list.v 159 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file collects general purpose definitions and theorems on lists that are not in the Coq standard library. *) `````` Robbert Krebbers committed Feb 13, 2016 5 ``````From Coq Require Export Permutation. `````` Robbert Krebbers committed Mar 10, 2016 6 ``````From iris.prelude Require Export numbers base option. `````` Robbert Krebbers committed Nov 11, 2015 7 8 9 10 `````` Arguments length {_} _. Arguments cons {_} _ _. Arguments app {_} _ _. `````` Robbert Krebbers committed Mar 21, 2016 11 12 13 14 `````` Instance: Params (@length) 1. Instance: Params (@cons) 1. Instance: Params (@app) 1. `````` Robbert Krebbers committed Nov 11, 2015 15 16 17 18 19 `````` Notation tail := tl. Notation take := firstn. Notation drop := skipn. `````` Robbert Krebbers committed Mar 21, 2016 20 ``````Arguments tail {_} _. `````` Robbert Krebbers committed Nov 11, 2015 21 22 23 ``````Arguments take {_} !_ !_ /. Arguments drop {_} !_ !_ /. `````` Robbert Krebbers committed Mar 21, 2016 24 25 26 27 28 29 30 ``````Instance: Params (@tail) 1. Instance: Params (@take) 1. Instance: Params (@drop) 1. Arguments Permutation {_} _ _. Arguments Forall_cons {_} _ _ _ _ _. `````` Robbert Krebbers committed Nov 11, 2015 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 ``````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 47 48 49 ``````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 50 ``````(** * Definitions *) `````` Robbert Krebbers committed Nov 18, 2015 51 52 53 54 55 56 ``````(** 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 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 ``````(** 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. `````` Robbert Krebbers committed Mar 21, 2016 87 ``````Instance: Params (@list_inserts) 1. `````` Robbert Krebbers committed Nov 11, 2015 88 89 90 91 92 93 94 95 96 97 98 99 100 101 `````` (** 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]) []. `````` Robbert Krebbers committed Mar 21, 2016 102 103 ``````Instance: Params (@option_list) 1. Instance maybe_list_singleton {A} : Maybe (λ x : A, [x]) := λ l, `````` Robbert Krebbers committed Nov 11, 2015 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 `````` 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. `````` Robbert Krebbers committed Mar 21, 2016 123 ``````Instance: Params (@list_find) 3. `````` Robbert Krebbers committed Nov 11, 2015 124 125 126 127 128 `````` (** 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. `````` Robbert Krebbers committed Mar 21, 2016 129 ``````Instance: Params (@replicate) 2. `````` Robbert Krebbers committed Nov 11, 2015 130 131 132 `````` (** The function [reverse l] returns the elements of [l] in reverse order. *) Definition reverse {A} (l : list A) : list A := rev_append l []. `````` Robbert Krebbers committed Mar 21, 2016 133 ``````Instance: Params (@reverse) 1. `````` Robbert Krebbers committed Nov 11, 2015 134 135 136 137 138 `````` (** 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. `````` Robbert Krebbers committed Mar 21, 2016 139 ``````Instance: Params (@last) 1. `````` Robbert Krebbers committed Nov 11, 2015 140 141 142 143 144 145 146 147 148 149 `````` (** 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 {_} !_ _ !_. `````` Robbert Krebbers committed Mar 21, 2016 150 ``````Instance: Params (@resize) 2. `````` Robbert Krebbers committed Nov 11, 2015 151 152 153 154 155 156 157 158 `````` (** 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. `````` Robbert Krebbers committed Mar 21, 2016 159 ``````Instance: Params (@reshape) 2. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` 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. `````` Robbert Krebbers committed Feb 13, 2016 241 242 ``````Hint Extern 0 (_ `prefix_of` _) => reflexivity. Hint Extern 0 (_ `suffix_of` _) => reflexivity. