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 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 general purpose definitions and theorems on lists that are not in the Coq standard library. *) Require Export Permutation. `````` Robbert Krebbers committed Nov 16, 2015 6 ``````Require Export prelude.numbers prelude.base prelude.decidable prelude.option. `````` Robbert Krebbers committed Nov 11, 2015 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 `````` Arguments length {_} _. Arguments cons {_} _ _. Arguments app {_} _ _. Arguments Permutation {_} _ _. Arguments Forall_cons {_} _ _ _ _ _. Notation tail := tl. Notation take := firstn. Notation drop := skipn. Arguments take {_} !_ !_ /. Arguments drop {_} !_ !_ /. Notation "(::)" := cons (only parsing) : C_scope. Notation "( x ::)" := (cons x) (only parsing) : C_scope. Notation "(:: l )" := (λ x, cons x l) (only parsing) : C_scope. Notation "(++)" := app (only parsing) : C_scope. Notation "( l ++)" := (app l) (only parsing) : C_scope. Notation "(++ k )" := (λ l, app l k) (only parsing) : C_scope. Infix "≡ₚ" := Permutation (at level 70, no associativity) : C_scope. Notation "(≡ₚ)" := Permutation (only parsing) : C_scope. Notation "( x ≡ₚ)" := (Permutation x) (only parsing) : C_scope. Notation "(≡ₚ x )" := (λ y, y ≡ₚ x) (only parsing) : C_scope. Notation "(≢ₚ)" := (λ x y, ¬x ≡ₚ y) (only parsing) : C_scope. Notation "x ≢ₚ y":= (¬x ≡ₚ y) (at level 70, no associativity) : C_scope. Notation "( x ≢ₚ)" := (λ y, x ≢ₚ y) (only parsing) : C_scope. Notation "(≢ₚ x )" := (λ y, y ≢ₚ x) (only parsing) : C_scope. `````` Robbert Krebbers committed Jan 12, 2016 37 38 39 ``````Instance maybe_cons {A} : Maybe2 (@cons A) := λ l, match l with x :: l => Some (x,l) | _ => None end. `````` Robbert Krebbers committed Nov 11, 2015 40 ``````(** * Definitions *) `````` Robbert Krebbers committed Nov 18, 2015 41 42 43 44 45 46 ``````(** Setoid equality lifted to lists *) Inductive list_equiv `{Equiv A} : Equiv (list A) := | nil_equiv : [] ≡ [] | cons_equiv x y l k : x ≡ y → l ≡ k → x :: l ≡ y :: k. Existing Instance list_equiv. `````` Robbert Krebbers committed Nov 11, 2015 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 ``````(** The operation [l !! i] gives the [i]th element of the list [l], or [None] in case [i] is out of bounds. *) Instance list_lookup {A} : Lookup nat A (list A) := fix go i l {struct l} : option A := let _ : Lookup _ _ _ := @go in match l with | [] => None | x :: l => match i with 0 => Some x | S i => l !! i end end. (** The operation [alter f i l] applies the function [f] to the [i]th element of [l]. In case [i] is out of bounds, the list is returned unchanged. *) Instance list_alter {A} : Alter nat A (list A) := λ f, fix go i l {struct l} := match l with | [] => [] | x :: l => match i with 0 => f x :: l | S i => x :: go i l end end. (** The operation [<[i:=x]> l] overwrites the element at position [i] with the value [x]. In case [i] is out of bounds, the list is returned unchanged. *) Instance list_insert {A} : Insert nat A (list A) := fix go i y l {struct l} := let _ : Insert _ _ _ := @go in match l with | [] => [] | x :: l => match i with 0 => y :: l | S i => x :: <[i:=y]>l end end. Fixpoint list_inserts {A} (i : nat) (k l : list A) : list A := match k with | [] => l | y :: k => <[i:=y]>(list_inserts (S i) k l) end. (** The operation [delete i l] removes the [i]th element of [l] and moves all consecutive elements one position ahead. In case [i] is out of bounds, the list is returned unchanged. *) Instance list_delete {A} : Delete nat (list A) := fix go (i : nat) (l : list A) {struct l} : list A := match l with | [] => [] | x :: l => match i with 0 => l | S i => x :: @delete _ _ go i l end end. (** The function [option_list o] converts an element [Some x] into the singleton list [[x]], and [None] into the empty list [[]]. *) Definition option_list {A} : option A → list A := option_rect _ (λ x, [x]) []. Definition list_singleton {A} (l : list A) : option A := match l with [x] => Some x | _ => None end. (** The function [filter P l] returns the list of elements of [l] that satisfies [P]. The order remains unchanged. *) Instance list_filter {A} : Filter A (list A) := fix go P _ l := let _ : Filter _ _ := @go in match l with | [] => [] | x :: l => if decide (P x) then x :: filter P l else filter P l end. (** The function [list_find P l] returns the first index [i] whose element satisfies the predicate [P]. *) Definition list_find {A} P `{∀ x, Decision (P x)} : list A → option (nat * A) := fix go l := match l with | [] => None | x :: l => if decide (P x) then Some (0,x) else prod_map S id <\$> go l end. (** The function [replicate n x] generates a list with length [n] of elements with value [x]. *) Fixpoint replicate {A} (n : nat) (x : A) : list A := match n with 0 => [] | S n => x :: replicate n x end. (** The function [reverse l] returns the elements of [l] in reverse order. *) Definition reverse {A} (l : list A) : list A := rev_append l []. (** The function [last l] returns the last element of the list [l], or [None] if the list [l] is empty. *) Fixpoint last {A} (l : list A) : option A := match l with [] => None | [x] => Some x | _ :: l => last l end. (** The function [resize n y l] takes the first [n] elements of [l] in case [length l ≤ n], and otherwise appends elements with value [x] to [l] to obtain a list of length [n]. *) Fixpoint resize {A} (n : nat) (y : A) (l : list A) : list A := match l with | [] => replicate n y | x :: l => match n with 0 => [] | S n => x :: resize n y l end end. Arguments resize {_} !_ _ !_. (** The function [reshape k l] transforms [l] into a list of lists whose sizes are specified by [k]. In case [l] is too short, the resulting list will be padded with empty lists. In case [l] is too long, it will be truncated. *) Fixpoint reshape {A} (szs : list nat) (l : list A) : list (list A) := match szs with | [] => [] | sz :: szs => take sz l :: reshape szs (drop sz l) end. Definition sublist_lookup {A} (i n : nat) (l : list A) : option (list A) := guard (i + n ≤ length l); Some (take n (drop i l)). Definition sublist_alter {A} (f : list A → list A) (i n : nat) (l : list A) : list A := take i l ++ f (take n (drop i l)) ++ drop (i + n) l. (** Functions to fold over a list. We redefine [foldl] with the arguments in the same order as in Haskell. *) Notation foldr := fold_right. Definition foldl {A B} (f : A → B → A) : A → list B → A := fix go a l := match l with [] => a | x :: l => go (f a x) l end. (** The monadic operations. *) Instance list_ret: MRet list := λ A x, x :: @nil A. Instance list_fmap : FMap list := λ A B f, fix go (l : list A) := match l with [] => [] | x :: l => f x :: go l end. Instance list_omap : OMap list := λ A B f, fix go (l : list A) := match l with | [] => [] | x :: l => match f x with Some y => y :: go l | None => go l end end. Instance list_bind : MBind list := λ A B f, fix go (l : list A) := match l with [] => [] | x :: l => f x ++ go l end. Instance list_join: MJoin list := fix go A (ls : list (list A)) : list A := match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end. Definition mapM `{MBind M, MRet M} {A B} (f : A → M B) : list A → M (list B) := fix go l := match l with [] => mret [] | x :: l => y ← f x; k ← go l; mret (y :: k) end. (** We define stronger variants of map and fold that allow the mapped function to use the index of the elements. *) Definition imap_go {A B} (f : nat → A → B) : nat → list A → list B := fix go (n : nat) (l : list A) := match l with [] => [] | x :: l => f n x :: go (S n) l end. Definition imap {A B} (f : nat → A → B) : list A → list B := imap_go f 0. Definition zipped_map {A B} (f : list A → list A → A → B) : list A → list A → list B := fix go l k := match k with [] => [] | x :: k => f l k x :: go (x :: l) k end. Definition imap2_go {A B C} (f : nat → A → B → C) : nat → list A → list B → list C:= fix go (n : nat) (l : list A) (k : list B) := match l, k with | [], _ |_, [] => [] | x :: l, y :: k => f n x y :: go (S n) l k end. Definition imap2 {A B C} (f : nat → A → B → C) : list A → list B → list C := imap2_go f 0. Inductive zipped_Forall {A} (P : list A → list A → A → Prop) : list A → list A → Prop := | zipped_Forall_nil l : zipped_Forall P l [] | zipped_Forall_cons l k x : P l k x → zipped_Forall P (x :: l) k → zipped_Forall P l (x :: k). Arguments zipped_Forall_nil {_ _} _. Arguments zipped_Forall_cons {_ _} _ _ _ _ _. (** The function [mask f βs l] applies the function [f] to elements in [l] at positions that are [true] in [βs]. *) Fixpoint mask {A} (f : A → A) (βs : list bool) (l : list A) : list A := match βs, l with | β :: βs, x :: l => (if β then f x else x) :: mask f βs l | _, _ => l end. (** The function [permutations l] yields all permutations of [l]. *) Fixpoint interleave {A} (x : A) (l : list A) : list (list A) := match l with | [] => [[x]]| y :: l => (x :: y :: l) :: ((y ::) <\$> interleave x l) end. Fixpoint permutations {A} (l : list A) : list (list A) := match l with [] => [[]] | x :: l => permutations l ≫= interleave x end. (** The predicate [suffix_of] holds if the first list is a suffix of the second. The predicate [prefix_of] holds if the first list is a prefix of the second. *) Definition suffix_of {A} : relation (list A) := λ l1 l2, ∃ k, l2 = k ++ l1. Definition prefix_of {A} : relation (list A) := λ l1 l2, ∃ k, l2 = l1 ++ k. Infix "`suffix_of`" := suffix_of (at level 70) : C_scope. Infix "`prefix_of`" := prefix_of (at level 70) : C_scope. Hint Extern 0 (?x `prefix_of` ?y) => reflexivity. Hint Extern 0 (?x `suffix_of` ?y) => reflexivity. Section prefix_suffix_ops. Context `{∀ x y : A, Decision (x = y)}. Definition max_prefix_of : list A → list A → list A * list A * list A := fix go l1 l2 := match l1, l2 with | [], l2 => ([], l2, []) | l1, [] => (l1, [], []) | x1 :: l1, x2 :: l2 => if decide_rel (=) x1 x2 then prod_map id (x1 ::) (go l1 l2) else (x1 :: l1, x2 :: l2, []) end. Definition max_suffix_of (l1 l2 : list A) : list A * list A * list A := match max_prefix_of (reverse l1) (reverse l2) with | (k1, k2, k3) => (reverse k1, reverse k2, reverse k3) end. Definition strip_prefix (l1 l2 : list A) := (max_prefix_of l1 l2).1.2. Definition strip_suffix (l1 l2 : list A) := (max_suffix_of l1 l2).1.2. End prefix_suffix_ops. (** A list [l1] is a sublist of [l2] if [l2] is obtained by removing elements from [l1] without changing the order. *) Inductive sublist {A} : relation (list A) := | sublist_nil : sublist [] [] | sublist_skip x l1 l2 : sublist l1 l2 → sublist (x :: l1) (x :: l2) | sublist_cons x l1 l2 : sublist l1 l2 → sublist l1 (x :: l2). Infix "`sublist`" := sublist (at level 70) : C_scope. Hint Extern 0 (?x `sublist` ?y) => reflexivity. (** A list [l2] contains a list [l1] if [l2] is obtained by removing elements from [l1] while possiblity changing the order. *) Inductive contains {A} : relation (list A) := | contains_nil : contains [] [] | contains_skip x l1 l2 : contains l1 l2 → contains (x :: l1) (x :: l2) | contains_swap x y l : contains (y :: x :: l) (x :: y :: l) | contains_cons x l1 l2 : contains l1 l2 → contains l1 (x :: l2) | contains_trans l1 l2 l3 : contains l1 l2 → contains l2 l3 → contains l1 l3. Infix "`contains`" := contains (at level 70) : C_scope. Hint Extern 0 (?x `contains` ?y) => reflexivity. Section contains_dec_help. Context {A} {dec : ∀ x y : A, Decision (x = y)}. Fixpoint list_remove (x : A) (l : list A) : option (list A) := match l with | [] => None | y :: l => if decide (x = y) then Some l else (y ::) <\$> list_remove x l end. Fixpoint list_remove_list (k : list A) (l : list A) : option (list A) := match k with | [] => Some l | x :: k => list_remove x l ≫= list_remove_list k end. End contains_dec_help. Inductive Forall3 {A B C} (P : A → B → C → Prop) : list A → list B → list C → Prop := | Forall3_nil : Forall3 P [] [] [] | Forall3_cons x y z l k k' : P x y z → Forall3 P l k k' → Forall3 P (x :: l) (y :: k) (z :: k'). (** Set operations on lists *) Section list_set. Context {A} {dec : ∀ x y : A, Decision (x = y)}. Global Instance elem_of_list_dec {dec : ∀ x y : A, Decision (x = y)} (x : A) : ∀ l, Decision (x ∈ l). Proof. refine ( fix go l := match l return Decision (x ∈ l) with | [] => right _ | y :: l => cast_if_or (decide (x = y)) (go l) end); clear go dec; subst; try (by constructor); abstract by inversion 1. Defined. Fixpoint remove_dups (l : list A) : list A := match l with | [] => [] | x :: l => if decide_rel (∈) x l then remove_dups l else x :: remove_dups l end. Fixpoint list_difference (l k : list A) : list A := match l with | [] => [] | x :: l => if decide_rel (∈) x k then list_difference l k else x :: list_difference l k end. Definition list_union (l k : list A) : list A := list_difference l k ++ k. Fixpoint list_intersection (l k : list A) : list A := match l with | [] => [] | x :: l => if decide_rel (∈) x k then x :: list_intersection l k else list_intersection l k end. Definition list_intersection_with (f : A → A → option A) : list A → list A → list A := fix go l k := match l with | [] => [] | x :: l => foldr (λ y, match f x y with None => id | Some z => (z ::) end) (go l k) k end. End list_set. (** * Basic tactics on lists *) (** The tactic [discriminate_list_equality] discharges a goal if it contains a list equality involving [(::)] and [(++)] of two lists that have a different length as one of its hypotheses. *) Tactic Notation "discriminate_list_equality" hyp(H) := apply (f_equal length) in H; repeat (csimpl in H || rewrite app_length in H); exfalso; lia. Tactic Notation "discriminate_list_equality" := match goal with | H : @eq (list _) _ _ |- _ => discriminate_list_equality H end. (** The tactic [simplify_list_equality] simplifies hypotheses involving equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies lookups in singleton lists. *) Lemma app_injective_1 {A} (l1 k1 l2 k2 : list A) : length l1 = length k1 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2. Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed. Lemma app_injective_2 {A} (l1 k1 l2 k2 : list A) : length l2 = length k2 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2. Proof. intros ? Hl. apply app_injective_1; auto. apply (f_equal length) in Hl. rewrite !app_length in Hl. lia. Qed. Ltac simplify_list_equality := repeat match goal with | _ => progress simplify_equality' | H : _ ++ _ = _ ++ _ |- _ => first [ apply app_inv_head in H | apply app_inv_tail in H | apply app_injective_1 in H; [destruct H|done] | apply app_injective_2 in H; [destruct H|done] ] | H : [?x] !! ?i = Some ?y |- _ => destruct i; [change (Some x = Some y) in H | discriminate] end. (** * General theorems *) Section general_properties. Context {A : Type}. Implicit Types x y z : A. Implicit Types l k : list A. `````` Robbert Krebbers committed Nov 18, 2015 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 ``````Section setoid. Context `{Equiv A} `{!Equivalence ((≡) : relation A)}. Global Instance map_equivalence : Equivalence ((≡) : relation (list A)). Proof. split. * intros l; induction l; constructor; auto. * induction 1; constructor; auto. * intros l1 l2 l3 Hl; revert l3. induction Hl; inversion_clear 1; constructor; try etransitivity; eauto. Qed. Global Instance cons_proper : Proper ((≡) ==> (≡) ==> (≡)) (@cons A). Proof. by constructor. Qed. Global Instance app_proper : Proper ((≡) ==> (≡) ==> (≡)) (@app A). Proof. induction 1 as [|x y l k ?? IH]; intros ?? Htl; simpl; auto. by apply cons_equiv, IH. Qed. Global Instance list_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (list A). `````` Robbert Krebbers committed Dec 15, 2015 386 `````` Proof. induction 1; f_equal; fold_leibniz; auto. Qed. `````` Robbert Krebbers committed Nov 18, 2015 387 388 ``````End setoid. `````` Robbert Krebbers committed Nov 11, 2015 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 ``````Global Instance: Injective2 (=) (=) (=) (@cons A). Proof. by injection 1. Qed. Global Instance: ∀ k, Injective (=) (=) (k ++). Proof. intros ???. apply app_inv_head. Qed. Global Instance: ∀ k, Injective (=) (=) (++ k). Proof. intros ???. apply app_inv_tail. Qed. Global Instance: Associative (=) (@app A). Proof. intros ???. apply app_assoc. Qed. Global Instance: LeftId (=) [] (@app A). Proof. done. Qed. Global Instance: RightId (=) [] (@app A). Proof. intro. apply app_nil_r. Qed. Lemma app_nil l1 l2 : l1 ++ l2 = [] ↔ l1 = [] ∧ l2 = []. Proof. split. apply app_eq_nil. by intros [-> ->]. Qed. Lemma app_singleton l1 l2 x : l1 ++ l2 = [x] ↔ l1 = [] ∧ l2 = [x] ∨ l1 = [x] ∧ l2 = []. Proof. split. apply app_eq_unit. by intros [[-> ->]|[-> ->]]. Qed. Lemma cons_middle x l1 l2 : l1 ++ x :: l2 = l1 ++ [x] ++ l2. Proof. done. Qed. Lemma list_eq l1 l2 : (∀ i, l1 !! i = l2 !! i) → l1 = l2. Proof. revert l2. induction l1; intros [|??] H. * done. * discriminate (H 0). * discriminate (H 0). * f_equal; [by injection (H 0)|]. apply (IHl1 _ \$ λ i, H (S i)). Qed. Global Instance list_eq_dec {dec : ∀ x y, Decision (x = y)} : ∀ l k, Decision (l = k) := list_eq_dec dec. Global Instance list_eq_nil_dec l : Decision (l = []). Proof. by refine match l with [] => left _ | _ => right _ end. Defined. Lemma list_singleton_reflect l : option_reflect (λ x, l = [x]) (length l ≠ 1) (list_singleton l). Proof. by destruct l as [|? []]; constructor. Defined. Definition nil_length : length (@nil A) = 0 := eq_refl. Definition cons_length x l : length (x :: l) = S (length l) := eq_refl. Lemma nil_or_length_pos l : l = [] ∨ length l ≠ 0. Proof. destruct l; simpl; auto with lia. Qed. Lemma nil_length_inv l : length l = 0 → l = []. Proof. by destruct l. Qed. Lemma lookup_nil i : @nil A !! i = None. Proof. by destruct i. Qed. Lemma lookup_tail l i : tail l !! i = l !! S i. Proof. by destruct l. Qed. Lemma lookup_lt_Some l i x : l !! i = Some x → i < length l. Proof. revert i. induction l; intros [|?] ?; simplify_equality'; auto with arith. Qed. Lemma lookup_lt_is_Some_1 l i : is_Some (l !! i) → i < length l. Proof. intros [??]; eauto using lookup_lt_Some. Qed. Lemma lookup_lt_is_Some_2 l i : i < length l → is_Some (l !! i). Proof. revert i. induction l; intros [|?] ?; simplify_equality'; eauto with lia. Qed. Lemma lookup_lt_is_Some l i : is_Some (l !! i) ↔ i < length l. Proof. split; auto using lookup_lt_is_Some_1, lookup_lt_is_Some_2. Qed. Lemma lookup_ge_None l i : l !! i = None ↔ length l ≤ i. Proof. rewrite eq_None_not_Some, lookup_lt_is_Some. lia. Qed. Lemma lookup_ge_None_1 l i : l !! i = None → length l ≤ i. Proof. by rewrite lookup_ge_None. Qed. Lemma lookup_ge_None_2 l i : length l ≤ i → l !! i = None. Proof. by rewrite lookup_ge_None. Qed. Lemma list_eq_same_length l1 l2 n : length l2 = n → length l1 = n → (∀ i x y, i < n → l1 !! i = Some x → l2 !! i = Some y → x = y) → l1 = l2. Proof. intros <- Hlen Hl; apply list_eq; intros i. destruct (l2 !! i) as [x|] eqn:Hx. * destruct (lookup_lt_is_Some_2 l1 i) as [y Hy]. { rewrite Hlen; eauto using lookup_lt_Some. } rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some. * by rewrite lookup_ge_None, Hlen, <-lookup_ge_None. Qed. Lemma lookup_app_l l1 l2 i : i < length l1 → (l1 ++ l2) !! i = l1 !! i. Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed. Lemma lookup_app_l_Some l1 l2 i x : l1 !! i = Some x → (l1 ++ l2) !! i = Some x. Proof. intros. rewrite lookup_app_l; eauto using lookup_lt_Some. Qed. Lemma lookup_app_r l1 l2 i : length l1 ≤ i → (l1 ++ l2) !! i = l2 !! (i - length l1). Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed. Lemma lookup_app_Some l1 l2 i x : (l1 ++ l2) !! i = Some x ↔ l1 !! i = Some x ∨ length l1 ≤ i ∧ l2 !! (i - length l1) = Some x. Proof. split. * revert i. induction l1 as [|y l1 IH]; intros [|i] ?; simplify_equality'; auto with lia. destruct (IH i) as [?|[??]]; auto with lia. * intros [?|[??]]; auto using lookup_app_l_Some. by rewrite lookup_app_r. Qed. Lemma list_lookup_middle l1 l2 x n : n = length l1 → (l1 ++ x :: l2) !! n = Some x. Proof. intros ->. by induction l1. Qed. Lemma list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l. Proof. by revert i; induction l; intros []; intros; f_equal'. Qed. Lemma alter_length f l i : length (alter f i l) = length l. Proof. revert i. by induction l; intros [|?]; f_equal'. Qed. Lemma insert_length l i x : length (<[i:=x]>l) = length l. Proof. revert i. by induction l; intros [|?]; f_equal'. Qed. Lemma list_lookup_alter f l i : alter f i l !! i = f <\$> l !! i. Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed. Lemma list_lookup_alter_ne f l i j : i ≠ j → alter f i l !! j = l !! j. Proof. revert i j. induction l; [done|]. intros [][] ?; csimpl; auto with congruence. Qed. Lemma list_lookup_insert l i x : i < length l → <[i:=x]>l !! i = Some x. Proof. revert i. induction l; intros [|?] ?; f_equal'; auto with lia. Qed. Lemma list_lookup_insert_ne l i j x : i ≠ j → <[i:=x]>l !! j = l !! j. Proof. revert i j. induction l; [done|]. intros [] [] ?; simpl; auto with congruence. Qed. Lemma list_lookup_insert_Some l i x j y : <[i:=x]>l !! j = Some y ↔ i = j ∧ x = y ∧ j < length l ∨ i ≠ j ∧ l !! j = Some y. Proof. destruct (decide (i = j)) as [->|]; [split|rewrite list_lookup_insert_ne by done; tauto]. * intros Hy. assert (j < length l). { rewrite <-(insert_length l j x); eauto using lookup_lt_Some. } rewrite list_lookup_insert in Hy by done; naive_solver. * intros [(?&?&?)|[??]]; rewrite ?list_lookup_insert; naive_solver. Qed. Lemma list_insert_commute l i j x y : i ≠ j → <[i:=x]>(<[j:=y]>l) = <[j:=y]>(<[i:=x]>l). Proof. revert i j. by induction l; intros [|?] [|?] ?; f_equal'; auto. Qed. Lemma list_lookup_other l i x : length l ≠ 1 → l !! i = Some x → ∃ j y, j ≠ i ∧ l !! j = Some y. Proof. intros. destruct i, l as [|x0 [|x1 l]]; simplify_equality'. `````` Robbert Krebbers committed Nov 11, 2015 520 521 `````` * by exists 1, x1. * by exists 0, x0. `````` Robbert Krebbers committed Nov 11, 2015 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 ``````Qed. Lemma alter_app_l f l1 l2 i : i < length l1 → alter f i (l1 ++ l2) = alter f i l1 ++ l2. Proof. revert i. induction l1; intros [|?] ?; f_equal'; auto with lia. Qed. Lemma alter_app_r f l1 l2 i : alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2. Proof. revert i. induction l1; intros [|?]; f_equal'; auto. Qed. Lemma alter_app_r_alt f l1 l2 i : length l1 ≤ i → alter f i (l1 ++ l2) = l1 ++ alter f (i - length l1) l2. Proof. intros. assert (i = length l1 + (i - length l1)) as Hi by lia. rewrite Hi at 1. by apply alter_app_r. Qed. Lemma list_alter_id f l i : (∀ x, f x = x) → alter f i l = l. Proof. intros ?. revert i. induction l; intros [|?]; f_equal'; auto. Qed. Lemma list_alter_ext f g l k i : (∀ x, l !! i = Some x → f x = g x) → l = k → alter f i l = alter g i k. Proof. intros H ->. revert i H. induction k; intros [|?] ?; f_equal'; auto. Qed. Lemma list_alter_compose f g l i : alter (f ∘ g) i l = alter f i (alter g i l). Proof. revert i. induction l; intros [|?]; f_equal'; auto. Qed. Lemma list_alter_commute f g l i j : i ≠ j → alter f i (alter g j l) = alter g j (alter f i l). Proof. revert i j. induction l; intros [|?][|?] ?; f_equal'; auto with lia. Qed. Lemma insert_app_l l1 l2 i x : i < length l1 → <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2. Proof. revert i. induction l1; intros [|?] ?; f_equal'; auto with lia. Qed. Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2. Proof. revert i. induction l1; intros [|?]; f_equal'; auto. Qed. Lemma insert_app_r_alt l1 l2 i x : length l1 ≤ i → <[i:=x]>(l1 ++ l2) = l1 ++ <[i - length l1:=x]>l2. Proof. intros. assert (i = length l1 + (i - length l1)) as Hi by lia. rewrite Hi at 1. by apply insert_app_r. Qed. Lemma delete_middle l1 l2 x : delete (length l1) (l1 ++ x :: l2) = l1 ++ l2. Proof. induction l1; f_equal'; auto. Qed. Lemma inserts_length l i k : length (list_inserts i k l) = length l. Proof. revert i. induction k; intros ?; csimpl; rewrite ?insert_length; auto. Qed. Lemma list_lookup_inserts l i k j : i ≤ j < i + length k → j < length l → list_inserts i k l !! j = k !! (j - i). Proof. revert i j. induction k as [|y k IH]; csimpl; intros i j ??; [lia|]. destruct (decide (i = j)) as [->|]. { by rewrite list_lookup_insert, Nat.sub_diag by (rewrite inserts_length; lia). } rewrite list_lookup_insert_ne, IH by lia. by replace (j - i) with (S (j - S i)) by lia. Qed. Lemma list_lookup_inserts_lt l i k j : j < i → list_inserts i k l !! j = l !! j. Proof. revert i j. induction k; intros i j ?; csimpl; rewrite ?list_lookup_insert_ne by lia; auto with lia. Qed. Lemma list_lookup_inserts_ge l i k j : i + length k ≤ j → list_inserts i k l !! j = l !! j. Proof. revert i j. induction k; csimpl; intros i j ?; rewrite ?list_lookup_insert_ne by lia; auto with lia. Qed. Lemma list_lookup_inserts_Some l i k j y : list_inserts i k l !! j = Some y ↔ (j < i ∨ i + length k ≤ j) ∧ l !! j = Some y ∨ i ≤ j < i + length k ∧ j < length l ∧ k !! (j - i) = Some y. Proof. destruct (decide (j < i)). { rewrite list_lookup_inserts_lt by done; intuition lia. } destruct (decide (i + length k ≤ j)). { rewrite list_lookup_inserts_ge by done; intuition lia. } split. * intros Hy. assert (j < length l). { rewrite <-(inserts_length l i k); eauto using lookup_lt_Some. } rewrite list_lookup_inserts in Hy by lia. intuition lia. * intuition. by rewrite list_lookup_inserts by lia. Qed. Lemma list_insert_inserts_lt l i j x k : i < j → <[i:=x]>(list_inserts j k l) = list_inserts j k (<[i:=x]>l). Proof. revert i j. induction k; intros i j ?; simpl; rewrite 1?list_insert_commute by lia; auto with f_equal. Qed. (** ** Properties of the [elem_of] predicate *) Lemma not_elem_of_nil x : x ∉ []. Proof. by inversion 1. Qed. Lemma elem_of_nil x : x ∈ [] ↔ False. Proof. intuition. by destruct (not_elem_of_nil x). Qed. Lemma elem_of_nil_inv l : (∀ x, x ∉ l) → l = []. Proof. destruct l. done. by edestruct 1; constructor. Qed. Lemma elem_of_not_nil x l : x ∈ l → l ≠ []. Proof. intros ? ->. by apply (elem_of_nil x). Qed. Lemma elem_of_cons l x y : x ∈ y :: l ↔ x = y ∨ x ∈ l. Proof. by split; [inversion 1; subst|intros [->|?]]; constructor. Qed. Lemma not_elem_of_cons l x y : x ∉ y :: l ↔ x ≠ y ∧ x ∉ l. Proof. rewrite elem_of_cons. tauto. Qed. Lemma elem_of_app l1 l2 x : x ∈ l1 ++ l2 ↔ x ∈ l1 ∨ x ∈ l2. Proof. induction l1. * split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x). * simpl. rewrite !elem_of_cons, IHl1. tauto. Qed. Lemma not_elem_of_app l1 l2 x : x ∉ l1 ++ l2 ↔ x ∉ l1 ∧ x ∉ l2. Proof. rewrite elem_of_app. tauto. Qed. Lemma elem_of_list_singleton x y : x ∈ [y] ↔ x = y. Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed. Global Instance elem_of_list_permutation_proper x : Proper ((≡ₚ) ==> iff) (x ∈). Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed. Lemma elem_of_list_split l x : x ∈ l → ∃ l1 l2, l = l1 ++ x :: l2. Proof. induction 1 as [x l|x y l ? [l1 [l2 ->]]]; [by eexists [], l|]. `````` Robbert Krebbers committed Nov 11, 2015 637 `````` by exists (y :: l1), l2. `````` Robbert Krebbers committed Nov 11, 2015 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 ``````Qed. Lemma elem_of_list_lookup_1 l x : x ∈ l → ∃ i, l !! i = Some x. Proof. induction 1 as [|???? IH]; [by exists 0 |]. destruct IH as [i ?]; auto. by exists (S i). Qed. Lemma elem_of_list_lookup_2 l i x : l !! i = Some x → x ∈ l. Proof. revert i. induction l; intros [|i] ?; simplify_equality'; constructor; eauto. Qed. Lemma elem_of_list_lookup l x : x ∈ l ↔ ∃ i, l !! i = Some x. Proof. firstorder eauto using elem_of_list_lookup_1, elem_of_list_lookup_2. Qed. Lemma elem_of_list_omap {B} (f : A → option B) l (y : B) : y ∈ omap f l ↔ ∃ x, x ∈ l ∧ f x = Some y. Proof. split. * induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst; setoid_rewrite elem_of_cons; naive_solver. * intros (x&Hx&?). by induction Hx; csimpl; repeat case_match; simplify_equality; try constructor; auto. Qed. (** ** Properties of the [NoDup] predicate *) Lemma NoDup_nil : NoDup (@nil A) ↔ True. Proof. split; constructor. Qed. Lemma NoDup_cons x l : NoDup (x :: l) ↔ x ∉ l ∧ NoDup l. Proof. split. by inversion 1. intros [??]. by constructor. Qed. Lemma NoDup_cons_11 x l : NoDup (x :: l) → x ∉ l. Proof. rewrite NoDup_cons. by intros [??]. Qed. Lemma NoDup_cons_12 x l : NoDup (x :: l) → NoDup l. Proof. rewrite NoDup_cons. by intros [??]. Qed. Lemma NoDup_singleton x : NoDup [x]. Proof. constructor. apply not_elem_of_nil. constructor. Qed. Lemma NoDup_app l k : NoDup (l ++ k) ↔ NoDup l ∧ (∀ x, x ∈ l → x ∉ k) ∧ NoDup k. Proof. induction l; simpl. * rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver. * rewrite !NoDup_cons. setoid_rewrite elem_of_cons. setoid_rewrite elem_of_app. naive_solver. Qed. Global Instance NoDup_proper: Proper ((≡ₚ) ==> iff) (@NoDup A). Proof. induction 1 as [|x l k Hlk IH | |]. * by rewrite !NoDup_nil. * by rewrite !NoDup_cons, IH, Hlk. * rewrite !NoDup_cons, !elem_of_cons. intuition. * intuition. Qed. Lemma NoDup_lookup l i j x : NoDup l → l !! i = Some x → l !! j = Some x → i = j. Proof. intros Hl. revert i j. induction Hl as [|x' l Hx Hl IH]. { intros; simplify_equality. } intros [|i] [|j] ??; simplify_equality'; eauto with f_equal; exfalso; eauto using elem_of_list_lookup_2. Qed. Lemma NoDup_alt l : NoDup l ↔ ∀ i j x, l !! i = Some x → l !! j = Some x → i = j. Proof. split; eauto using NoDup_lookup. induction l as [|x l IH]; intros Hl; constructor. * rewrite elem_of_list_lookup. intros [i ?]