list.v 160 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
2 3 4
(* 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. *)
5
From Coq Require Export Permutation.
6
From stdpp Require Export numbers base option.
Robbert Krebbers's avatar
Robbert Krebbers committed
7

8
Arguments length {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
9 10
Arguments cons {_} _ _.
Arguments app {_} _ _.
11 12 13 14

Instance: Params (@length) 1.
Instance: Params (@cons) 1.
Instance: Params (@app) 1.
Robbert Krebbers's avatar
Robbert Krebbers committed
15

16 17 18
Notation tail := tl.
Notation take := firstn.
Notation drop := skipn.
19

20
Arguments tail {_} _.
21 22 23
Arguments take {_} !_ !_ /.
Arguments drop {_} !_ !_ /.

24 25 26 27 28 29
Instance: Params (@tail) 1.
Instance: Params (@take) 1.
Instance: Params (@drop) 1.

Arguments Permutation {_} _ _.
Arguments Forall_cons {_} _ _ _ _ _.
30
Remove Hints Permutation_cons : typeclass_instances.
31

Robbert Krebbers's avatar
Robbert Krebbers committed
32 33 34 35 36 37 38
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.

39 40 41 42 43 44 45 46 47
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's avatar
Robbert Krebbers committed
48 49 50
Instance maybe_cons {A} : Maybe2 (@cons A) := λ l,
  match l with x :: l => Some (x,l) | _ => None end.

51
(** * Definitions *)
52 53 54 55 56 57
(** 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.

58 59
(** The operation [l !! i] gives the [i]th element of the list [l], or [None]
in case [i] is out of bounds. *)
60 61
Instance list_lookup {A} : Lookup nat A (list A) :=
  fix go i l {struct l} : option A := let _ : Lookup _ _ _ := @go in
62
  match l with
63
  | [] => None | x :: l => match i with 0 => Some x | S i => l !! i end
64
  end.
65 66 67

(** 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. *)
68
Instance list_alter {A} : Alter nat A (list A) := λ f,
69
  fix go i l {struct l} :=
70 71
  match l with
  | [] => []
72
  | x :: l => match i with 0 => f x :: l | S i => x :: go i l end
73
  end.
74

75 76
(** 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. *)
77 78
Instance list_insert {A} : Insert nat A (list A) :=
  fix go i y l {struct l} := let _ : Insert _ _ _ := @go in
79 80 81 82
  match l with
  | [] => []
  | x :: l => match i with 0 => y :: l | S i => x :: <[i:=y]>l end
  end.
83 84 85 86 87
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.
88
Instance: Params (@list_inserts) 1.
89

90 91 92
(** 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. *)
93 94
Instance list_delete {A} : Delete nat (list A) :=
  fix go (i : nat) (l : list A) {struct l} : list A :=
95 96
  match l with
  | [] => []
97
  | x :: l => match i with 0 => l | S i => x :: @delete _ _ go i l end
98
  end.
99 100 101

(** The function [option_list o] converts an element [Some x] into the
singleton list [[x]], and [None] into the empty list [[]]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
102
Definition option_list {A} : option A → list A := option_rect _ (λ x, [x]) [].
103 104
Instance: Params (@option_list) 1.
Instance maybe_list_singleton {A} : Maybe (λ x : A, [x]) := λ l,
105
  match l with [x] => Some x | _ => None end.
Robbert Krebbers's avatar
Robbert Krebbers committed
106 107 108 109

(** 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) :=
110
  fix go P _ l := let _ : Filter _ _ := @go in
Robbert Krebbers's avatar
Robbert Krebbers committed
111 112
  match l with
  | [] => []
113
  | x :: l => if decide (P x) then x :: filter P l else filter P l
114 115 116 117
  end.

(** The function [list_find P l] returns the first index [i] whose element
satisfies the predicate [P]. *)
118
Definition list_find {A} P `{∀ x, Decision (P x)} : list A → option (nat * A) :=
119 120
  fix go l :=
  match l with
121 122
  | [] => None
  | x :: l => if decide (P x) then Some (0,x) else prod_map S id <$> go l
123
  end.
124
Instance: Params (@list_find) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
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 :=
129
  match n with 0 => [] | S n => x :: replicate n x end.
130
Instance: Params (@replicate) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
131 132 133

(** The function [reverse l] returns the elements of [l] in reverse order. *)
Definition reverse {A} (l : list A) : list A := rev_append l [].
134
Instance: Params (@reverse) 1.
Robbert Krebbers's avatar
Robbert Krebbers committed
135