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` 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. `````` Robbert Krebbers committed Feb 13, 2016 270 ``````Hint Extern 0 (_ `sublist` _) => reflexivity. `````` Robbert Krebbers committed Nov 11, 2015 271 272 273 274 275 276 277 278 279 280 `````` (** 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. `````` Robbert Krebbers committed Feb 13, 2016 281 ``````Hint Extern 0 (_ `contains` _) => reflexivity. `````` Robbert Krebbers committed Nov 11, 2015 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 `````` 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 *) `````` Robbert Krebbers committed Mar 04, 2016 303 304 305 ``````Definition included {A} (l1 l2 : list A) := ∀ x, x ∈ l1 → x ∈ l2. Infix "`included`" := included (at level 70) : C_scope. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````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 *) `````` Robbert Krebbers committed Feb 17, 2016 349 ``````(** The tactic [discriminate_list] discharges a goal if it contains `````` Robbert Krebbers committed Nov 11, 2015 350 351 ``````a list equality involving [(::)] and [(++)] of two lists that have a different length as one of its hypotheses. *) `````` Robbert Krebbers committed Feb 17, 2016 352 ``````Tactic Notation "discriminate_list" hyp(H) := `````` Robbert Krebbers committed Nov 11, 2015 353 354 `````` apply (f_equal length) in H; repeat (csimpl in H || rewrite app_length in H); exfalso; lia. `````` Robbert Krebbers committed Feb 17, 2016 355 356 ``````Tactic Notation "discriminate_list" := match goal with H : @eq (list _) _ _ |- _ => discriminate_list H end. `````` Robbert Krebbers committed Nov 11, 2015 357 `````` `````` Robbert Krebbers committed Feb 17, 2016 358 ``````(** The tactic [simplify_list_eq] simplifies hypotheses involving `````` Robbert Krebbers committed Nov 11, 2015 359 360 ``````equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies lookups in singleton lists. *) `````` Robbert Krebbers committed Feb 11, 2016 361 ``````Lemma app_inj_1 {A} (l1 k1 l2 k2 : list A) : `````` Robbert Krebbers committed Nov 11, 2015 362 363 `````` length l1 = length k1 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2. Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed. `````` Robbert Krebbers committed Feb 11, 2016 364 ``````Lemma app_inj_2 {A} (l1 k1 l2 k2 : list A) : `````` Robbert Krebbers committed Nov 11, 2015 365 366 `````` length l2 = length k2 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2. Proof. `````` Robbert Krebbers committed Feb 11, 2016 367 `````` intros ? Hl. apply app_inj_1; auto. `````` Robbert Krebbers committed Nov 11, 2015 368 369 `````` apply (f_equal length) in Hl. rewrite !app_length in Hl. lia. Qed. `````` Robbert Krebbers committed Feb 17, 2016 370 ``````Ltac simplify_list_eq := `````` Robbert Krebbers committed Nov 11, 2015 371 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 372 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Nov 11, 2015 373 374 `````` | H : _ ++ _ = _ ++ _ |- _ => first [ apply app_inv_head in H | apply app_inv_tail in H `````` Robbert Krebbers committed Feb 11, 2016 375 376 `````` | apply app_inj_1 in H; [destruct H|done] | apply app_inj_2 in H; [destruct H|done] ] `````` Robbert Krebbers committed Nov 11, 2015 377 378 379 380 381 382 383 384 385 386 `````` | 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 Feb 11, 2016 387 ``````Global Instance: Inj2 (=) (=) (=) (@cons A). `````` Robbert Krebbers committed Nov 11, 2015 388 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 11, 2016 389 ``````Global Instance: ∀ k, Inj (=) (=) (k ++). `````` Robbert Krebbers committed Nov 11, 2015 390 ``````Proof. intros ???. apply app_inv_head. Qed. `````` Robbert Krebbers committed Feb 11, 2016 391 ``````Global Instance: ∀ k, Inj (=) (=) (++ k). `````` Robbert Krebbers committed Nov 11, 2015 392 ``````Proof. intros ???. apply app_inv_tail. Qed. `````` Robbert Krebbers committed Feb 11, 2016 393 ``````Global Instance: Assoc (=) (@app A). `````` Robbert Krebbers committed Nov 11, 2015 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 ``````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. `````` Robbert Krebbers committed Mar 21, 2016 409 `````` revert l2. induction l1 as [|x l1 IH]; intros [|y l2] H. `````` Robbert