A}. Global Instance: LeftAbsorb (=) ∅ (∩). Proof. intros ?. unfold_leibniz. apply (left_absorb _ _). Qed. Global Instance: RightAbsorb (=) ∅ (∩). Proof. intros ?. unfold_leibniz. apply (right_absorb _ _). Qed. `````` Robbert Krebbers committed Jan 16, 2016 593 594 595 596 597 598 599 600 `````` Lemma union_intersection_l_L (X Y Z : A) : X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z). Proof. unfold_leibniz; apply union_intersection_l. Qed. Lemma union_intersection_r_L (X Y Z : A) : (X ∩ Y) ∪ Z = (X ∪ Z) ∩ (Y ∪ Z). Proof. unfold_leibniz; apply union_intersection_r. Qed. Lemma intersection_union_l_L (X Y Z : A) : X ∩ (Y ∪ Z) ≡ (X ∩ Y) ∪ (X ∩ Z). Proof. unfold_leibniz; apply intersection_union_l. Qed. Lemma intersection_union_r_L (X Y Z : A) : (X ∪ Y) ∩ Z ≡ (X ∩ Z) ∪ (Y ∩ Z). Proof. unfold_leibniz; apply intersection_union_r. Qed. `````` Robbert Krebbers committed Nov 11, 2015 601 602 `````` End leibniz. End lattice.``````