. by feed pose proof (Hl (S i) 0 x); auto. * apply IH. intros i j x' ??. by apply (injective S), (Hl (S i) (S j) x'). Qed. Section no_dup_dec. Context `{!∀ x y, Decision (x = y)}. Global Instance NoDup_dec: ∀ l, Decision (NoDup l) := fix NoDup_dec l := match l return Decision (NoDup l) with | [] => left NoDup_nil_2 | x :: l => match decide_rel (∈) x l with | left Hin => right (λ H, NoDup_cons_11 _ _ H Hin) | right Hin => match NoDup_dec l with | left H => left (NoDup_cons_2 _ _ Hin H) | right H => right (H ∘ NoDup_cons_12 _ _) end end end. Lemma elem_of_remove_dups l x : x ∈ remove_dups l ↔ x ∈ l. Proof. split; induction l; simpl; repeat case_decide; rewrite ?elem_of_cons; intuition (simplify_equality; auto). Qed. Lemma NoDup_remove_dups l : NoDup (remove_dups l). Proof. induction l; simpl; repeat case_decide; try constructor; auto. by rewrite elem_of_remove_dups. Qed. End no_dup_dec. (** ** Set operations on lists *) Section list_set. Context {dec : ∀ x y, Decision (x = y)}. Lemma elem_of_list_difference l k x : x ∈ list_difference l k ↔ x ∈ l ∧ x ∉ k. Proof. split; induction l; simpl; try case_decide; rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence. Qed. Lemma NoDup_list_difference l k : NoDup l → NoDup (list_difference l k). Proof. induction 1; simpl; try case_decide. * constructor. * done. * constructor. rewrite elem_of_list_difference; intuition. done. Qed. Lemma elem_of_list_union l k x : x ∈ list_union l k ↔ x ∈ l ∨ x ∈ k. Proof. unfold list_union. rewrite elem_of_app, elem_of_list_difference. intuition. case (decide (x ∈ k)); intuition. Qed. Lemma NoDup_list_union l k : NoDup l → NoDup k → NoDup (list_union l k). Proof. intros. apply NoDup_app. repeat split. * by apply NoDup_list_difference. * intro. rewrite elem_of_list_difference. intuition. * done. Qed. Lemma elem_of_list_intersection l k x : x ∈ list_intersection l k ↔ x ∈ l ∧ x ∈ k. Proof. split; induction l; simpl; repeat case_decide; rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence. Qed. Lemma NoDup_list_intersection l k : NoDup l → NoDup (list_intersection l k). Proof. induction 1; simpl; try case_decide. * constructor. * constructor. rewrite elem_of_list_intersection; intuition. done. * done. Qed. Lemma elem_of_list_intersection_with f l k x : x ∈ list_intersection_with f l k ↔ ∃ x1 x2, x1 ∈ l ∧ x2 ∈ k ∧ f x1 x2 = Some x. Proof. split. * induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|]. intros Hx. setoid_rewrite elem_of_cons. cut ((∃ x2, x2 ∈ k ∧ f x1 x2 = Some x) ∨ x ∈ list_intersection_with f l k); [naive_solver|]. clear IH. revert Hx. generalize (list_intersection_with f l k). induction k; simpl; [by auto|]. case_match; setoid_rewrite elem_of_cons; naive_solver. * intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl. + generalize (list_intersection_with f l k). induction Hx2; simpl; [by rewrite Hx; left |]. case_match; simpl; try setoid_rewrite elem_of_cons; auto. + generalize (IH Hx). clear Hx IH Hx2. generalize (list_intersection_with f l k). induction k; simpl; intros; [done|]. case_match; simpl; rewrite ?elem_of_cons; auto. Qed. End list_set. (** ** Properties of the [filter] function *) Section filter. Context (P : A → Prop) `{∀ x, Decision (P x)}. Lemma elem_of_list_filter l x : x ∈ filter P l ↔ P x ∧ x ∈ l. Proof. unfold filter. induction l; simpl; repeat case_decide; rewrite ?elem_of_nil, ?elem_of_cons; naive_solver. Qed. Lemma NoDup_filter l : NoDup l → NoDup (filter P l). Proof. unfold filter. induction 1; simpl; repeat case_decide; rewrite ?NoDup_nil, ?NoDup_cons, ?elem_of_list_filter; tauto. Qed. End filter. (** ** Properties of the [find] function *) Section find. Context (P : A → Prop) `{∀ x, Decision (P x)}. Lemma list_find_Some l i x : list_find P l = Some (i,x) → l !! i = Some x ∧ P x. Proof. revert i; induction l; intros [] ?; repeat (match goal with x : prod _ _ |- _ => destruct x end || simplify_option_equality); eauto. Qed. Lemma list_find_elem_of l x : x ∈ l → P x → is_Some (list_find P l). Proof. induction 1 as [|x y l ? IH]; intros; simplify_option_equality; eauto. by destruct IH as [[i x'] ->]; [|exists (S i, x')]. Qed. End find. (** ** Properties of the [reverse] function *) Lemma reverse_nil : reverse [] = @nil A. Proof. done. Qed. Lemma reverse_singleton x : reverse [x] = [x]. Proof. done. Qed. Lemma reverse_cons l x : reverse (x :: l) = reverse l ++ [x]. Proof. unfold reverse. by rewrite <-!rev_alt. Qed. Lemma reverse_snoc l x : reverse (l ++ [x]) = x :: reverse l. Proof. unfold reverse. by rewrite <-!rev_alt, rev_unit. Qed. Lemma reverse_app l1 l2 : reverse (l1 ++ l2) = reverse l2 ++ reverse l1. Proof. unfold reverse. rewrite <-!rev_alt. apply rev_app_distr. Qed. Lemma reverse_length l : length (reverse l) = length l. Proof. unfold reverse. rewrite <-!rev_alt. apply rev_length. Qed. Lemma reverse_involutive l : reverse (reverse l) = l. Proof. unfold reverse. rewrite <-!rev_alt. apply rev_involutive. Qed. Lemma elem_of_reverse_2 x l : x ∈ l → x ∈ reverse l. Proof. induction 1; rewrite reverse_cons, elem_of_app, ?elem_of_list_singleton; intuition. Qed. Lemma elem_of_reverse x l : x ∈ reverse l ↔ x ∈ l. Proof. split; auto using elem_of_reverse_2. intros. rewrite <-(reverse_involutive l). by apply elem_of_reverse_2. Qed. Global Instance: Injective (=) (=) (@reverse A). Proof. intros l1 l2 Hl. by rewrite <-(reverse_involutive l1), <-(reverse_involutive l2), Hl. Qed. Lemma sum_list_with_app (f : A → nat) l k : sum_list_with f (l ++ k) = sum_list_with f l + sum_list_with f k. Proof. induction l; simpl; lia. Qed. Lemma sum_list_with_reverse (f : A → nat) l : sum_list_with f (reverse l) = sum_list_with f l. Proof. induction l; simpl; rewrite ?reverse_cons, ?sum_list_with_app; simpl; lia. Qed. (** ** Properties of the [last] function *) Lemma last_snoc x l : last (l ++ [x]) = Some x. Proof. induction l as [|? []]; simpl; auto. Qed. Lemma last_reverse l : last (reverse l) = head l. Proof. by destruct l as [|x l]; rewrite ?reverse_cons, ?last_snoc. Qed. Lemma head_reverse l : head (reverse l) = last l. Proof. by rewrite <-last_reverse, reverse_involutive. Qed. (** ** Properties of the [take] function *) Definition take_drop i l : take i l ++ drop i l = l := firstn_skipn i l. Lemma take_drop_middle l i x : l !! i = Some x → take i l ++ x :: drop (S i) l = l. Proof. revert i x. induction l; intros [|?] ??; simplify_equality'; f_equal; auto. Qed. Lemma take_nil n : take n (@nil A) = []. Proof. by destruct n. Qed. Lemma take_app l k : take (length l) (l ++ k) = l. Proof. induction l; f_equal'; auto. Qed. Lemma take_app_alt l k n : n = length l → take n (l ++ k) = l. Proof. intros ->. by apply take_app. Qed. Lemma take_app3_alt l1 l2 l3 n : n = length l1 → take n ((l1 ++ l2) ++ l3) = l1. Proof. intros ->. by rewrite <-(associative_L (++)), take_app. Qed. Lemma take_app_le l k n : n ≤ length l → take n (l ++ k) = take n l. Proof. revert n. induction l; intros [|?] ?; f_equal'; auto with lia. Qed. Lemma take_plus_app l k n m : length l = n → take (n + m) (l ++ k) = l ++ take m k. Proof. intros <-. induction l; f_equal'; auto. Qed. Lemma take_app_ge l k n : length l ≤ n → take n (l ++ k) = l ++ take (n - length l) k. Proof. revert n. induction l; intros [|?] ?; f_equal'; auto with lia. Qed. Lemma take_ge l n : length l ≤ n → take n l = l. Proof. revert n. induction l; intros [|?] ?; f_equal'; auto with lia. Qed. Lemma take_take l n m : take n (take m l) = take (min n m) l. Proof. revert n m. induction l; intros [|?] [|?]; f_equal'; auto. Qed. Lemma take_idempotent l n : take n (take n l) = take n l. Proof. by rewrite take_take, Min.min_idempotent. Qed. Lemma take_length l n : length (take n l) = min n (length l). Proof. revert n. induction l; intros [|?]; f_equal'; done. Qed. Lemma take_length_le l n : n ≤ length l → length (take n l) = n. Proof. rewrite take_length. apply Min.min_l. Qed. Lemma take_length_ge l n : length l ≤ n → length (take n l) = length l. Proof. rewrite take_length. apply Min.min_r. Qed. Lemma take_drop_commute l n m : take n (drop m l) = drop m (take (m + n) l). Proof. revert n m. induction l; intros [|?][|?]; simpl; auto using take_nil with lia. Qed. Lemma lookup_take l n i : i < n → take n l !! i = l !! i. Proof. revert n i. induction l; intros [|n] [|i] ?; simpl; auto with lia. Qed. Lemma lookup_take_ge l n i : n ≤ i → take n l !! i = None. Proof. revert n i. induction l; intros [|?] [|?] ?; simpl; auto with lia. Qed. Lemma take_alter f l n i : n ≤ i → take n (alter f i l) = take n l. Proof. intros. apply list_eq. intros j. destruct (le_lt_dec n j). * by rewrite !lookup_take_ge. * by rewrite !lookup_take, !list_lookup_alter_ne by lia. Qed. Lemma take_insert l n i x : n ≤ i → take n (<[i:=x]>l) = take n l. Proof. intros. apply list_eq. intros j. destruct (le_lt_dec n j). * by rewrite !lookup_take_ge. * by rewrite !lookup_take, !list_lookup_insert_ne by lia. Qed. (** ** Properties of the [drop] function *) Lemma drop_0 l : drop 0 l = l. Proof. done. Qed. Lemma drop_nil n : drop n (@nil A) = []. Proof. by destruct n. Qed. Lemma drop_length l n : length (drop n l) = length l - n. Proof. revert n. by induction l; intros [|i]; f_equal'. Qed. Lemma drop_ge l n : length l ≤ n → drop n l = []. Proof. revert n. induction l; intros [|??]; simpl in *; auto with lia. Qed. Lemma drop_all l : drop (length l) l = []. Proof. by apply drop_ge. Qed. Lemma drop_drop l n1 n2 : drop n1 (drop n2 l) = drop (n2 + n1) l. Proof. revert n2. induction l; intros [|?]; simpl; rewrite ?drop_nil; auto. Qed. Lemma drop_app_le l k n : n ≤ length l → drop n (l ++ k) = drop n l ++ k. Proof. revert n. induction l; intros [|?]; simpl; auto with lia. Qed. Lemma drop_app l k : drop (length l) (l ++ k) = k. Proof. by rewrite drop_app_le, drop_all. Qed. Lemma drop_app_alt l k n : n = length l → drop n (l ++ k) = k. Proof. intros ->. by apply drop_app. Qed. Lemma drop_app3_alt l1 l2 l3 n : n = length l1 → drop n ((l1 ++ l2) ++ l3) = l2 ++ l3. Proof. intros ->. by rewrite <-(associative_L (++)), drop_app. Qed. Lemma drop_app_ge l k n : length l ≤ n → drop n (l ++ k) = drop (n - length l) k. Proof. intros. rewrite <-(Nat.sub_add (length l) n) at 1 by done. by rewrite Nat.add_comm, <-drop_drop, drop_app. Qed. Lemma drop_plus_app l k n m : length l = n → drop (n + m) (l ++ k) = drop m k. Proof. intros <-. by rewrite <-drop_drop, drop_app. Qed. Lemma lookup_drop l n i : drop n l !! i = l !! (n + i). Proof. revert n i. induction l; intros [|i] ?; simpl; auto. Qed. Lemma drop_alter f l n i : i < n → drop n (alter f i l) = drop n l. Proof. intros. apply list_eq. intros j. by rewrite !lookup_drop, !list_lookup_alter_ne by lia. Qed. Lemma drop_insert l n i x : i < n → drop n (<[i:=x]>l) = drop n l. Proof. intros. apply list_eq. intros j. by rewrite !lookup_drop, !list_lookup_insert_ne by lia. Qed. Lemma delete_take_drop l i : delete i l = take i l ++ drop (S i) l. Proof. revert i. induction l; intros [|?]; f_equal'; auto. Qed. Lemma take_take_drop l n m : take n l ++ take m (drop n l) = take (n + m) l. Proof. revert n m. induction l; intros [|?] [|?]; f_equal'; auto. Qed. Lemma drop_take_drop l n m : n ≤ m → drop n (take m l) ++ drop m l = drop n l. Proof. revert n m. induction l; intros [|?] [|?] ?; f_equal'; auto using take_drop with lia. Qed. (** ** Properties of the [replicate] function *) Lemma replicate_length n x : length (replicate n x) = n. Proof. induction n; simpl; auto. Qed. Lemma lookup_replicate n x y i : replicate n x !! i = Some y ↔ y = x ∧ i < n. Proof. split. * revert i. induction n; intros [|?]; naive_solver auto with lia. * intros [-> Hi]. revert i Hi. induction n; intros [|?]; naive_solver auto with lia. Qed. Lemma lookup_replicate_1 n x y i : replicate n x !! i = Some y → y = x ∧ i < n. Proof. by rewrite lookup_replicate. Qed. Lemma lookup_replicate_2 n x i : i < n → replicate n x !! i = Some x. Proof. by rewrite lookup_replicate. Qed. Lemma lookup_replicate_None n x i : n ≤ i ↔ replicate n x !! i = None. Proof. rewrite eq_None_not_Some, Nat.le_ngt. split. * intros Hin [x' Hx']; destruct Hin. rewrite lookup_replicate in Hx'; tauto. * intros Hx ?. destruct Hx. exists x; auto using lookup_replicate_2. Qed. Lemma insert_replicate x n i : <[i:=x]>(replicate n x) = replicate n x. Proof. revert i. induction n; intros [|?]; f_equal'; auto. Qed. Lemma elem_of_replicate_inv x n y : x ∈ replicate n y → x = y. Proof. induction n; simpl; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed. Lemma replicate_S n x : replicate (S n) x = x :: replicate n x. Proof. done. Qed. Lemma replicate_plus n m x : replicate (n + m) x = replicate n x ++ replicate m x. Proof. induction n; f_equal'; auto. Qed. Lemma take_replicate n m x : take n (replicate m x) = replicate (min n m) x. Proof. revert m. by induction n; intros [|?]; f_equal'. Qed. Lemma take_replicate_plus n m x : take n (replicate (n + m) x) = replicate n x. Proof. by rewrite take_replicate, min_l by lia. Qed. Lemma drop_replicate n m x : drop n (replicate m x) = replicate (m - n) x. Proof. revert m. by induction n; intros [|?]; f_equal'. Qed. Lemma drop_replicate_plus n m x : drop n (replicate (n + m) x) = replicate m x. Proof. rewrite drop_replicate. f_equal. lia. Qed. Lemma replicate_as_elem_of x n l : replicate n x = l ↔ length l = n ∧ ∀ y, y ∈ l → y = x. Proof. split; [intros <-; eauto using elem_of_replicate_inv, replicate_length|]. intros [<- Hl]. symmetry. induction l as [|y l IH]; f_equal'. * apply Hl. by left. * apply IH. intros ??. apply Hl. by right. Qed. Lemma reverse_replicate n x : reverse (replicate n x) = replicate n x. Proof. symmetry. apply replicate_as_elem_of. rewrite reverse_length, replicate_length. split; auto. intros y. rewrite elem_of_reverse. by apply elem_of_replicate_inv. Qed. Lemma replicate_false βs n : length βs = n → replicate n false =.