Krebbers committed Feb 17, 2016 410 411 412 `````` - done. - discriminate (H 0). - discriminate (H 0). `````` Robbert Krebbers committed Mar 21, 2016 413 `````` - f_equal; [by injection (H 0)|]. apply (IH _ \$ λ i, H (S i)). `````` Robbert Krebbers committed Nov 11, 2015 414 415 416 417 418 419 ``````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 : `````` Robbert Krebbers committed Mar 21, 2016 420 `````` option_reflect (λ x, l = [x]) (length l ≠ 1) (maybe (λ x, [x]) l). `````` Robbert Krebbers committed Nov 11, 2015 421 422 423 424 425 426 427 428 429 430 431 432 433 ``````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. `````` Robbert Krebbers committed Mar 21, 2016 434 ``````Proof. revert i. induction l; intros [|?] ?; naive_solver auto with arith. Qed. `````` Robbert Krebbers committed Nov 11, 2015 435 436 437 ``````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). `````` Robbert Krebbers committed Mar 21, 2016 438 ``````Proof. revert i. induction l; intros [|?] ?; naive_solver eauto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 439 440 441 442 443 444 445 446 447 448 449 450 451 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 452 `````` - destruct (lookup_lt_is_Some_2 l1 i) as [y Hy]. `````` Robbert Krebbers committed Nov 11, 2015 453 454 `````` { rewrite Hlen; eauto using lookup_lt_Some. } rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some. `````` Robbert Krebbers committed Feb 17, 2016 455 `````` - by rewrite lookup_ge_None, Hlen, <-lookup_ge_None. `````` Robbert Krebbers committed Nov 11, 2015 456 457 ``````Qed. Lemma lookup_app_l l1 l2 i : i < length l1 → (l1 ++ l2) !! i = l1 !! i. `````` Robbert Krebbers committed Mar 21, 2016 458 ``````Proof. revert i. induction l1; intros [|?]; naive_solver auto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 459 460 461 462 463 464 465 466 467 468 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 469 `````` - revert i. induction l1 as [|y l1 IH]; intros [|i] ?; `````` Robbert Krebbers committed Feb 17, 2016 470 `````` simplify_eq/=; auto with lia. `````` Robbert Krebbers committed Nov 11, 2015 471 `````` destruct (IH i) as [?|[??]]; auto with lia. `````` Robbert Krebbers committed Feb 17, 2016 472 `````` - intros [?|[??]]; auto using lookup_app_l_Some. by rewrite lookup_app_r. `````` Robbert Krebbers committed Nov 11, 2015 473 474 475 476 477 478 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 479 ``````Proof. by revert i; induction l; intros []; intros; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 480 ``````Lemma alter_length f l i : length (alter f i l) = length l. `````` Robbert Krebbers committed Feb 17, 2016 481 ``````Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 482 ``````Lemma insert_length l i x : length (<[i:=x]>l) = length l. `````` Robbert Krebbers committed Feb 17, 2016 483 ``````Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 484 485 486 ``````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. `````` Robbert Krebbers committed Mar 21, 2016 487 ``````Proof. revert i j. induction l; [done|]. intros [] []; naive_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 488 ``````Lemma list_lookup_insert l i x : i < length l → <[i:=x]>l !! i = Some x. `````` Robbert Krebbers committed Feb 17, 2016 489 ``````Proof. revert i. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 490 ``````Lemma list_lookup_insert_ne l i j x : i ≠ j → <[i:=x]>l !! j = l !! j. `````` Robbert Krebbers committed Mar 21, 2016 491 ``````Proof. revert i j. induction l; [done|]. intros [] []; naive_solver. Qed. `````` Robbert Krebbers committed Nov 11, 2015 492 493 494 495 496 497 ``````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]. `````` Robbert Krebbers committed Feb 17, 2016 498 `````` - intros Hy. assert (j < length l). `````` Robbert Krebbers committed Nov 11, 2015 499 500 `````` { rewrite <-(insert_length l j x); eauto using lookup_lt_Some. } rewrite list_lookup_insert in Hy by done; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 501 `````` - intros [(?&?&?)|[??]]; rewrite ?list_lookup_insert; naive_solver. `````` Robbert Krebbers committed Nov 11, 2015 502 503 504 ``````Qed. Lemma list_insert_commute l i j x y : i ≠ j → <[i:=x]>(<[j:=y]>l) = <[j:=y]>(<[i:=x]>l). `````` Robbert Krebbers committed Feb 17, 2016 505 ``````Proof. revert i j. by induction l; intros [|?] [|?] ?; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 506 507 508 ``````Lemma list_lookup_other l i x : length l ≠ 1 → l !! i = Some x → ∃ j y, j ≠ i ∧ l !! j = Some y. Proof. `````` Robbert Krebbers committed Feb 17, 2016 509 `````` intros. destruct i, l as [|x0 [|x1 l]]; simplify_eq/=. `````` Robbert Krebbers committed Feb 17, 2016 510 511 `````` - by exists 1, x1. - by exists 0, x0. `````` Robbert Krebbers committed Nov 11, 2015 512 513 514 ``````Qed. Lemma alter_app_l f l1 l2 i : i < length l1 → alter f i (l1 ++ l2) = alter f i l1 ++ l2. `````` Robbert Krebbers committed Feb 17, 2016 515 ``````Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 516 517 ``````Lemma alter_app_r f l1 l2 i : alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2. `````` Robbert Krebbers committed Feb 17, 2016 518 ``````Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 519 520 521 522 523 524 525 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 526 ``````Proof. intros ?. revert i. induction l; intros [|?]; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 527 528 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 529 ``````Proof. intros H ->. revert i H. induction k; intros [|?] ?; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 530 531 ``````Lemma list_alter_compose f g l i : alter (f ∘ g) i l = alter f i (alter g i l). `````` Robbert Krebbers committed Feb 17, 2016 532 ``````Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 533 534 ``````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). `````` Robbert Krebbers committed Feb 17, 2016 535 ``````Proof. revert i j. induction l; intros [|?][|?] ?; f_equal/=; auto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 536 537 ``````Lemma insert_app_l l1 l2 i x : i < length l1 → <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2. `````` Robbert Krebbers committed Feb 17, 2016 538 ``````Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 539 ``````Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2. `````` Robbert Krebbers committed Feb 17, 2016 540 ``````Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 541 542 543 544 545 546 547 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 548 ``````Proof. induction l1; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 587 `````` - intros Hy. assert (j < length l). `````` Robbert Krebbers committed Nov 11, 2015 588 589 `````` { rewrite <-(inserts_length l i k); eauto using lookup_lt_Some. } rewrite list_lookup_inserts in Hy by lia. intuition lia. `````` Robbert Krebbers committed Feb 17, 2016 590 `````` - intuition. by rewrite list_lookup_inserts by lia. `````` Robbert Krebbers committed Nov 11, 2015 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 615 616 `````` - split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x). - simpl. rewrite !elem_of_cons, IHl1. tauto. `````` Robbert Krebbers committed Nov 11, 2015 617 618 619 620 621 622 623 624 625 626 ``````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 627 `````` by exists (y :: l1), l2. `````` Robbert Krebbers committed Nov 11, 2015 628 629 630 631 632 633 634 635 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 636 `````` revert i. induction l; intros [|i] ?; simplify_eq/=; constructor; eauto. `````` Robbert Krebbers committed Nov 11, 2015 637 638 639 640 641 642 643 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 644 `````` - induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst; `````` Robbert Krebbers committed Nov 11, 2015 645 `````` setoid_rewrite elem_of_cons; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 646 `````` - intros (x&Hx&?). by induction Hx; csimpl; repeat case_match; `````` Robbert Krebbers committed Feb 17, 2016 647 `````` simplify_eq; try constructor; auto. `````` Robbert Krebbers committed Nov 11, 2015 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 