>* βs. Proof. intros <-. by induction βs; simpl; constructor. Qed. (** ** Properties of the [resize] function *) Lemma resize_spec l n x : resize n x l = take n l ++ replicate (n - length l) x. Proof. revert n. induction l; intros [|?]; f_equal'; auto. Qed. Lemma resize_0 l x : resize 0 x l = []. Proof. by destruct l. Qed. Lemma resize_nil n x : resize n x [] = replicate n x. Proof. rewrite resize_spec. rewrite take_nil. f_equal'. lia. Qed. Lemma resize_ge l n x : length l ≤ n → resize n x l = l ++ replicate (n - length l) x. Proof. intros. by rewrite resize_spec, take_ge. Qed. Lemma resize_le l n x : n ≤ length l → resize n x l = take n l. Proof. intros. rewrite resize_spec, (proj2 (Nat.sub_0_le _ _)) by done. simpl. by rewrite (right_id_L [] (++)). Qed. Lemma resize_all l x : resize (length l) x l = l. Proof. intros. by rewrite resize_le, take_ge. Qed. Lemma resize_all_alt l n x : n = length l → resize n x l = l. Proof. intros ->. by rewrite resize_all. Qed. Lemma resize_plus l n m x : resize (n + m) x l = resize n x l ++ resize m x (drop n l). Proof. revert n m. induction l; intros [|?] [|?]; f_equal'; auto. * by rewrite Nat.add_0_r, (right_id_L [] (++)). * by rewrite replicate_plus. Qed. Lemma resize_plus_eq l n m x : length l = n → resize (n + m) x l = l ++ replicate m x. Proof. intros <-. by rewrite resize_plus, resize_all, drop_all, resize_nil. Qed. Lemma resize_app_le l1 l2 n x : n ≤ length l1 → resize n x (l1 ++ l2) = resize n x l1. Proof. intros. by rewrite !resize_le, take_app_le by (rewrite ?app_length; lia). Qed. Lemma resize_app l1 l2 n x : n = length l1 → resize n x (l1 ++ l2) = l1. Proof. intros ->. by rewrite resize_app_le, resize_all. Qed. Lemma resize_app_ge l1 l2 n x : length l1 ≤ n → resize n x (l1 ++ l2) = l1 ++ resize (n - length l1) x l2. Proof. intros. rewrite !resize_spec, take_app_ge, (associative_L (++)) by done. do 2 f_equal. rewrite app_length. lia. Qed. Lemma resize_length l n x : length (resize n x l) = n. Proof. rewrite resize_spec, app_length, replicate_length, take_length. lia. Qed. Lemma resize_replicate x n m : resize n x (replicate m x) = replicate n x. Proof. revert m. induction n; intros [|?]; f_equal'; auto. Qed. Lemma resize_resize l n m x : n ≤ m → resize n x (resize m x l) = resize n x l. Proof. revert n m. induction l; simpl. * intros. by rewrite !resize_nil, resize_replicate. * intros [|?] [|?] ?; f_equal'; auto with lia. Qed. Lemma resize_idempotent l n x : resize n x (resize n x l) = resize n x l. Proof. by rewrite resize_resize. Qed. Lemma resize_take_le l n m x : n ≤ m → resize n x (take m l) = resize n x l. Proof. revert n m. induction l; intros [|?][|?] ?; f_equal'; auto with lia. Qed. Lemma resize_take_eq l n x : resize n x (take n l) = resize n x l. Proof. by rewrite resize_take_le. Qed. Lemma take_resize l n m x : take n (resize m x l) = resize (min n m) x l. Proof. revert n m. induction l; intros [|?][|?]; f_equal'; auto using take_replicate. Qed. Lemma take_resize_le l n m x : n ≤ m → take n (resize m x l) = resize n x l. Proof. intros. by rewrite take_resize, Min.min_l. Qed. Lemma take_resize_eq l n x : take n (resize n x l) = resize n x l. Proof. intros. by rewrite take_resize, Min.min_l. Qed. Lemma take_resize_plus l n m x : take n (resize (n + m) x l) = resize n x l. Proof. by rewrite take_resize, min_l by lia. Qed. Lemma drop_resize_le l n m x : n ≤ m → drop n (resize m x l) = resize (m - n) x (drop n l). Proof. revert n m. induction l; simpl. * intros. by rewrite drop_nil, !resize_nil, drop_replicate. * intros [|?] [|?] ?; simpl; try case_match; auto with lia. Qed. Lemma drop_resize_plus l n m x : drop n (resize (n + m) x l) = resize m x (drop n l). Proof. rewrite drop_resize_le by lia. f_equal. lia. Qed. Lemma lookup_resize l n x i : i < n → i < length l → resize n x l !! i = l !! i. Proof. intros ??. destruct (decide (n < length l)). * by rewrite resize_le, lookup_take by lia. * by rewrite resize_ge, lookup_app_l by lia. Qed. Lemma lookup_resize_new l n x i : length l ≤ i → i < n → resize n x l !! i = Some x. Proof. intros ??. rewrite resize_ge by lia. replace i with (length l + (i - length l)) by lia. by rewrite lookup_app_r, lookup_replicate_2 by lia. Qed. Lemma lookup_resize_old l n x i : n ≤ i → resize n x l !! i = None. Proof. intros ?. apply lookup_ge_None_2. by rewrite resize_length. Qed. End general_properties. Section more_general_properties. Context {A : Type}. Implicit Types x y z : A. Implicit Types l k : list A. (** ** Properties of the [reshape] function *) Lemma reshape_length szs l : length (reshape szs l) = length szs. Proof. revert l. by induction szs; intros; f_equal'. Qed. Lemma join_reshape szs l : sum_list szs = length l → mjoin (reshape szs l) = l. Proof. revert l. induction szs as [|sz szs IH]; simpl; intros l Hl; [by destruct l|]. by rewrite IH, take_drop by (rewrite drop_length; lia). Qed. Lemma sum_list_replicate n m : sum_list (replicate m n) = m * n. Proof. induction m; simpl; auto. Qed. (** ** Properties of [sublist_lookup] and [sublist_alter] *) Lemma sublist_lookup_length l i n k : sublist_lookup i n l = Some k → length k = n. Proof. unfold sublist_lookup; intros; simplify_option_equality. rewrite take_length, drop_length; lia. Qed. Lemma sublist_lookup_all l n : length l = n → sublist_lookup 0 n l = Some l. Proof. intros. unfold sublist_lookup; case_option_guard; [|lia]. by rewrite take_ge by (rewrite drop_length; lia). Qed. Lemma sublist_lookup_Some l i n : i + n ≤ length l → sublist_lookup i n l = Some (take n (drop i l)). Proof. by unfold sublist_lookup; intros; simplify_option_equality. Qed. Lemma sublist_lookup_None l i n : length l < i + n → sublist_lookup i n l = None. Proof. by unfold sublist_lookup; intros; simplify_option_equality by lia. Qed. Lemma sublist_eq l k n : (n | length l) → (n | length k) → (∀ i, sublist_lookup (i * n) n l = sublist_lookup (i * n) n k) → l = k. Proof. revert l k. assert (∀ l i, n ≠ 0 → (n | length l) → ¬n * i `div` n + n ≤ length l → length l ≤ i). { intros l i ? [j ->] Hjn. apply Nat.nlt_ge; contradict Hjn. rewrite <-Nat.mul_succ_r, (Nat.mul_comm n). apply Nat.mul_le_mono_r, Nat.le_succ_l, Nat.div_lt_upper_bound; lia. } intros l k Hl Hk Hlookup. destruct (decide (n = 0)) as [->|]. { by rewrite (nil_length_inv l), (nil_length_inv k) by eauto using Nat.divide_0_l. } apply list_eq; intros i. specialize (Hlookup (i `div` n)). rewrite (Nat.mul_comm _ n) in Hlookup. unfold sublist_lookup in *; simplify_option_equality; [|by rewrite !lookup_ge_None_2 by auto]. apply (f_equal (!! i `mod` n)) in Hlookup. by rewrite !lookup_take, !lookup_drop, <-!Nat.div_mod in Hlookup by (auto using Nat.mod_upper_bound with lia). Qed. Lemma sublist_eq_same_length l k j n : length l = j * n → length k = j * n → (∀ i,i < j → sublist_lookup (i * n) n l = sublist_lookup (i * n) n k) → l = k. Proof. intros Hl Hk ?. destruct (decide (n = 0)) as [->|]. { by rewrite (nil_length_inv l), (nil_length_inv k) by lia. } apply sublist_eq with n; [by exists j|by exists j|]. intros i. destruct (decide (i < j)); [by auto|]. assert (∀ m, m = j * n → m < i * n + n). { intros ? ->. replace (i * n + n) with (S i * n) by lia. apply Nat.mul_lt_mono_pos_r; lia. } by rewrite !sublist_lookup_None by auto. Qed. Lemma sublist_lookup_reshape l i n m : 0 < n → length l = m * n → reshape (replicate m n) l !! i = sublist_lookup (i * n) n l. Proof. intros Hn Hl. unfold sublist_lookup. apply option_eq; intros x; split. * intros Hx. case_option_guard as Hi. { f_equal. clear Hi. revert i l Hl Hx. induction m as [|m IH]; intros [|i] l ??; simplify_equality'; auto. rewrite <-drop_drop. apply IH; rewrite ?drop_length; auto with lia. } destruct Hi. rewrite Hl, <-Nat.mul_succ_l. apply Nat.mul_le_mono_r, Nat.le_succ_l. apply lookup_lt_Some in Hx. by rewrite reshape_length, replicate_length in Hx. * intros Hx. case_option_guard as Hi; simplify_equality'. revert i l Hl Hi. induction m as [|m IH]; [auto with lia|]. intros [|i] l ??; simpl; [done|]. rewrite <-drop_drop. rewrite IH; rewrite ?drop_length; auto with lia. Qed. Lemma sublist_lookup_compose l1 l2 l3 i n j m : sublist_lookup i n l1 = Some l2 → sublist_lookup j m l2 = Some l3 → sublist_lookup (i + j) m l1 = Some l3. Proof. unfold sublist_lookup; intros; simplify_option_equality; repeat match goal with | H : _ ≤ length _ |- _ => rewrite take_length, drop_length in H end; rewrite ?take_drop_commute, ?drop_drop, ?take_take, ?Min.min_l, Nat.add_assoc by lia; auto with lia. Qed. Lemma sublist_alter_length f l i n k : sublist_lookup i n l = Some k → length (f k) = n → length (sublist_alter f i n l) = length l. Proof. unfold sublist_alter, sublist_lookup. intros Hk ?; simplify_option_equality. rewrite !app_length, Hk, !take_length, !drop_length; lia. Qed. Lemma sublist_lookup_alter f l i n k : sublist_lookup i n l = Some k → length (f k) = n → sublist_lookup i n (sublist_alter f i n l) = f <\$> sublist_lookup i n l. Proof. unfold sublist_lookup. intros Hk ?. erewrite sublist_alter_length by eauto. unfold sublist_alter; simplify_option_equality. by rewrite Hk, drop_app_alt, take_app_alt by (rewrite ?take_length; lia). Qed. Lemma sublist_lookup_alter_ne f l i j n k : sublist_lookup j n l = Some k → length (f k) = n → i + n ≤ j ∨ j + n ≤ i → sublist_lookup i n (sublist_alter f j n l) = sublist_lookup i n l. Proof. unfold sublist_lookup. intros Hk Hi ?. erewrite sublist_alter_length by eauto. unfold sublist_alter; simplify_option_equality; f_equal; rewrite Hk. apply list_eq; intros ii. destruct (decide (ii < length (f k))); [|by rewrite !lookup_take_ge by lia]. rewrite !lookup_take, !lookup_drop by done. destruct (decide (i + ii < j)). { by rewrite lookup_app_l, lookup_take by (rewrite ?take_length; lia). } rewrite lookup_app_r by (rewrite take_length; lia). rewrite take_length_le, lookup_app_r, lookup_drop by lia. f_equal; lia. Qed. Lemma sublist_alter_all f l n : length l = n → sublist_alter f 0 n l = f l. Proof. intros <-. unfold sublist_alter; simpl. by rewrite drop_all, (right_id_L [] (++)), take_ge. Qed. Lemma sublist_alter_compose f g l i n k : sublist_lookup i n l = Some k → length (f k) = n → length (g k) = n → sublist_alter (f ∘ g) i n l = sublist_alter f i n (sublist_alter g i n l). Proof. unfold sublist_alter, sublist_lookup. intros Hk ??; simplify_option_equality. by rewrite !take_app_alt, drop_app_alt, !(associative_L (++)), drop_app_alt, take_app_alt by (rewrite ?app_length, ?take_length, ?Hk; lia). Qed. (** ** Properties of the [mask] function *) Lemma mask_nil f βs : mask f βs (@nil A) = []. Proof. by destruct βs. Qed. Lemma mask_length f βs l : length (mask f βs l) = length l. Proof. revert βs. induction l; intros [|??]; f_equal'; auto. Qed. Lemma mask_true f l n : length l ≤ n → mask f (replicate n true) l = f <\$> l. Proof. revert n. induction l; intros [|?] ?; f_equal'; auto with lia. Qed. Lemma mask_false f l n : mask f (replicate n false) l = l. Proof. revert l. induction n; intros [|??]; f_equal'; auto. Qed. Lemma mask_app f βs1 βs2 l : mask f (βs1 ++ βs2) l = mask f βs1 (take (length βs1) l) ++ mask f βs2 (drop (length βs1) l). Proof. revert l. induction βs1;intros [|??]; f_equal'; auto using mask_nil. Qed. Lemma mask_app_2 f βs l1 l2 : mask f βs (l1 ++ l2) = mask f (take (length l1) βs) l1 ++ mask f (drop (length l1) βs) l2. Proof. revert βs. induction l1; intros [|??]; f_equal'; auto. Qed. Lemma take_mask f βs l n : take n (mask f βs l) = mask f (take n βs) (take n l). Proof. revert n βs. induction l; intros [|?] [|[] ?]; f_equal'; auto. Qed. Lemma drop_mask f βs l n : drop n (mask f βs l) = mask f (drop n βs) (drop n l). Proof. revert n βs. induction l; intros [|?] [|[] ?]; f_equal'; auto using mask_nil. Qed. Lemma sublist_lookup_mask f βs l i n : sublist_lookup i n (mask f βs l) = mask f (take n (drop i βs)) <\$> sublist_lookup i n l. Proof. unfold sublist_lookup; rewrite mask_length; simplify_option_equality; auto. by rewrite drop_mask, take_mask. Qed. Lemma mask_mask f g βs1 βs2 l : (∀ x, f (g x) = f x) → βs1 =.>* βs2 → mask f βs2 (mask g βs1 l) = mask f βs2 l. Proof. intros ? Hβs. revert l. by induction Hβs as [|[] []]; intros [|??]; f_equal'. Qed. Lemma lookup_mask f βs l i : βs !! i = Some true → mask f βs l !! i = f <\$> l !! i. Proof. revert i βs. induction l; intros [] [] ?; simplify_equality'; f_equal; auto. Qed. Lemma lookup_mask_notin f βs l i : βs !! i ≠ Some true → mask f βs l !! i = l !! i. Proof. revert i βs. induction l; intros [] [|[]] ?; simplify_equality'; auto. Qed. (** ** Properties of the [seq] function *) Lemma fmap_seq j n : S <\$> seq j n = seq (S j) n. Proof. revert j. induction n; intros; f_equal'; auto. Qed. Lemma lookup_seq j n i : i < n → seq j n !! i = Some (j + i). Proof. revert j i. induction n as [|n IH]; intros j [|i] ?; simpl; auto with lia. rewrite IH; auto with lia. Qed. Lemma lookup_seq_ge j n i : n ≤ i → seq j n !! i = None. Proof. revert j i. induction n; intros j [|i] ?; simpl; auto with lia. Qed. Lemma lookup_seq_inv j n i j' : seq j n !! i = Some j' → j' = j + i ∧ i < n. Proof. destruct (le_lt_dec n i); [by rewrite lookup_seq_ge|]. rewrite lookup_seq by done. intuition congruence. Qed. (** ** Properties of the [Permutation] predicate *) Lemma Permutation_nil l : l ≡ₚ [] ↔ l = []. Proof. split. by intro; apply Permutation_nil. by intros ->. Qed. Lemma Permutation_singleton l x : l ≡ₚ [x] ↔ l = [x]. Proof. split. by intro; apply Permutation_length_1_inv. by intros ->. Qed. Definition Permutation_skip := @perm_skip A. Definition Permutation_swap := @perm_swap A. Definition Permutation_singleton_inj := @Permutation_length_1 A. Global Existing Instance Permutation_app'. Global Instance: Proper ((≡ₚ) ==> (=)) (@length A). Proof. induction 1; simpl; auto with lia. Qed. Global Instance: Commutative (≡ₚ) (@app A). Proof. intros l1. induction l1 as [|x l1 IH]; intros l2; simpl. * by rewrite (right_id_L [] (++)). * rewrite Permutation_middle, IH. simpl. by rewrite Permutation_middle. Qed. Global Instance: ∀ x : A, Injective (≡ₚ) (≡ₚ) (x ::). Proof. red. eauto using Permutation_cons_inv. Qed. Global Instance: ∀ k : list A, Injective (≡ₚ) (≡ₚ) (k ++). Proof. red. induction k as [|x k IH]; intros l1 l2; simpl; auto. intros. by apply IH, (injective (x ::)). Qed. Global Instance: ∀ k : list A, Injective (≡ₚ) (≡ₚ) (++ k). Proof. intros k l1 l2. rewrite !(commutative (++) _ k). by apply (injective (k ++)). Qed. Lemma replicate_Permutation n x l : replicate n x ≡ₚ l → replicate n x = l. Proof. intros Hl. apply replicate_as_elem_of. split. * by rewrite <-Hl, replicate_length. * intros y. rewrite <-Hl. by apply elem_of_replicate_inv. Qed. Lemma reverse_Permutation l : reverse l ≡ₚ l. Proof. induction l as [|x l IH]; [done|]. by rewrite reverse_cons, (commutative (++)), IH. Qed. `````` Robbert Krebbers committed Jan 20, 2016 1375 1376 1377 1378 1379 ``````Lemma delete_Permutation l i x : l !! i = Some x → l ≡ₚ x :: delete i l. Proof. revert i; induction l as [|y l IH]; intros [|i] ?; simplify_equality'; auto. by rewrite Permutation_swap, <-(IH i). Qed. `````` Robbert Krebbers committed Nov 11, 2015 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 `````` (** ** Properties of the [prefix_of] and [suffix_of] predicates *) Global Instance: PreOrder (@prefix_of A). Proof. split. * intros ?. eexists []. by rewrite (right_id_L [] (++)). * intros ???[k1->] [k2->]. exists (k1 ++ k2). by rewrite (associative_L (++)). Qed. Lemma prefix_of_nil l : [] `prefix_of` l. Proof. by exists l. Qed. Lemma prefix_of_nil_not x l : ¬x :: l `prefix_of` []. Proof. by intros [k ?]. Qed. Lemma prefix_of_cons x l1 l2 : l1 `prefix_of` l2 → x :: l1 `prefix_of` x :: l2. Proof. intros [k ->]. by exists k. Qed. Lemma prefix_of_cons_alt x y l1 l2 : x = y → l1 `prefix_of` l2 → x :: l1 `prefix_of` y :: l2. Proof. intros ->. apply prefix_of_cons. Qed. Lemma prefix_of_cons_inv_1 x y l1 l2 : x :: l1 `prefix_of` y :: l2 → x = y. Proof. by intros [k ?]; simplify_equality'. Qed. Lemma prefix_of_cons_inv_2 x y l1 l2 : x :: l1 `prefix_of` y :: l2 → l1 `prefix_of` l2. Proof. intros [k ?]; simplify_equality'. by exists k. Qed. Lemma prefix_of_app k l1 l2 : l1 `prefix_of` l2 → k ++ l1 `prefix_of` k ++ l2. Proof. intros [k' ->]. exists k'. by rewrite (associative_L (++)). Qed. Lemma prefix_of_app_alt k1 k2 l1 l2 : k1 = k2 → l1 `prefix_of` l2 → k1 ++ l1 `prefix_of` k2 ++ l2. Proof. intros ->. apply prefix_of_app. Qed. Lemma prefix_of_app_l l1 l2 l3 : l1 ++ l3 `prefix_of` l2 → l1 `prefix_of` l2. Proof. intros [k ->]. exists (l3 ++ k). by rewrite (associative_L (++)). Qed. Lemma prefix_of_app_r l1 l2 l3 : l1 `prefix_of` l2 → l1 `prefix_of` l2 ++ l3. Proof. intros [k ->]. exists (k ++ l3). by rewrite (associative_L (++)). Qed. Lemma prefix_of_length l1 l2 : l1 `prefix_of` l2 → length l1 ≤ length l2. Proof. intros [? ->]. rewrite app_length. lia. Qed. Lemma prefix_of_snoc_not l x : ¬l ++ [x] `prefix_of` l. Proof. intros [??]. discriminate_list_equality. Qed. Global Instance: PreOrder (@suffix_of A). Proof. split. * intros ?. by eexists []. * intros ???[k1->] [k2->]. exists (k2 ++ k1). by rewrite (associative_L (++)). Qed. Global Instance prefix_of_dec `{∀ x y, Decision (x = y)} : ∀ l1 l2, Decision (l1 `prefix_of` l2) := fix go l1 l2 := match l1, l2 return { l1 `prefix_of` l2 } + { ¬l1 `prefix_of` l2 } with | [], _ => left (prefix_of_nil _) | _, [] => right (prefix_of_nil_not _ _) | x :: l1, y :: l2 => match decide_rel (=) x y with | left Hxy => match go l1 l2 with | left Hl1l2 => left (prefix_of_cons_alt _ _ _ _ Hxy Hl1l2) | right Hl1l2 => right (Hl1l2 ∘ prefix_of_cons_inv_2 _ _ _ _) end | right Hxy => right (Hxy ∘ prefix_of_cons_inv_1 _ _ _ _) end end. Section prefix_ops. Context `{∀ x y, Decision (x = y)}. Lemma max_prefix_of_fst l1 l2 : l1 = (max_prefix_of l1 l2).2 ++ (max_prefix_of l1 l2).1.1. Proof. revert l2. induction l1; intros [|??]; simpl; repeat case_decide; f_equal'; auto. Qed. Lemma max_prefix_of_fst_alt l1 l2 k1 k2 k3 : max_prefix_of l1 l2 = (k1, k2, k3) → l1 = k3 ++ k1. Proof. intros. pose proof (max_prefix_of_fst l1 l2). by destruct (max_prefix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_prefix_of_fst_prefix l1 l2 : (max_prefix_of l1 l2).2 `prefix_of` l1. Proof. eexists. apply max_prefix_of_fst. Qed. Lemma max_prefix_of_fst_prefix_alt l1 l2 k1 k2 k3 : max_prefix_of l1 l2 = (k1, k2, k3) → k3 `prefix_of` l1. Proof. eexists. eauto using max_prefix_of_fst_alt. Qed. Lemma max_prefix_of_snd l1 l2 : l2 = (max_prefix_of l1 l2).2 ++ (max_prefix_of l1 l2).1.2. Proof. revert l2. induction l1; intros [|??]; simpl; repeat case_decide; f_equal'; auto. Qed. Lemma max_prefix_of_snd_alt l1 l2 k1 k2 k3 : max_prefix_of l1 l2 = (k1, k2, k3) → l2 = k3 ++ k2. Proof. intro. pose proof (max_prefix_of_snd l1 l2). by destruct (max_prefix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_prefix_of_snd_prefix l1 l2 : (max_prefix_of l1 l2).2 `prefix_of` l2. Proof. eexists. apply max_prefix_of_snd. Qed. Lemma max_prefix_of_snd_prefix_alt l1 l2 k1 k2 k3 : max_prefix_of l1 l2 = (k1,k2,k3) → k3 `prefix_of` l2. Proof. eexists. eauto using max_prefix_of_snd_alt. Qed. Lemma max_prefix_of_max l1 l2 k : k `prefix_of` l1 → k `prefix_of` l2 → k `prefix_of` (max_prefix_of l1 l2).2. Proof. intros [l1' ->] [l2' ->]. by induction k; simpl; repeat case_decide; simpl; auto using prefix_of_nil, prefix_of_cons. Qed. Lemma max_prefix_of_max_alt l1 l2 k1 k2 k3 k : max_prefix_of l1 l2 = (k1,k2,k3) → k `prefix_of` l1 → k `prefix_of` l2 → k `prefix_of` k3. Proof. intro. pose proof (max_prefix_of_max l1 l2 k). by destruct (max_prefix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_prefix_of_max_snoc l1 l2 k1 k2 k3 x1 x2 : max_prefix_of l1 l2 = (x1 :: k1, x2 :: k2, k3) → x1 ≠ x2. Proof. intros Hl ->. destruct (prefix_of_snoc_not k3 x2). eapply max_prefix_of_max_alt; eauto. * rewrite (max_prefix_of_fst_alt _ _ _ _ _ Hl). apply prefix_of_app, prefix_of_cons, prefix_of_nil. * rewrite (max_prefix_of_snd_alt _ _ _ _ _ Hl). apply prefix_of_app, prefix_of_cons, prefix_of_nil. Qed. End prefix_ops. Lemma prefix_suffix_reverse l1 l2 : l1 `prefix_of` l2 ↔ reverse l1 `suffix_of` reverse l2. Proof. split; intros [k E]; exists (reverse k). * by rewrite E, reverse_app. * by rewrite <-(reverse_involutive l2), E, reverse_app, reverse_involutive. Qed. Lemma suffix_prefix_reverse l1 l2 : l1 `suffix_of` l2 ↔ reverse l1 `prefix_of` reverse l2. Proof. by rewrite prefix_suffix_reverse, !reverse_involutive. Qed. Lemma suffix_of_nil l : [] `suffix_of` l. Proof. exists l. by rewrite (right_id_L [] (++)). Qed. Lemma suffix_of_nil_inv l : l `suffix_of` [] → l = []. Proof. by intros [[|?] ?]; simplify_list_equality. Qed. Lemma suffix_of_cons_nil_inv x l : ¬x :: l `suffix_of` []. Proof. by intros [[] ?]. Qed. Lemma suffix_of_snoc l1 l2 x : l1 `suffix_of` l2 → l1 ++ [x] `suffix_of` l2 ++ [x]. Proof. intros [k ->]. exists k. by rewrite (associative_L (++)). Qed. Lemma suffix_of_snoc_alt x y l1 l2 : x = y → l1 `suffix_of` l2 → l1 ++ [x] `suffix_of` l2 ++ [y]. Proof. intros ->. apply suffix_of_snoc. Qed. Lemma suffix_of_app l1 l2 k : l1 `suffix_of` l2 → l1 ++ k `suffix_of` l2 ++ k. Proof. intros [k' ->]. exists k'. by rewrite (associative_L (++)). Qed. Lemma suffix_of_app_alt l1 l2 k1 k2 : k1 = k2 → l1 `suffix_of` l2 → l1 ++ k1 `suffix_of` l2 ++ k2. Proof. intros ->. apply suffix_of_app. Qed. Lemma suffix_of_snoc_inv_1 x y l1 l2 : l1 ++ [x] `suffix_of` l2 ++ [y] → x = y. Proof. intros [k' E]. rewrite (associative_L (++)) in E. by simplify_list_equality. Qed. Lemma suffix_of_snoc_inv_2 x y l1 l2 : l1 ++ [x] `suffix_of` l2 ++ [y] → l1 `suffix_of` l2. Proof. intros [k' E]. exists k'. rewrite (associative_L (++)) in E. by simplify_list_equality. Qed. Lemma suffix_of_app_inv l1 l2 k : l1 ++ k `suffix_of` l2 ++ k → l1 `suffix_of` l2. Proof. intros [k' E]. exists k'. rewrite (associative_L (++)) in E. by simplify_list_equality. Qed. Lemma suffix_of_cons_l l1 l2 x : x :: l1 `suffix_of` l2 → l1 `suffix_of` l2. Proof. intros [k ->]. exists (k ++ [x]). by rewrite <-(associative_L (++)). Qed. Lemma suffix_of_app_l l1 l2 l3 : l3 ++ l1 `suffix_of` l2 → l1 `suffix_of` l2. Proof. intros [k ->]. exists (k ++ l3). by rewrite <-(associative_L (++)). Qed. Lemma suffix_of_cons_r l1 l2 x : l1 `suffix_of` l2 → l1 `suffix_of` x :: l2. Proof. intros [k ->]. by exists (x :: k). Qed. Lemma suffix_of_app_r l1 l2 l3 : l1 `suffix_of` l2 → l1 `suffix_of` l3 ++ l2. Proof. intros [k ->]. exists (l3 ++ k). by rewrite (associative_L (++)). Qed. Lemma suffix_of_cons_inv l1 l2 x y : x :: l1 `suffix_of` y :: l2 → x :: l1 = y :: l2 ∨ x :: l1 `suffix_of` l2. Proof. intros [[|? k] E]; [by left|]. right. simplify_equality'. by apply suffix_of_app_r. Qed. Lemma suffix_of_length l1 l2 : l1 `suffix_of` l2 → length l1 ≤ length l2. Proof. intros [? ->]. rewrite app_length. lia. Qed. Lemma suffix_of_cons_not x l : ¬x :: l `suffix_of` l. Proof. intros [??]. discriminate_list_equality. Qed. Global Instance suffix_of_dec `{∀ x y, Decision (x = y)} l1 l2 : Decision (l1 `suffix_of` l2). Proof. refine (cast_if (decide_rel prefix_of (reverse l1) (reverse l2))); abstract (by rewrite suffix_prefix_reverse). Defined. Section max_suffix_of. Context `{∀ x y, Decision (x = y)}. Lemma max_suffix_of_fst l1 l2 : l1 = (max_suffix_of l1 l2).1.1 ++ (max_suffix_of l1 l2).2. Proof. rewrite <-(reverse_involutive l1) at 1. rewrite (max_prefix_of_fst (reverse l1) (reverse l2)). unfold max_suffix_of. destruct (max_prefix_of (reverse l1) (reverse l2)) as ((?&?)&?); simpl. by rewrite reverse_app. Qed. Lemma max_suffix_of_fst_alt l1 l2 k1 k2 k3 : max_suffix_of l1 l2 = (k1, k2, k3) → l1 = k1 ++ k3. Proof. intro. pose proof (max_suffix_of_fst l1 l2). by destruct (max_suffix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_suffix_of_fst_suffix l1 l2 : (max_suffix_of l1 l2).2 `suffix_of` l1. Proof. eexists. apply max_suffix_of_fst. Qed. Lemma max_suffix_of_fst_suffix_alt l1 l2 k1 k2 k3 : max_suffix_of l1 l2 = (k1, k2, k3) → k3 `suffix_of` l1. Proof. eexists. eauto using max_suffix_of_fst_alt. Qed. Lemma max_suffix_of_snd l1 l2 : l2 = (max_suffix_of l1 l2).1.2 ++ (max_suffix_of l1 l2).2. Proof. rewrite <-(reverse_involutive l2) at 1. rewrite (max_prefix_of_snd (reverse l1) (reverse l2)). unfold max_suffix_of. destruct (max_prefix_of (reverse l1) (reverse l2)) as ((?&?)&?); simpl. by rewrite reverse_app. Qed. Lemma max_suffix_of_snd_alt l1 l2 k1 k2 k3 : max_suffix_of l1 l2 = (k1,k2,k3) → l2 = k2 ++ k3. Proof. intro. pose proof (max_suffix_of_snd l1 l2). by destruct (max_suffix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_suffix_of_snd_suffix l1 l2 : (max_suffix_of l1 l2).2 `suffix_of` l2. Proof. eexists. apply max_suffix_of_snd. Qed. Lemma max_suffix_of_snd_suffix_alt l1 l2 k1 k2 k3 : max_suffix_of l1 l2 = (k1,k2,k3) → k3 `suffix_of` l2. Proof. eexists. eauto using max_suffix_of_snd_alt. Qed. Lemma max_suffix_of_max l1 l2 k : k `suffix_of` l1 → k `suffix_of` l2 → k `suffix_of` (max_suffix_of l1 l2).2. Proof. generalize (max_prefix_of_max (reverse l1) (reverse l2)). rewrite !suffix_prefix_reverse. unfold max_suffix_of. destruct (max_prefix_of (reverse l1) (reverse l2)) as ((?&?)&?); simpl. rewrite reverse_involutive. auto. Qed. Lemma max_suffix_of_max_alt l1 l2 k1 k2 k3 k : max_suffix_of l1 l2 = (k1, k2, k3) → k `suffix_of` l1 → k `suffix_of` l2 → k `suffix_of` k3. Proof. intro. pose proof (max_suffix_of_max l1 l2 k). by destruct (max_suffix_of l1 l2) as [[]?]; simplify_equality. Qed. Lemma max_suffix_of_max_snoc l1 l2 k1 k2 k3 x1 x2 : max_suffix_of l1 l2 = (k1 ++ [x1], k2 ++ [x2], k3) → x1 ≠ x2. Proof. intros Hl ->. destruct (suffix_of_cons_not x2 k3). eapply max_suffix_of_max_alt; eauto. * rewrite (max_suffix_of_fst_alt _ _ _ _ _ Hl). by apply (suffix_of_app [x2]), suffix_of_app_r. * rewrite (max_suffix_of_snd_alt _ _ _ _ _ Hl). by apply (suffix_of_app [x2]), suffix_of_app_r. Qed. End max_suffix_of. (** ** Properties of the [sublist] predicate *) Lemma sublist_length l1 l2 : l1 `sublist` l2 → length l1 ≤ length l2. Proof. induction 1; simpl; auto with arith. Qed. Lemma sublist_nil_l l : [] `sublist` l. Proof. induction l; try constructor; auto. Qed. Lemma sublist_nil_r l : l `sublist` [] ↔ l = []. Proof. split. by inversion 1. intros ->. constructor. Qed. Lemma sublist_app l1 l2 k1 k2 : l1 `sublist` l2 → k1 `sublist` k2 → l1 ++ k1 `sublist` l2 ++ k2. Proof. induction 1; simpl; try constructor; auto. Qed. Lemma sublist_inserts_l k l1 l2 : l1 `sublist` l2 → l1 `sublist` k ++ l2. Proof. induction k; try constructor; auto. Qed. Lemma sublist_inserts_r k l1 l2 : l1 `sublist` l2 → l1 `sublist` l2 ++ k. Proof. induction 1; simpl; try constructor; auto using sublist_nil_l. Qed. Lemma sublist_cons_r x l k : l `sublist` x :: k ↔ l `sublist` k ∨ ∃ l', l = x :: l' ∧ l' `sublist` k. Proof. split. inversion 1; eauto. intros [?|(?&->&?)]; constructor; auto. Qed. Lemma sublist_cons_l x l k : x :: l `sublist` k ↔ ∃ k1 k2, k = k1 ++ x :: k2 ∧ l `sublist` k2. Proof. split. * intros Hlk. induction k as [|y k IH]; inversion Hlk. + eexists [], k. by repeat constructor. `````` Robbert Krebbers committed Nov 11, 2015 1659 `````` + destruct IH as (k1&k2&->&?); auto. by exists (y :: k1), k2. `````` Robbert Krebbers committed Nov 11, 2015 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 `````` * intros (k1&k2&->&?). by apply sublist_inserts_l, sublist_skip. Qed. Lemma sublist_app_r l k1 k2 : l `sublist` k1 ++ k2 ↔ ∃ l1 l2, l = l1 ++ l2 ∧ l1 `sublist` k1 ∧ l2 `sublist` k2. Proof. split. * revert l k2. induction k1 as [|y k1 IH]; intros l k2; simpl. { eexists [], l. by repeat constructor. } rewrite sublist_cons_r. intros [?|(l' & ? &?)]; subst. + destruct (IH l k2) as (l1&l2&?&?&?); trivial; subst. `````` Robbert Krebbers committed Nov 11, 2015 1671 `````` exists l1, l2. auto using sublist_cons. `````` Robbert Krebbers committed Nov 11, 2015 1672 `````` + destruct (IH l' k2) as (l1&l2&?&?&?); trivial; subst. `````` Robbert Krebbers committed Nov 11, 2015 1673 `````` exists (y :: l1), l2. auto using sublist_skip. `````` Robbert Krebbers committed Nov 11, 2015 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 `````` * intros (?&?&?&?&?); subst. auto using sublist_app. Qed. Lemma sublist_app_l l1 l2 k : l1 ++ l2 `sublist` k ↔ ∃ k1 k2, k = k1 ++ k2 ∧ l1 `sublist` k1 ∧ l2 `sublist` k2. Proof. split. * revert l2 k. induction l1 as [|x l1 IH]; intros l2 k; simpl. { eexists [], k. by repeat constructor. } rewrite sublist_cons_l. intros (k1 & k2 &?&?); subst. destruct (IH l2 k2) as (h1 & h2 &?&?&?); trivial; subst. `````` Robbert Krebbers committed Nov 11, 2015 1685 `````` exists (k1 ++ x :: h1), h2. rewrite <-(associative_L (++)). `````` Robbert Krebbers committed Nov 11, 2015 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 `````` auto using sublist_inserts_l, sublist_skip. * intros (?&?&?&?&?); subst. auto using sublist_app. Qed. Lemma sublist_app_inv_l k l1 l2 : k ++ l1 `sublist` k ++ l2 → l1 `sublist` l2. Proof. induction k as [|y k IH]; simpl; [done |]. rewrite sublist_cons_r. intros [Hl12|(?&?&?)]; [|simplify_equality; eauto]. rewrite sublist_cons_l in Hl12. destruct Hl12 as (k1&k2&Hk&?). apply IH. rewrite Hk. eauto using sublist_inserts_l, sublist_cons. Qed. Lemma sublist_app_inv_r k l1 l2 : l1 ++ k `sublist` l2 ++ k → l1 `sublist` l2. Proof. revert l1 l2. induction k as [|y k IH]; intros l1 l2. { by rewrite !(right_id_L [] (++)). } intros. feed pose proof (IH (l1 ++ [y]) (l2 ++ [y])) as Hl12. { by rewrite <-!(associative_L (++)). } rewrite sublist_app_l in Hl12. destruct Hl12 as (k1&k2&E&?