664 665 `````` - rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver. - rewrite !NoDup_cons. `````` Robbert Krebbers committed Nov 11, 2015 666 667 668 669 670 `````` 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 | |]. `````` Robbert Krebbers committed Feb 17, 2016 671 672 673 674 `````` - by rewrite !NoDup_nil. - by rewrite !NoDup_cons, IH, Hlk. - rewrite !NoDup_cons, !elem_of_cons. intuition. - intuition. `````` Robbert Krebbers committed Nov 11, 2015 675 676 677 678 679 ``````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]. `````` Robbert Krebbers committed Feb 17, 2016 680 681 `````` { intros; simplify_eq. } intros [|i] [|j] ??; simplify_eq/=; eauto with f_equal; `````` Robbert Krebbers committed Nov 11, 2015 682 683 684 685 686 687 688 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 689 `````` - rewrite elem_of_list_lookup. intros [i ?]. `````` Robbert Krebbers committed Nov 11, 2015 690 `````` by feed pose proof (Hl (S i) 0 x); auto. `````` Robbert Krebbers committed Feb 17, 2016 691 `````` - apply IH. intros i j x' ??. by apply (inj S), (Hl (S i) (S j) x'). `````` Robbert Krebbers committed Nov 11, 2015 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 ``````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; `````` Robbert Krebbers committed Feb 17, 2016 713 `````` rewrite ?elem_of_cons; intuition (simplify_eq; auto). `````` Robbert Krebbers committed Nov 11, 2015 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 733 734 735 `````` - constructor. - done. - constructor. rewrite elem_of_list_difference; intuition. done. `````` Robbert Krebbers committed Nov 11, 2015 736 737 738 739 740 741 742 743 744 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 745 746 747 `````` - by apply NoDup_list_difference. - intro. rewrite elem_of_list_difference. intuition. - done. `````` Robbert Krebbers committed Nov 11, 2015 748 749 750 751 752 753 754 755 756 757 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 758 759 760 `````` - constructor. - constructor. rewrite elem_of_list_intersection; intuition. done. - done. `````` Robbert Krebbers committed Nov 11, 2015 761 762 763 764 765 766 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 767 `````` - induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|]. `````` Robbert Krebbers committed Nov 11, 2015 768 769 770 771 772 773 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 774 `````` - intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` + 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. `````` Robbert Krebbers committed Mar 21, 2016 806 807 808 `````` revert i; induction l; intros [] ?; repeat first [ match goal with x : prod _ _ |- _ => destruct x end | simplify_option_eq ]; eauto. `````` Robbert Krebbers committed Nov 11, 2015 809 810 811 `````` Qed. Lemma list_find_elem_of l x : x ∈ l → P x → is_Some (list_find P l). Proof. `````` Robbert Krebbers committed Feb 17, 2016 812 `````` induction 1 as [|x y l ? IH]; intros; simplify_option_eq; eauto. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` 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. `````` Robbert Krebbers committed Feb 11, 2016 842 ``````Global Instance: Inj (=) (=) (@reverse A). `````` Robbert Krebbers committed Nov 11, 2015 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 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 869 `````` revert i x. induction l; intros [|?] ??; simplify_eq/=; f_equal; auto. `````` Robbert Krebbers committed Nov 11, 2015 870 871 872 873 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 874 ``````Proof. induction l; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 875 876 877 ``````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. `````` Robbert Krebbers committed Feb 11, 2016 878 ``````Proof. intros ->. by rewrite <-(assoc_L (++)), take_app. Qed. `````` Robbert Krebbers committed Nov 11, 2015 879 ``````Lemma take_app_le l k n : n ≤ length l → take n (l ++ k) = take n l. `````` Robbert Krebbers committed Feb 17, 2016 880 ``````Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 881 882 ``````Lemma take_plus_app l k n m : length l = n → take (n + m) (l ++ k) = l ++ take m k. `````` Robbert Krebbers committed Feb 17, 2016 883 ``````Proof. intros <-. induction l; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 884 885 ``````Lemma take_app_ge l k n : length l ≤ n → take n (l ++ k) = l ++ take (n - length l) k. `````` Robbert Krebbers committed Feb 17, 2016 886 ``````Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 887 ``````Lemma take_ge l n : length l ≤ n → take n l = l. `````` Robbert Krebbers committed Feb 17, 2016 888 ``````Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed. `````` Robbert Krebbers committed Nov 11, 2015 889 ``````Lemma take_take l n m : take n (take m l) = take (min n m) l. `````` Robbert Krebbers committed Feb 17, 2016 890 ``````Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 891 ``````Lemma take_idemp l n : take n (take n l) = take n l. `````` Robbert Krebbers committed Nov 11, 2015 892 893 ``````Proof. by rewrite take_take, Min.min_idempotent. Qed. Lemma take_length l n : length (take n l) = min n (length l). `````` Robbert Krebbers committed Feb 17, 2016 894 ``````Proof. revert n. induction l; intros [|?]; f_equal/=; done. Qed. `````` Robbert Krebbers committed Nov 11, 2015 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 ``````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). `````` Robbert Krebbers committed Feb 17, 2016 910 911 `````` - by rewrite !lookup_take_ge. - by rewrite !lookup_take, !list_lookup_alter_ne by lia. `````` Robbert Krebbers committed Nov 11, 2015 912 913 914 915 ``````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). `````` Robbert Krebbers committed Feb 17, 2016 916 917 `````` - by rewrite !lookup_take_ge. - by rewrite !lookup_take, !list_lookup_insert_ne by lia. `````` Robbert Krebbers committed Nov 11, 2015 918 919 920 921 922 923 924 925 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 926 ``````Proof. revert n. by induction l; intros [|i]; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 ``````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. `````` Robbert Krebbers committed Feb 11, 2016 942 ``````Proof. intros ->. by rewrite <-(assoc_L (++)), drop_app. Qed. `````` Robbert Krebbers committed Nov 11, 2015 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 965 ``````Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 966 ``````Lemma take_take_drop l n m : take n l ++ take m (drop n l) = take (n + m) l. `````` Robbert Krebbers committed Feb 17, 2016 967 ``````Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 968 969 970 ``````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 [|?] [|?] ?; `````` Robbert Krebbers committed Feb 17, 2016 971 `````` f_equal/=; auto using take_drop with lia. `````` Robbert Krebbers committed Nov 11, 2015 972 973 974 975 976 977 978 979 980 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 981 982 `````` - revert i. induction n; intros [|?]; naive_solver auto with lia. - intros [-> Hi]. revert i Hi. `````` Robbert Krebbers committed Nov 11, 2015 983 984 985 986 987 988 989 990 991 992 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 993 994 `````` - intros Hin [x' Hx']; destruct Hin. rewrite lookup_replicate in Hx'; tauto. - intros Hx ?. destruct Hx. exists x; auto using lookup_replicate_2. `````` Robbert Krebbers committed Nov 11, 2015 995 996 ``````Qed. Lemma insert_replicate x n i : <[i:=x]>(replicate n x) = replicate n x. `````` Robbert Krebbers committed Feb 17, 2016 997 ``````Proof. revert i. induction n; intros [|?]; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 998 999 1000 1001 1002 1003 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 1004 ``````Proof. induction n; f_equal/=; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 1005 ``````Lemma take_replicate n m x : take n (replicate m x) = replicate (min n m) x. `````` Robbert Krebbers committed Feb 17, 2016 1006 ``````Proof. revert m. by induction n; intros [|?]; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 1007 1008 1009 ``````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<``````