&Hk2). destruct k2 as [|z k2] using rev_ind; [inversion Hk2|]. rewrite (associative_L (++)) in E; simplify_list_equality. eauto using sublist_inserts_r. Qed. Global Instance: PartialOrder (@sublist A). Proof. split; [split|]. * intros l. induction l; constructor; auto. * intros l1 l2 l3 Hl12. revert l3. induction Hl12. + auto using sublist_nil_l. + intros ?. rewrite sublist_cons_l. intros (?&?&?&?); subst. eauto using sublist_inserts_l, sublist_skip. + intros ?. rewrite sublist_cons_l. intros (?&?&?&?); subst. eauto using sublist_inserts_l, sublist_cons. * intros l1 l2 Hl12 Hl21. apply sublist_length in Hl21. induction Hl12; f_equal'; auto with arith. apply sublist_length in Hl12. lia. Qed. Lemma sublist_take l i : take i l `sublist` l. Proof. rewrite <-(take_drop i l) at 2. by apply sublist_inserts_r. Qed. Lemma sublist_drop l i : drop i l `sublist` l. Proof. rewrite <-(take_drop i l) at 2. by apply sublist_inserts_l. Qed. Lemma sublist_delete l i : delete i l `sublist` l. Proof. revert i. by induction l; intros [|?]; simpl; constructor. Qed. Lemma sublist_foldr_delete l is : foldr delete l is `sublist` l. Proof. induction is as [|i is IH]; simpl; [done |]. transitivity (foldr delete l is); auto using sublist_delete. Qed. Lemma sublist_alt l1 l2 : l1 `sublist` l2 ↔ ∃ is, l1 = foldr delete l2 is. Proof. split; [|intros [is ->]; apply sublist_foldr_delete]. intros Hl12. cut (∀ k, ∃ is, k ++ l1 = foldr delete (k ++ l2) is). { intros help. apply (help []). } induction Hl12 as [|x l1 l2 _ IH|x l1 l2 _ IH]; intros k. * by eexists []. * destruct (IH (k ++ [x])) as [is His]. exists is. by rewrite <-!(associative_L (++)) in His. * destruct (IH k) as [is His]. exists (is ++ [length k]). rewrite fold_right_app. simpl. by rewrite delete_middle. Qed. Lemma Permutation_sublist l1 l2 l3 : l1 ≡ₚ l2 → l2 `sublist` l3 → ∃ l4, l1 `sublist` l4 ∧ l4 ≡ₚ l3. Proof. intros Hl1l2. revert l3. induction Hl1l2 as [|x l1 l2 ? IH|x y l1|l1 l1' l2 ? IH1 ? IH2]. * intros l3. by exists l3. * intros l3. rewrite sublist_cons_l. intros (l3'&l3''&?&?); subst. destruct (IH l3'') as (l4&?&Hl4); auto. exists (l3' ++ x :: l4). split. by apply sublist_inserts_l, sublist_skip. by rewrite Hl4. * intros l3. rewrite sublist_cons_l. intros (l3'&l3''&?& Hl3); subst. rewrite sublist_cons_l in Hl3. destruct Hl3 as (l5'&l5''&?& Hl5); subst. exists (l3' ++ y :: l5' ++ x :: l5''). split. - by do 2 apply sublist_inserts_l, sublist_skip. - by rewrite !Permutation_middle, Permutation_swap. * intros l3 ?. destruct (IH2 l3) as (l3'&?&?); trivial. destruct (IH1 l3') as (l3'' &?&?); trivial. exists l3''. split. done. etransitivity; eauto. Qed. Lemma sublist_Permutation l1 l2 l3 : l1 `sublist` l2 → l2 ≡ₚ l3 → ∃ l4, l1 ≡ₚ l4 ∧ l4 `sublist` l3. Proof. intros Hl1l2 Hl2l3. revert l1 Hl1l2. induction Hl2l3 as [|x l2 l3 ? IH|x y l2|l2 l2' l3 ? IH1 ? IH2]. * intros l1. by exists l1. * intros l1. rewrite sublist_cons_r. intros [?|(l1'&l1''&?)]; subst. { destruct (IH l1) as (l4&?&?); trivial. exists l4. split. done. by constructor. } destruct (IH l1') as (l4&?&Hl4); auto. exists (x :: l4). split. by constructor. by constructor. * intros l1. rewrite sublist_cons_r. intros [Hl1|(l1'&l1''&Hl1)]; subst. { exists l1. split; [done|]. rewrite sublist_cons_r in Hl1. destruct Hl1 as [?|(l1'&?&?)]; subst; by repeat constructor. } rewrite sublist_cons_r in Hl1. destruct Hl1 as [?|(l1''&?&?)]; subst. + exists (y :: l1'). by repeat constructor. + exists (x :: y :: l1''). by repeat constructor. * intros l1 ?. destruct (IH1 l1) as (l3'&?&?); trivial. destruct (IH2 l3') as (l3'' &?&?); trivial. exists l3''. split; [|done]. etransitivity; eauto. Qed. (** Properties of the [contains] predicate *) Lemma contains_length l1 l2 : l1 `contains` l2 → length l1 ≤ length l2. Proof. induction 1; simpl; auto with lia. Qed. Lemma contains_nil_l l : [] `contains` l. Proof. induction l; constructor; auto. Qed. Lemma contains_nil_r l : l `contains` [] ↔ l = []. Proof. split; [|intros ->; constructor]. intros Hl. apply contains_length in Hl. destruct l; simpl in *; auto with lia. Qed. Global Instance: PreOrder (@contains A). Proof. split. * intros l. induction l; constructor; auto. * red. apply contains_trans. Qed. Lemma Permutation_contains l1 l2 : l1 ≡ₚ l2 → l1 `contains` l2. Proof. induction 1; econstructor; eauto. Qed. Lemma sublist_contains l1 l2 : l1 `sublist` l2 → l1 `contains` l2. Proof. induction 1; constructor; auto. Qed. Lemma contains_Permutation l1 l2 : l1 `contains` l2 → ∃ k, l2 ≡ₚ l1 ++ k. Proof. induction 1 as [|x y l ? [k Hk]| |x l1 l2 ? [k Hk]|l1 l2 l3 ? [k Hk] ? [k' Hk']]. * by eexists []. * exists k. by rewrite Hk. * eexists []. rewrite (right_id_L [] (++)). by constructor. * exists (x :: k). by rewrite Hk, Permutation_middle. * exists (k ++ k'). by rewrite Hk', Hk, (associative_L (++)). Qed. Lemma contains_Permutation_length_le l1 l2 : length l2 ≤ length l1 → l1 `contains` l2 → l1 ≡ₚ l2. Proof. intros Hl21 Hl12. destruct (contains_Permutation l1 l2) as [[|??] Hk]; auto. * by rewrite Hk, (right_id_L [] (++)). * rewrite Hk, app_length in Hl21; simpl in Hl21; lia. Qed. Lemma contains_Permutation_length_eq l1 l2 : length l2 = length l1 → l1 `contains` l2 → l1 ≡ₚ l2. Proof. intro. apply contains_Permutation_length_le. lia. Qed. Global Instance: Proper ((≡ₚ) ==> (≡ₚ) ==> iff) (@contains A). Proof. intros l1 l2 ? k1 k2 ?. split; intros. * transitivity l1. by apply Permutation_contains. transitivity k1. done. by apply Permutation_contains. * transitivity l2. by apply Permutation_contains. transitivity k2. done. by apply Permutation_contains. Qed. Global Instance: AntiSymmetric (≡ₚ) (@contains A). Proof. red. auto using contains_Permutation_length_le, contains_length. Qed. Lemma contains_take l i : take i l `contains` l. Proof. auto using sublist_take, sublist_contains. Qed. Lemma contains_drop l i : drop i l `contains` l. Proof. auto using sublist_drop, sublist_contains. Qed. Lemma contains_delete l i : delete i l `contains` l. Proof. auto using sublist_delete, sublist_contains. Qed. Lemma contains_foldr_delete l is : foldr delete l is `sublist` l. Proof. auto using sublist_foldr_delete, sublist_contains. Qed. Lemma contains_sublist_l l1 l3 : l1 `contains` l3 ↔ ∃ l2, l1 `sublist` l2 ∧ l2 ≡ₚ l3. Proof. split. { intros Hl13. elim Hl13; clear l1 l3 Hl13. * by eexists []. * intros x l1 l3 ? (l2&?&?). exists (x :: l2). by repeat constructor. * intros x y l. exists (y :: x :: l). by repeat constructor. * intros x l1 l3 ? (l2&?&?). exists (x :: l2). by repeat constructor. * intros l1 l3 l5 ? (l2&?&?) ? (l4&?&?). destruct (Permutation_sublist l2 l3 l4) as (l3'&?&?); trivial. exists l3'. split; etransitivity; eauto. } intros (l2&?&?). transitivity l2; auto using sublist_contains, Permutation_contains. Qed. Lemma contains_sublist_r l1 l3 : l1 `contains` l3 ↔ ∃ l2, l1 ≡ₚ l2 ∧ l2 `sublist` l3. Proof. rewrite contains_sublist_l. split; intros (l2&?&?); eauto using sublist_Permutation, Permutation_sublist. Qed. Lemma contains_inserts_l k l1 l2 : l1 `contains` l2 → l1 `contains` k ++ l2. Proof. induction k; try constructor; auto. Qed. Lemma contains_inserts_r k l1 l2 : l1 `contains` l2 → l1 `contains` l2 ++ k. Proof. rewrite (commutative (++)). apply contains_inserts_l. Qed. Lemma contains_skips_l k l1 l2 : l1 `contains` l2 → k ++ l1 `contains` k ++ l2. Proof. induction k; try constructor; auto. Qed. Lemma contains_skips_r k l1 l2 : l1 `contains` l2 → l1 ++ k `contains` l2 ++ k. Proof. rewrite !(commutative (++) _ k). apply contains_skips_l. Qed. Lemma contains_app l1 l2 k1 k2 : l1 `contains` l2 → k1 `contains` k2 → l1 ++ k1 `contains` l2 ++ k2. Proof. transitivity (l1 ++ k2); auto using contains_skips_l, contains_skips_r. Qed. Lemma contains_cons_r x l k : l `contains` x :: k ↔ l `contains` k ∨ ∃ l', l ≡ₚ x :: l' ∧ l' `contains` k. Proof. split. * rewrite contains_sublist_r. intros (l'&E&Hl'). rewrite sublist_cons_r in Hl'. destruct Hl' as [?|(?&?&?)]; subst. + left. rewrite E. eauto using sublist_contains. + right. eauto using sublist_contains. * intros [?|(?&E&?)]; [|rewrite E]; by constructor. Qed. Lemma contains_cons_l x l k : x :: l `contains` k ↔ ∃ k', k ≡ₚ x :: k' ∧ l `contains` k'. Proof. split. * rewrite contains_sublist_l. intros (l'&Hl'&E). rewrite sublist_cons_l in Hl'. destruct Hl' as (k1&k2&?&?); subst. exists (k1 ++ k2). split; eauto using contains_inserts_l, sublist_contains. by rewrite Permutation_middle. * intros (?&E&?). rewrite E. by constructor. Qed. Lemma contains_app_r l k1 k2 : l `contains` k1 ++ k2 ↔ ∃ l1 l2, l ≡ₚ l1 ++ l2 ∧ l1 `contains` k1 ∧ l2 `contains` k2. Proof. split. * rewrite contains_sublist_r. intros (l'&E&Hl'). rewrite sublist_app_r in Hl'. destruct Hl' as (l1&l2&?&?&?); subst. `````` Robbert Krebbers committed Nov 11, 2015 1903 `````` exists l1, l2. eauto using sublist_contains. `````` Robbert Krebbers committed Nov 11, 2015 1904 1905 1906 1907 1908 1909 1910 1911 1912 `````` * intros (?&?&E&?&?). rewrite E. eauto using contains_app. Qed. Lemma contains_app_l l1 l2 k : l1 ++ l2 `contains` k ↔ ∃ k1 k2, k ≡ₚ k1 ++ k2 ∧ l1 `contains` k1 ∧ l2 `contains` k2. Proof. split. * rewrite contains_sublist_l. intros (l'&Hl'&E). rewrite sublist_app_l in Hl'. destruct Hl' as (k1&k2&?&?&?); subst. `````` Robbert Krebbers committed Nov 11, 2015 1913 `````` exists k1, k2. split. done. eauto using sublist_contains. `````` Robbert Krebbers committed Nov 11, 2015 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145 2146 2147 2148 2149 2150 2151 2152 2153 2154 2155 2156 2157 2158 2159 2160 2161 2162 2163 2164 2165 2166 2167 2168 2169 2170 2171 2172 2173 2174 2175 2176 2177 2178 2179 2180 2181 2182 2183 2184 2185 2186 2187 2188 2189 2190 2191 2192 2193 2194 2195 2196 2197 2198 2199 2200 2201 2202 2203 2204 2205 2206 2207 2208 2209 2210 2211 2212 2213 2214 2215 2216 2217 2218 2219 2220 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310 2311 2312 2313 2314 2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325 2326 2327 2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 2340 2341 2342 2343 2344 2345 2346 2347 2348 2349 2350 2351 2352 2353 2354 2355 2356 2357 2358 2359 2360 2361 2362 2363 2364 2365 2366 2367 2368 2369 2370 2371 2372 2373 2374 2375 2376 2377 2378 2379 2380 2381 2382 2383 2384 2385 2386 2387 2388 2389 2390 2391 2392 2393 2394 2395 2396 2397 2398 2399 2400 2401 2402 2403 2404 2405 2406 2407 2408 2409 2410 2411 2412 2413 2414 2415 2416 2417 2418 2419 2420 2421 2422 2423 2424 2425 2426 2427 2428 2429 2430 2431 2432 2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444 2445 2446 2447 2448 2449 2450 2451 2452 2453 2454 2455 2456 2457 2458 2459 2460 2461 2462 2463 2464 2465 2466 2467 2468 2469 2470 2471 2472 2473 2474 2475 2476 2477 2478 2479 2480 2481 2482 2483 2484 2485 2486 2487 2488 2489 2490 2491 2492 2493 2494 2495 2496 2497 2498 2499 2500 2501 2502 2503 2504 2505 2506 2507 2508 2509 2510 2511 2512 2513 2514 2515 2516 2517 2518 2519 2520 2521 2522 2523 2524 2525 2526 2527 2528 2529 `````` * intros (?&?&E&?&?). rewrite E. eauto using contains_app. Qed. Lemma contains_app_inv_l l1 l2 k : k ++ l1 `contains` k ++ l2 → l1 `contains` l2. Proof. induction k as [|y k IH]; simpl; [done |]. rewrite contains_cons_l. intros (?&E&?). apply Permutation_cons_inv in E. apply IH. by rewrite E. Qed. Lemma contains_app_inv_r l1 l2 k : l1 ++ k `contains` l2 ++ k → l1 `contains` l2. Proof. revert l1 l2. induction k as [|y k IH]; intros l1 l2. { by rewrite !(right_id_L [] (++)). } intros. feed pose proof (IH (l1 ++ [y]) (l2 ++ [y])) as Hl12. { by rewrite <-!(associative_L (++)). } rewrite contains_app_l in Hl12. destruct Hl12 as (k1&k2&E1&?&Hk2). rewrite contains_cons_l in Hk2. destruct Hk2 as (k2'&E2&?). rewrite E2, (Permutation_cons_append k2'), (associative_L (++)) in E1. apply Permutation_app_inv_r in E1. rewrite E1. eauto using contains_inserts_r. Qed. Lemma contains_cons_middle x l k1 k2 : l `contains` k1 ++ k2 → x :: l `contains` k1 ++ x :: k2. Proof. rewrite <-Permutation_middle. by apply contains_skip. Qed. Lemma contains_app_middle l1 l2 k1 k2 : l2 `contains` k1 ++ k2 → l1 ++ l2 `contains` k1 ++ l1 ++ k2. Proof. rewrite !(associative (++)), (commutative (++) k1 l1), <-(associative_L (++)). by apply contains_skips_l. Qed. Lemma contains_middle l k1 k2 : l `contains` k1 ++ l ++ k2. Proof. by apply contains_inserts_l, contains_inserts_r. Qed. Lemma Permutation_alt l1 l2 : l1 ≡ₚ l2 ↔ length l1 = length l2 ∧ l1 `contains` l2. Proof. split. * by intros Hl; rewrite Hl. * intros [??]; auto using contains_Permutation_length_eq. Qed. Lemma NoDup_contains l k : NoDup l → (∀ x, x ∈ l → x ∈ k) → l `contains` k. Proof. intros Hl. revert k. induction Hl as [|x l Hx ? IH]. { intros k Hk. by apply contains_nil_l. } intros k Hlk. destruct (elem_of_list_split k x) as (l1&l2&?); subst. { apply Hlk. by constructor. } rewrite <-Permutation_middle. apply contains_skip, IH. intros y Hy. rewrite elem_of_app. specialize (Hlk y). rewrite elem_of_app, !elem_of_cons in Hlk. by destruct Hlk as [?|[?|?]]; subst; eauto. Qed. Lemma NoDup_Permutation l k : NoDup l → NoDup k → (∀ x, x ∈ l ↔ x ∈ k) → l ≡ₚ k. Proof. intros. apply (anti_symmetric contains); apply NoDup_contains; naive_solver. Qed. Section contains_dec. Context `{∀ x y, Decision (x = y)}. Lemma list_remove_Permutation l1 l2 k1 x : l1 ≡ₚ l2 → list_remove x l1 = Some k1 → ∃ k2, list_remove x l2 = Some k2 ∧ k1 ≡ₚ k2. Proof. intros Hl. revert k1. induction Hl as [|y l1 l2 ? IH|y1 y2 l|l1 l2 l3 ? IH1 ? IH2]; simpl; intros k1 Hk1. * done. * case_decide; simplify_equality; eauto. destruct (list_remove x l1) as [l|] eqn:?; simplify_equality. destruct (IH l) as (?&?&?); simplify_option_equality; eauto. * simplify_option_equality; eauto using Permutation_swap. * destruct (IH1 k1) as (k2&?&?); trivial. destruct (IH2 k2) as (k3&?&?); trivial. exists k3. split; eauto. by transitivity k2. Qed. Lemma list_remove_Some l k x : list_remove x l = Some k → l ≡ₚ x :: k. Proof. revert k. induction l as [|y l IH]; simpl; intros k ?; [done |]. simplify_option_equality; auto. by rewrite Permutation_swap, <-IH. Qed. Lemma list_remove_Some_inv l k x : l ≡ₚ x :: k → ∃ k', list_remove x l = Some k' ∧ k ≡ₚ k'. Proof. intros. destruct (list_remove_Permutation (x :: k) l k x) as (k'&?&?). * done. * simpl; by case_decide. * by exists k'. Qed. Lemma list_remove_list_contains l1 l2 : l1 `contains` l2 ↔ is_Some (list_remove_list l1 l2). Proof. split. * revert l2. induction l1 as [|x l1 IH]; simpl. { intros l2 _. by exists l2. } intros l2. rewrite contains_cons_l. intros (k&Hk&?). destruct (list_remove_Some_inv l2 k x) as (k2&?&Hk2); trivial. simplify_option_equality. apply IH. by rewrite <-Hk2. * intros [k Hk]. revert l2 k Hk. induction l1 as [|x l1 IH]; simpl; intros l2 k. { intros. apply contains_nil_l. } destruct (list_remove x l2) as [k'|] eqn:?; intros; simplify_equality. rewrite contains_cons_l. eauto using list_remove_Some. Qed. Global Instance contains_dec l1 l2 : Decision (l1 `contains` l2). Proof. refine (cast_if (decide (is_Some (list_remove_list l1 l2)))); abstract (rewrite list_remove_list_contains; tauto). Defined. Global Instance Permutation_dec l1 l2 : Decision (l1 ≡ₚ l2). Proof. refine (cast_if_and (decide (length l1 = length l2)) (decide (l1 `contains` l2))); abstract (rewrite Permutation_alt; tauto). Defined. End contains_dec. End more_general_properties. (** ** Properties of the [Forall] and [Exists] predicate *) Lemma Forall_Exists_dec {A} {P Q : A → Prop} (dec : ∀ x, {P x} + {Q x}) : ∀ l, {Forall P l} + {Exists Q l}. Proof. refine ( fix go l := match l return {Forall P l} + {Exists Q l} with | [] => left _ | x :: l => cast_if_and (dec x) (go l) end); clear go; intuition. Defined. Section Forall_Exists. Context {A} (P : A → Prop). Definition Forall_nil_2 := @Forall_nil A. Definition Forall_cons_2 := @Forall_cons A. Lemma Forall_forall l : Forall P l ↔ ∀ x, x ∈ l → P x. Proof. split; [induction 1; inversion 1; subst; auto|]. intros Hin; induction l as [|x l IH]; constructor; [apply Hin; constructor|]. apply IH. intros ??. apply Hin. by constructor. Qed. Lemma Forall_nil : Forall P [] ↔ True. Proof. done. Qed. Lemma Forall_cons_1 x l : Forall P (x :: l) → P x ∧ Forall P l. Proof. by inversion 1. Qed. Lemma Forall_cons x l : Forall P (x :: l) ↔ P x ∧ Forall P l. Proof. split. by inversion 1. intros [??]. by constructor. Qed. Lemma Forall_singleton x : Forall P [x] ↔ P x. Proof. rewrite Forall_cons, Forall_nil; tauto. Qed. Lemma Forall_app_2 l1 l2 : Forall P l1 → Forall P l2 → Forall P (l1 ++ l2). Proof. induction 1; simpl; auto. Qed. Lemma Forall_app l1 l2 : Forall P (l1 ++ l2) ↔ Forall P l1 ∧ Forall P l2. Proof. split; [induction l1; inversion 1; intuition|]. intros [??]; auto using Forall_app_2. Qed. Lemma Forall_true l : (∀ x, P x) → Forall P l. Proof. induction l; auto. Qed. Lemma Forall_impl (Q : A → Prop) l : Forall P l → (∀ x, P x → Q x) → Forall Q l. Proof. intros H ?. induction H; auto. Defined. Global Instance Forall_proper: Proper (pointwise_relation _ (↔) ==> (=) ==> (↔)) (@Forall A). Proof. split; subst; induction 1; constructor; by firstorder auto. Qed. Lemma Forall_iff l (Q : A → Prop) : (∀ x, P x ↔ Q x) → Forall P l ↔ Forall Q l. Proof. intros H. apply Forall_proper. red; apply H. done. Qed. Lemma Forall_not l : length l ≠ 0 → Forall (not ∘ P) l → ¬Forall P l. Proof. by destruct 2; inversion 1. Qed. Lemma Forall_and {Q} l : Forall (λ x, P x ∧ Q x) l ↔ Forall P l ∧ Forall Q l. Proof. split; [induction 1; constructor; naive_solver|]. intros [Hl Hl']; revert Hl'; induction Hl; inversion_clear 1; auto. Qed. Lemma Forall_and_l {Q} l : Forall (λ x, P x ∧ Q x) l → Forall P l. Proof. rewrite Forall_and; tauto. Qed. Lemma Forall_and_r {Q} l : Forall (λ x, P x ∧ Q x) l → Forall Q l. Proof. rewrite Forall_and; tauto. Qed. Lemma Forall_delete l i : Forall P l → Forall P (delete i l). Proof. intros H. revert i. by induction H; intros [|i]; try constructor. Qed. Lemma Forall_lookup l : Forall P l ↔ ∀ i x, l !! i = Some x → P x. Proof. rewrite Forall_forall. setoid_rewrite elem_of_list_lookup. naive_solver. Qed. Lemma Forall_lookup_1 l i x : Forall P l → l !! i = Some x → P x. Proof. rewrite Forall_lookup. eauto. Qed. Lemma Forall_lookup_2 l : (∀ i x, l !! i = Some x → P x) → Forall P l. Proof. by rewrite Forall_lookup. Qed. Lemma Forall_tail l : Forall P l → Forall P (tail l). Proof. destruct 1; simpl; auto. Qed. Lemma Forall_alter f l i : Forall P l → (∀ x, l!!i = Some x → P x → P (f x)) → Forall P (alter f i l). Proof. intros Hl. revert i. induction Hl; simpl; intros [|i]; constructor; auto. Qed. Lemma Forall_alter_inv f l i : Forall P (alter f i l) → (∀ x, l!!i = Some x → P (f x) → P x) → Forall P l. Proof. revert i. induction l; intros [|?]; simpl; inversion_clear 1; constructor; eauto. Qed. Lemma Forall_insert l i x : Forall P l → P x → Forall P (<[i:=x]>l). Proof. rewrite list_insert_alter; auto using Forall_alter. Qed. Lemma Forall_inserts l i k : Forall P l → Forall P k → Forall P (list_inserts i k l). Proof. intros Hl Hk; revert i. induction Hk; simpl; auto using Forall_insert. Qed. Lemma Forall_replicate n x : P x → Forall P (replicate n x). Proof. induction n; simpl; constructor; auto. Qed. Lemma Forall_replicate_eq n (x : A) : Forall (x =) (replicate n x). Proof. induction n; simpl; constructor; auto. Qed. Lemma Forall_take n l : Forall P l → Forall P (take n l). Proof. intros Hl. revert n. induction Hl; intros [|?]; simpl; auto. Qed. Lemma Forall_drop n l : Forall P l → Forall P (drop n l). Proof. intros Hl. revert n. induction Hl; intros [|?]; simpl; auto. Qed. Lemma Forall_resize n x l : P x → Forall P l → Forall P (resize n x l). Proof. intros ? Hl. revert n. induction Hl; intros [|?]; simpl; auto using Forall_replicate. Qed. Lemma Forall_resize_inv n x l : length l ≤ n → Forall P (resize n x l) → Forall P l. Proof. intros ?. rewrite resize_ge, Forall_app by done. by intros []. Qed. Lemma Forall_sublist_lookup l i n k : sublist_lookup i n l = Some k → Forall P l → Forall P k. Proof. unfold sublist_lookup. intros; simplify_option_equality. auto using Forall_take, Forall_drop. Qed. Lemma Forall_sublist_alter f l i n k : Forall P l → sublist_lookup i n l = Some k → Forall P (f k) → Forall P (sublist_alter f i n l). Proof. unfold sublist_alter, sublist_lookup. intros; simplify_option_equality. auto using Forall_app_2, Forall_drop, Forall_take. Qed. Lemma Forall_sublist_alter_inv f l i n k : sublist_lookup i n l = Some k → Forall P (sublist_alter f i n l) → Forall P (f k). Proof. unfold sublist_alter, sublist_lookup. intros ?; simplify_option_equality. rewrite !Forall_app; tauto. Qed. Lemma Forall_reshape l szs : Forall P l → Forall (Forall P) (reshape szs l). Proof. revert l. induction szs; simpl; auto using Forall_take, Forall_drop. Qed. Lemma Forall_rev_ind (Q : list A → Prop) : Q [] → (∀ x l, P x → Forall P l → Q l → Q (l ++ [x])) → ∀ l, Forall P l → Q l. Proof. intros ?? l. induction l using rev_ind; auto. rewrite Forall_app, Forall_singleton; intros [??]; auto. Qed. Lemma Exists_exists l : Exists P l ↔ ∃ x, x ∈ l ∧ P x. Proof. split. * induction 1 as [x|y ?? [x [??]]]; exists x; by repeat constructor. * intros [x [Hin ?]]. induction l; [by destruct (not_elem_of_nil x)|]. inversion Hin; subst. by left. right; auto. Qed. Lemma Exists_inv x l : Exists P (x :: l) → P x ∨ Exists P l. Proof. inversion 1; intuition trivial. Qed. Lemma Exists_app l1 l2 : Exists P (l1 ++ l2) ↔ Exists P l1 ∨ Exists P l2. Proof. split. * induction l1; inversion 1; intuition. * intros [H|H]; [induction H | induction l1]; simpl; intuition. Qed. Lemma Exists_impl (Q : A → Prop) l : Exists P l → (∀ x, P x → Q x) → Exists Q l. Proof. intros H ?. induction H; auto. Defined. Global Instance Exists_proper: Proper (pointwise_relation _ (↔) ==> (=) ==> (↔)) (@Exists A). Proof. split; subst; induction 1; constructor; by firstorder auto. Qed. Lemma Exists_not_Forall l : Exists (not ∘ P) l → ¬Forall P l. Proof. induction 1; inversion_clear 1; contradiction. Qed. Lemma Forall_not_Exists l : Forall (not ∘ P) l → ¬Exists P l. Proof. induction 1; inversion_clear 1; contradiction. Qed. Lemma Forall_list_difference `{∀ x y : A, Decision (x = y)} l k : Forall P l → Forall P (list_difference l k). Proof. rewrite !Forall_forall. intros ? x; rewrite elem_of_list_difference; naive_solver. Qed. Lemma Forall_list_union `{∀ x y : A, Decision (x = y)} l k : Forall P l → Forall P k → Forall P (list_union l k). Proof. intros. apply Forall_app; auto using Forall_list_difference. Qed. Lemma Forall_list_intersection `{∀ x y : A, Decision (x = y)} l k : Forall P l → Forall P (list_intersection l k). Proof. rewrite !Forall_forall. intros ? x; rewrite elem_of_list_intersection; naive_solver. Qed. Context {dec : ∀ x, Decision (P x)}. Lemma not_Forall_Exists l : ¬Forall P l → Exists (not ∘ P) l. Proof. intro. destruct (Forall_Exists_dec dec l); intuition. Qed. Lemma not_Exists_Forall l : ¬Exists P l → Forall (not ∘ P) l. Proof. by destruct (Forall_Exists_dec (λ x, swap_if (decide (P x))) l). Qed. Global Instance Forall_dec l : Decision (Forall P l) := match Forall_Exists_dec dec l with | left H => left H | right H => right (Exists_not_Forall _ H) end. Global Instance Exists_dec l : Decision (Exists P l) := match Forall_Exists_dec (λ x, swap_if (decide (P x))) l with | left H => right (Forall_not_Exists _ H) | right H => left H end. End Forall_Exists. Lemma replicate_as_Forall {A} (x : A) n l : replicate n x = l ↔ length l = n ∧ Forall (x =) l. Proof. rewrite replicate_as_elem_of, Forall_forall. naive_solver. Qed. Lemma replicate_as_Forall_2 {A} (x : A) n l : length l = n → Forall (x =) l → replicate n x = l. Proof. by rewrite replicate_as_Forall. Qed. Lemma Forall_swap {A B} (Q : A → B → Prop) l1 l2 : Forall (λ y, Forall (Q y) l1) l2 ↔ Forall (λ x, Forall (flip Q x) l2) l1. Proof. repeat setoid_rewrite Forall_forall. simpl. split; eauto. Qed. Lemma Forall_seq (P : nat → Prop) i n : Forall P (seq i n) ↔ ∀ j, i ≤ j < i + n → P j. Proof. rewrite Forall_lookup. split. * intros H j [??]. apply (H (j - i)). rewrite lookup_seq; auto with f_equal lia. * intros H j x Hj. apply lookup_seq_inv in Hj. destruct Hj; subst. auto with lia. Qed. (** ** Properties of the [Forall2] predicate *) Section Forall2. Context {A B} (P : A → B → Prop). Implicit Types x : A. Implicit Types y : B. Implicit Types l : list A. Implicit Types k : list B. Lemma Forall2_true l k : (∀ x y, P x y) → length l = length k → Forall2 P l k. Proof. intro. revert k. induction l; intros [|??] ?; simplify_equality'; auto. Qed. Lemma Forall2_same_length l k : Forall2 (λ _ _, True) l k ↔ length l = length k. Proof. split; [by induction 1; f_equal'|]. revert k. induction l; intros [|??] ?; simplify_equality'; auto. Qed. Lemma Forall2_length l k : Forall2 P l k → length l = length k. Proof. by induction 1; f_equal'. Qed. Lemma Forall2_length_l l k n : Forall2 P l k → length l = n → length k = n. Proof. intros ? <-; symmetry. by apply Forall2_length. Qed. Lemma Forall2_length_r l k n : Forall2 P l k → length k = n → length l = n. Proof. intros ? <-. by apply Forall2_length. Qed. Lemma Forall2_nil_inv_l k : Forall2 P [] k → k = []. Proof. by inversion 1. Qed. Lemma Forall2_nil_inv_r l : Forall2 P l [] → l = []. Proof. by inversion 1. Qed. Lemma Forall2_cons_inv x l y k : Forall2 P (x :: l) (y :: k) → P x y ∧ Forall2 P l k. Proof. by inversion 1. Qed. Lemma Forall2_cons_inv_l x l k : Forall2 P (x :: l) k → ∃ y k', P x y ∧ Forall2 P l k' ∧ k = y :: k'. Proof. inversion 1; subst; eauto. Qed. Lemma Forall2_cons_inv_r l k y : Forall2 P l (y :: k) → ∃ x l', P x y ∧ Forall2 P l' k ∧ l = x :: l'. Proof. inversion 1; subst; eauto. Qed. Lemma Forall2_cons_nil_inv x l : Forall2 P (x :: l) [] → False. Proof. by inversion 1. Qed. Lemma Forall2_nil_cons_inv y k : Forall2 P [] (y :: k) → False. Proof. by inversion 1. Qed. Lemma Forall2_app_l l1 l2 k : Forall2 P l1 (take (length l1) k) → Forall2 P l2 (drop (length l1) k) → Forall2 P (l1 ++ l2) k. Proof. intros. rewrite <-(take_drop (length l1) k). by apply Forall2_app. Qed. Lemma Forall2_app_r l k1 k2 : Forall2 P (take (length k1) l) k1 → Forall2 P (drop (length k1) l) k2 → Forall2 P l (k1 ++ k2). Proof. intros. rewrite <-(take_drop (length k1) l). by apply Forall2_app. Qed. Lemma Forall2_app_inv l1 l2 k1 k2 : length l1 = length k1 → Forall2 P (l1 ++ l2) (k1 ++ k2) → Forall2 P l1 k1 ∧ Forall2 P l2 k2. Proof. rewrite <-Forall2_same_length. induction 1; inversion 1; naive_solver. Qed. Lemma Forall2_app_inv_l l1 l2 k : Forall2 P (l1 ++ l2) k ↔ ∃ k1 k2, Forall2 P l1 k1 ∧ Forall2 P l2 k2 ∧ k = k1 ++ k2. Proof. split; [|intros (?&?&?&?&->); by apply Forall2_app]. revert k. induction l1; inversion 1; naive_solver. Qed. Lemma Forall2_app_inv_r l k1 k2 : Forall2 P l (k1 ++ k2) ↔ ∃ l1 l2, Forall2 P l1 k1 ∧ Forall2 P l2 k2 ∧ l = l1 ++ l2. Proof. split; [|intros (?&?&?&?&->); by apply Forall2_app]. revert l. induction k1; inversion 1; naive_solver. Qed. Lemma Forall2_flip l k : Forall2 (flip P) k l ↔ Forall2 P l k. Proof. split; induction 1; constructor; auto. Qed. Lemma Forall2_impl (Q : A → B → Prop) l k : Forall2 P l k → (∀ x y, P x y → Q x y) → Forall2 Q l k. Proof. intros H ?. induction H; auto. Defined. Lemma Forall2_unique l k1 k2 : Forall2 P l k1 → Forall2 P l k2 → (∀ x y1 y2, P x y1 → P x y2 → y1 = y2) → k1 = k2. Proof. intros H. revert k2. induction H; inversion_clear 1; intros; f_equal; eauto. Qed. Lemma Forall2_Forall_l (Q : A → Prop) l k : Forall2 P l k → Forall (λ y, ∀ x, P x y → Q x) k → Forall Q l. Proof. induction 1; inversion_clear 1; eauto. Qed. Lemma Forall2_Forall_r (Q : B → Prop) l k : Forall2 P l k → Forall (λ x, ∀ y, P x y → Q y) l → Forall Q k. Proof. induction 1; inversion_clear 1; eauto. Qed. Lemma Forall2_lookup_lr l k i x y : Forall2 P l k → l !! i = Some x → k !! i = Some y → P x y. Proof. intros H. revert i. induction H; intros [|?] ??; simplify_equality'; eauto. Qed. Lemma Forall2_lookup_l l k i x : Forall2 P l k → l !! i = Some x → ∃ y, k !! i = Some y ∧ P x y. Proof. intros H. revert i. induction H; intros [|?] ?; simplify_equality'; eauto. Qed. Lemma Forall2_lookup_r l k i y : Forall2 P l k → k !! i = Some y → ∃ x, l !! i = Some x ∧ P x y. Proof. intros H. revert i. induction H; intros [|?] ?; simplify_equality'; eauto. Qed. Lemma Forall2_lookup_2 l k : length l = length k → (∀ i x y, l !! i = Some x → k !! i = Some y → P x y) → Forall2 P l k. Proof. rewrite <-Forall2_same_length. intros Hl Hlookup. induction Hl as [|?????? IH]; constructor; [by apply (Hlookup 0)|]. apply IH. apply (λ i, Hlookup (S i)). Qed. Lemma Forall2_lookup l k : Forall2 P l k ↔ length l = length k ∧ (∀ i x y, l !! i = Some x → k !! i = Some y → P x y). Proof. naive_solver eauto using Forall2_length, Forall2_lookup_lr,Forall2_lookup_2. Qed. Lemma Forall2_tail l k : Forall2 P l k → Forall2 P (tail l) (tail k). Proof. destruct 1; simpl; auto. Qed. Lemma Forall2_alter_l f l k i : Forall2 P l k → (∀ x y, l !! i = Some x → k !! i = Some y → P x y → P (f x) y) → Forall2 P (alter f i l) k. Proof. intros Hl. revert i. induction Hl; intros [|]; constructor; auto. Qed. Lemma Forall2_alter_r f l k i : Forall2 P l k → (∀ x y, l !! i = Some x → k !! i = Some y → P x y → P x (f y)) → Forall2 P l (alter f i k). Proof. intros Hl. revert i. induction Hl; intros [|]; constructor; auto. Qed. Lemma Forall2_alter f g l k i : Forall2 P l k → (∀ x y, l !! i = Some x → k !! i = Some y → P x y → P (f x) (g y)) → Forall2 P (alter f i l) (alter g i k). Proof. intros Hl. revert i. induction Hl; intros [|]; constructor; auto. Qed. Lemma Forall2_insert l k x y i : Forall2 P l k → P x y → Forall2 P (<[i:=x]> l) (<[i:=y]> k). Proof. intros Hl. revert i. induction Hl; intros [|]; constructor; auto. Qed. Lemma Forall2_delete l k i : Forall2 P l k → Forall2 P (delete i l) (delete i k). Proof. intros Hl. revert i. induction Hl; intros [|]; simpl; intuition. Qed. Lemma Forall2_replicate_l k n x : length k = n → Forall (P x) k → Forall2 P (replicate n x) k. Proof. intros <-. induction 1; simpl; auto. Qed. Lemma Forall2_replicate_r l n y : length l = n → Forall (flip P y) l → Forall2 P l (replicate n y). Proof. intros <-. induction 1; simpl; auto. Qed. Lemma Forall2_replicate n x y : P x y → Forall2 P (replicate n x) (replicate n y). Proof. induction n; simpl; constructor; auto. Qed. Lemma Forall2_take l k n : Forall2 P l k → Forall2 P (take n l) (take n k). Proof. intros Hl. revert n. induction Hl; intros [|?]; simpl; auto. Qed. Lemma Forall2_drop l k n : Forall2 P l k → Forall2 P (drop n l) (drop n k). Proof. intros Hl. revert n. induction Hl; intros [|?]; simpl; auto. Qed. Lemma Forall2_resize l k x y n : P x y → Forall2 P l k → Forall2 P (resize n x l) (resize n y k). Proof. intros. rewrite !resize_spec, (Forall2_length l k) by done. auto using Forall2_app, Forall2_take, Forall2_replicate. Qed. Lemma Forall2_resize_l l k x y n m : P x y → Forall (flip P y) l → Forall2 P (resize n x l) k → Forall2 P (resize m x l) (resize m y k). Proof. intros. destruct (decide (m ≤ n)). { rewrite <-(resize_resize l m n) by done. by apply Forall2_resize. } intros. assert (n = length k); subst. { by rewrite <-(Forall2_length (resize n x l) k), resize_length. } rewrite (le_plus_minus (length k) m), !resize_plus, resize_all, drop_all, resize_nil by lia. auto using Forall2_app, Forall2_replicate_r, Forall_resize, Forall_drop, resize_length. Qed. Lemma Forall2_resize_r l k x y n m : P x y → Forall (P x) k → Forall2 P l (resize n y k) → Forall2 P (resize m x l) (resize m y k). Proof. intros. destruct (decide (m ≤ n)). { rewrite <-(resize_resize k m n) by done. by apply Forall2_resize. } assert (n = length l); subst. { by rewrite (Forall2_length l (resize n y k)), resize_length. } rewrite (le_plus_minus (length l) m), !resize_plus, resize_all, drop_all, resize_nil by lia. auto using Forall2_app, Forall2_replicate_l, Forall_resize, Forall_drop, resize_length. Qed. Lemma Forall2_resize_r_flip l k x y n m : P x y → Forall (P x) k → length k = m → Forall2 P l (resize n y k) → Forall2 P (resize m x l) k. Proof. intros ?? <- ?. rewrite <-(resize_all k y) at 2. apply Forall2_resize_r with n; auto using Forall_true. Qed. Lemma Forall2_sublist_lookup_l l k n i l' : Forall2 P l k → sublist_lookup n i l = Some l' → ∃ k', sublist_lookup n i k = Some k' ∧ Forall2 P l' k'. Proof. unfold sublist_lookup. intros Hlk Hl. exists (take i (drop n k)); simplify_option_equality. * auto using Forall2_take, Forall2_drop. * apply Forall2_length in Hlk; lia. Qed. Lemma Forall2_sublist_lookup_r l k n i k' : Forall2 P l k → sublist_lookup n i k = Some k' → ∃ l', sublist_lookup n i l = Some l' ∧ Forall2 P l' k'. Proof. intro. unfold sublist_lookup. erewrite Forall2_length by eauto; intros; simplify_option_equality. eauto using Forall2_take, Forall2_drop. Qed. Lemma Forall2_sublist_alter f g l k i n l' k' : Forall2 P l k → sublist_lookup i n l = Some l' → sublist_lookup i n k = Some k' → Forall2 P (f l') (g k') → Forall2 P (sublist_alter f i n l) (sublist_alter g i n k). Proof. intro. unfold sublist_alter, sublist_lookup. erewrite Forall2_length by eauto; intros; simplify_option_equality. auto using Forall2_app, Forall2_drop, Forall2_take. Qed. Lemma Forall2_sublist_alter_l f l k i n l' k' : Forall2 P l k → sublist_lookup i n l = Some l' → sublist_lookup i n k = Some k' → Forall2 P (f l') k' → Forall2 P (sublist_alter f i n l) k. Proof. intro. unfold sublist_lookup, sublist_alter. erewrite <-Forall2_length by eauto; intros; simplify_option_equality. apply Forall2_app_l; rewrite ?take_length_le by lia; auto using Forall2_take. apply Forall2_app_l; erewrite Forall2_length, take_length, drop_length, <-Forall2_length, Min.min_l by eauto with lia; [done|]. rewrite drop_drop; auto using Forall2_drop. Qed. Lemma Forall2_transitive {C} (Q : B → C → Prop) (R : A → C → Prop) l k lC : (∀ x y z, P x y → Q y z → R x z) → Forall2 P l k → Forall2 Q k lC → Forall2 R l lC. Proof. intros ? Hl. revert lC. induction Hl; inversion_clear 1; eauto. Qed. Lemma Forall2_Forall (Q : A → A → Prop) l : Forall (λ x, Q x x) l → Forall2 Q l l. Proof. induction 1; constructor; auto. Qed. Global Instance Forall2_dec `{dec : ∀ x y, Decision (P x y)} : ∀ l k, Decision (Forall2 P l k). Proof. refine ( fix go l k : Decision (Forall2 P l k) := match l, k with | [], [] => left _ | x :: l, y :: k => cast_if_and (decide (P x y)) (go l k) | _, _ => right _ end); clear dec go; abstract first [by constructor | by inversion 1]. Defined. End Forall2. Section Forall2_order. Context {A} (R : relation A). Global Instance: Reflexive R → Reflexive (Forall2 R). Proof. intros ? l. induction l; by constructor. Qed. Global Instance: Symmetric R → Symmetric (Forall2 R). Proof. intros. induction 1; constructor; auto. Qed. Global Instance: Transitive R → Transitive (Forall2 R). Proof. intros ????. apply Forall2_transitive. by apply @transitivity. Qed. Global Instance: Equivalence R → Equivalence (Forall2 R). Proof. split; apply _. Qed. Global Instance: PreOrder R → PreOrder (Forall2 R). Proof. split; apply _. Qed. Global Instance: AntiSymmetric (=) R → AntiSymmetric (=) (Forall2 R). Proof. induction 2; inversion_clear 1; f_equal; auto. Qed. Global Instance: Proper (R ==> Forall2 R ==> Forall2 R) (::). Proof. by constructor. Qed. Global Instance: Proper (Forall2 R ==> Forall2 R ==> Forall2 R) (++). Proof. repeat intro. eauto using Forall2_app. Qed. Global Instance: Proper (Forall2 R ==> Forall2 R) (delete i). Proof. repeat intro. eauto using Forall2_delete. Qed. Global Instance: Proper (R ==> Forall2 R) (replicate n). Proof. repeat intro. eauto using Forall2_replicate. Qed. Global Instance: Proper (Forall2 R ==> Forall2 R) (take n). Proof. repeat intro. eauto using Forall2_take. Qed. Global Instance: Proper (Forall2 R ==> Forall2 R) (drop n). Proof. repeat intro. eauto using Forall2_drop. Qed. Global Instance: Proper (R ==> Forall2 R ==> Forall2 R) (resize n). Proof. repeat intro. eauto using Forall2_resize. Qed. End Forall2_order. Section Forall3. Context {A B C} (P : A → B → C → Prop). Hint Extern 0 (Forall3 _ _ _ _) => constructor. `````` Robbert Krebbers committed Jan 20, 2016 2530 `````` Lemma Forall3_app l1 l2 k1 k2 k1' k2' : `````` Robbert Krebbers committed Nov 11, 2015 2531 2532 2533 `````` Forall3 P l1 k1 k1' → Forall3 P l2 k2 k2' → Forall3 P (l1 ++ l2) (k1 ++ k2) (k1' ++ k2'). Proof. induction 1; simpl; auto. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2534 2535 2536 2537 2538 2539 2540 2541 2542 2543 2544 2545 2546 2547 2548 2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 2562 `````` Lemma Forall3_cons_inv_l x l k k' : Forall3 P (x :: l) k k' → ∃ y k2 z k2', k = y :: k2 ∧ k' = z :: k2' ∧ P x y z ∧ Forall3 P l k2 k2'. Proof. inversion_clear 1; naive_solver. Qed. Lemma Forall3_app_inv_l l1 l2 k k' : Forall3 P (l1 ++ l2) k k' → ∃ k1 k2 k1' k2', k = k1 ++ k2 ∧ k' = k1' ++ k2' ∧ Forall3 P l1 k1 k1' ∧ Forall3 P l2 k2 k2'. Proof. revert k k'. induction l1 as [|x l1 IH]; simpl; inversion_clear 1. * by repeat eexists; eauto. * by repeat eexists; eauto. * edestruct IH as (?&?&?&?&?&?&?&?); eauto; naive_solver. Qed. Lemma Forall3_cons_inv_m l y k k' : Forall3 P l (y :: k) k' → ∃ x l2 z k2', l = x :: l2 ∧ k' = z :: k2' ∧ P x y z ∧ Forall3 P l2 k k2'. Proof. inversion_clear 1; naive_solver. Qed. Lemma Forall3_app_inv_m l k1 k2 k' : Forall3 P l (k1 ++ k2) k' → ∃ l1 l2 k1' k2', l = l1 ++ l2 ∧ k' = k1' ++ k2' ∧ Forall3 P l1 k1 k1' ∧ Forall3 P l2 k2 k2'. Proof. revert l k'. induction k1 as [|x k1 IH]; simpl; inversion_clear 1. * by repeat eexists; eauto. * by repeat eexists; eauto. * edestruct IH as (?&?&?&?&?&?&?&?); eauto; naive_solver. Qed. Lemma Forall3_cons_inv_r l k z k' : Forall3 P l k (z :: k') → ∃ x l2 y k2, l = x :: l2 ∧ k = y :: k2 ∧ P x y z ∧ Forall3 P l2 k2 k'. `````` Robbert Krebbers committed Nov 11, 2015 2563 `````` Proof. inversion_clear 1; naive_solver. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2564 2565 2566 `````` Lemma Forall3_app_inv_r l k k1' k2' : Forall3 P l k (k1' ++ k2') → ∃ l1 l2 k1 k2, l = l1 ++ l2 ∧ k = k1 ++ k2 ∧ Forall3 P l1 k1 k1' ∧ Forall3 P l2 k2 k2'. `````` Robbert Krebbers committed Nov 11, 2015 2567 `````` Proof. `````` Robbert Krebbers committed Jan 20, 2016 2568 `````` revert l k. induction k1' as [|x k1' IH]; simpl; inversion_clear 1. `````` Robbert Krebbers committed Nov 11, 2015 2569 2570 2571 2572 `````` * by repeat eexists; eauto. * by repeat eexists; eauto. * edestruct IH as (?&?&?&?&?&?&?&?); eauto; naive_solver. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2573 2574 2575 2576 `````` Lemma Forall3_impl (Q : A → B → C → Prop) l k k' : Forall3 P l k k' → (∀ x y z, P x y z → Q x y z) → Forall3 Q l k k'. Proof. intros Hl ?; induction Hl; auto. Defined. Lemma Forall3_length_lm l k k' : Forall3 P l k k' → length l = length k. `````` Robbert Krebbers committed Nov 11, 2015 2577 `````` Proof. by induction 1; f_equal'. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2578 `````` Lemma Forall3_length_lr l k k' : Forall3 P l k k' → length l = length k'. `````` Robbert Krebbers committed Nov 11, 2015 2579 `````` Proof. by induction 1; f_equal'. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2580 2581 2582 `````` Lemma Forall3_lookup_lmr l k k' i x y z : Forall3 P l k k' → l !! i = Some x → k !! i = Some y → k' !! i = Some z → P x y z. `````` Robbert Krebbers committed Nov 11, 2015 2583 2584 2585 `````` Proof. intros H. revert i. induction H; intros [|?] ???; simplify_equality'; eauto. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2586 2587 2588 `````` Lemma Forall3_lookup_l l k k' i x : Forall3 P l k k' → l !! i = Some x → ∃ y z, k !! i = Some y ∧ k' !! i = Some z ∧ P x y z. `````` Robbert Krebbers committed Nov 11, 2015 2589 2590 2591 `````` Proof. intros H. revert i. induction H; intros [|?] ?; simplify_equality'; eauto. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2592 2593 2594 `````` Lemma Forall3_lookup_m l k k' i y : Forall3 P l k k' → k !! i = Some y → ∃ x z, l !! i = Some x ∧ k' !! i = Some z ∧ P x y z. `````` Robbert Krebbers committed Nov 11, 2015 2595 2596 2597 `````` Proof. intros H. revert i. induction H; intros [|?] ?; simplify_equality'; eauto. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2598 2599 2600 `````` Lemma Forall3_lookup_r l k k' i z : Forall3 P l k k' → k' !! i = Some z → ∃ x y, l !! i = Some x ∧ k !! i = Some y ∧ P x y z. `````` Robbert Krebbers committed Nov 11, 2015 2601 2602 2603 `````` Proof. intros H. revert i. induction H; intros [|?] ?; simplify_equality'; eauto. Qed. `````` Robbert Krebbers committed Jan 20, 2016 2604 2605 2606 `````` Lemma Forall3_alter_lm f g l k k' i : Forall3 P l k k' → (∀ x y z, l !! i = Some x → k !! i = Some y → k' !! i = Some z → `````` Robbert Krebbers committed Nov 11, 2015 2607 `````` P x y z → P (f x) (g y) z) → `````` Robbert Krebbers committed Jan 20, 2016 2608 `````` Forall3 P (alter f i l) (alter g i k) k'. `````` Robbert Krebbers committed Nov 11, 2015 2609 2610 2611 2612 2613 2614 2615 2616 2617 2618 2619 2620 2621 2622 2623 2624 2625 2626 2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637 2638 2639 2640 2641 `````` Proof. intros Hl. revert i. induction Hl; intros [|]; auto. Qed. End Forall3. (** * Properties of the monadic operations *) Section fmap. Context {A B : Type} (f : A → B). Lemma list_fmap_id (l : list A) : id <\$> l = l. Proof. induction l; f_equal'; auto. Qed. Lemma list_fmap_compose {C} (g : B → C) l : g ∘ f <\$> l = g <\$> f <\$> l. Proof. induction l; f_equal'; auto. Qed. Lemma list_fmap_ext (g : A → B) (l1 l2 : list A) : (∀ x, f x = g x) → l1 = l2 → fmap f l1 = fmap g l2. Proof. intros ? <-. induction l1; f_equal'; auto. Qed. Global Instance: Injective (=) (=) f → Injective (=) (=) (fmap f). Proof. intros ? l1. induction l1 as [|x l1 IH]; [by intros [|??]|]. intros [|??]; intros; f_equal'; simplify_equality; auto. Qed. Definition fmap_nil : f <\$> [] = [] := eq_refl. Definition fmap_cons x l : f <\$> x :: l = f x :: f <\$> l := eq_refl. Lemma fmap_app l1 l2 : f <\$> l1 ++ l2 = (f <\$> l1) ++ (f <\$> l2). Proof. by induction l1; f_equal'. Qed. Lemma fmap_nil_inv k : f <\$> k = [] → k = []. Proof. by destruct k. Qed. Lemma fmap_cons_inv y l k : f <\$> l = y :: k → ∃ x l', y = f x ∧ k = f <\$> l' ∧ l = x :: l'. Proof. intros. destruct l; simplify_equality'; eauto. Qed. Lemma fmap_app_inv l k1 k2 : f <\$> l = k1 ++ k2 → ∃ l1 l2, k1 = f <\$> l1 ∧ k2 = f <\$> l2 ∧ l = l1 ++ l2. Proof. revert l. induction k1 as [|y k1 IH]; simpl; [intros l ?; by eexists [],l|]. intros [|x l] ?; simplify_equality'. `````` Robbert Krebbers committed Nov 11, 2015 2642 `````` destruct (IH l) as (l1&l2&->&->&->); [done|]. by exists (x :: l1), l2. `````` Robbert Krebbers committed Nov 11, 2015 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652 2653 2654 2655 2656 2657 2658 2659 2660 2661 2662 2663 2664 2665 2666 2667 2668 2669 2670 2671 2672 2673 2674 2675 2676 2677 2678 2679 2680 2681 2682 2683 2684 2685 2686 2687 2688 2689 2690 2691 2692 2693 2694 2695 2696 2697 2698 2699 2700 2701 2702 2703 2704 2705 2706 2707 2708 2709 2710 2711 2712 2713 2714 2715 2716 2717 2718 2719 2720 2721 2722 2723 2724 2725 2726 2727 2728 2729 2730 2731 2732 2733 2734 2735 2736 2737 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747 2748 2749 2750 2751 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 2781 2782 2783 2784 2785 2786 2787 2788 2789 2790 2791 2792 2793 2794 2795 2796 2797 2798 2799 2800 2801 2802 2803 2804 2805 2806 2807 2808 2809 2810 2811 2812 2813 2814 2815 2816 2817 2818 2819 2820 2821 2822 2823 2824 2825 2826 2827 2828 2829 2830 2831 2832 2833 2834 2835 2836 2837 2838 2839 2840 2841 2842 2843 2844 2845 2846 2847 2848 2849 2850 2851 2852 2853 2854 2855 2856 2857 2858 2859 2860 2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886 2887 2888 2889 2890 2891 2892