136 137 138 139
(** 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.
140
Instance: Params (@last) 1.
141

Robbert Krebbers's avatar
Robbert Krebbers committed
142 143 144 145 146 147
(** 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
148
  | x :: l => match n with 0 => [] | S n => x :: resize n y l end
Robbert Krebbers's avatar
Robbert Krebbers committed
149 150
  end.
Arguments resize {_} !_ _ !_.
151
Instance: Params (@resize) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
152

153 154 155
(** 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. *)
156 157
Fixpoint reshape {A} (szs : list nat) (l : list A) : list (list A) :=
  match szs with
158
  | [] => [] | sz :: szs => take sz l :: reshape szs (drop sz l)
159
  end.
160
Instance: Params (@reshape) 2.
161

162
Definition sublist_lookup {A} (i n : nat) (l : list A) : option (list A) :=
163 164 165 166
  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.
167

168 169 170 171
(** 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 :=
172
  fix go a l := match l with [] => a | x :: l => go (f a x) l end.
173 174 175

(** The monadic operations. *)
Instance list_ret: MRet list := λ A x, x :: @nil A.
176 177
Instance list_fmap : FMap list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x :: go l end.
178 179 180 181 182 183
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.
184 185
Instance list_bind : MBind list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x ++ go l end.
186 187
Instance list_join: MJoin list :=
  fix go A (ls : list (list A)) : list A :=
188
  match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end.
189
Definition mapM `{MBind M, MRet M} {A B} (f : A → M B) : list A → M (list B) :=
190
  fix go l :=
191
  match l with [] => mret [] | x :: l => y ← f x; k ← go l; mret (y :: k) end.
192 193 194 195 196

(** 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) :=
197
  match l with [] => [] | x :: l => f n x :: go (S n) l end.
198
Definition imap {A B} (f : nat → A → B) : list A → list B := imap_go f 0.
199 200
Arguments imap : simpl never.

201 202 203 204
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.

Robbert Krebbers's avatar
Robbert Krebbers committed
205 206 207 208 209 210 211 212 213
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.

214 215 216 217 218 219 220
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 {_ _} _ _ _ _ _.
221

222 223 224 225 226 227 228
(** 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.
229 230 231 232

(** The function [permutations l] yields all permutations of [l]. *)
Fixpoint interleave {A} (x : A) (l : list A) : list (list A) :=
  match l with
233
  | [] => [[x]]| y :: l => (x :: y :: l) :: ((y ::) <$> interleave x l)
234 235
  end.
Fixpoint permutations {A} (l : list A) : list (list A) :=
236
  match l with [] => [[]] | x :: l => permutations l ≫= interleave x end.
237

238 239
(** 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. *)
240 241
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.
242 243
Infix "`suffix_of`" := suffix_of (at level 70) : C_scope.
Infix "`prefix_of`" := prefix_of (at level 70) : C_scope.
244 245
Hint Extern 0 (_ `prefix_of` _) => reflexivity.
Hint Extern 0 (_ `suffix_of` _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
246

247 248 249 250 251 252 253 254
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 =>
255
      if decide_rel (=) x1 x2
256
      then prod_map id (x1 ::) (go l1 l2) else (x1 :: l1, x2 :: l2, [])
257 258 259 260 261
    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.
262 263
  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.
264
End prefix_suffix_ops.
Robbert Krebbers's avatar
Robbert Krebbers committed
265

266
(** A list [l1] is a sublist of [l2] if [l2] is obtained by removing elements
267 268 269
from [l1] without changing the order. *)
Inductive sublist {A} : relation (list A) :=
  | sublist_nil : sublist [] []
270
  | sublist_skip x l1 l2 : sublist l1 l2 → sublist (x :: l1) (x :: l2)
271
  | sublist_cons x l1 l2 : sublist l1 l2 → sublist l1 (x :: l2).
272
Infix "`sublist`" := sublist (at level 70) : C_scope.
273
Hint Extern 0 (_ `sublist` _) => reflexivity.
274 275

(** A list [l2] contains a list [l1] if [l2] is obtained by removing elements
276
from [l1] while possiblity changing the order. *)
277 278 279 280
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)
281
  | contains_cons x l1 l2 : contains l1 l2 → contains l1 (x :: l2)
282 283
  | contains_trans l1 l2 l3 : contains l1 l2 → contains l2 l3 → contains l1 l3.
Infix "`contains`" := contains (at level 70) : C_scope.
284
Hint Extern 0 (_ `contains` _) => reflexivity.
285 286 287 288 289 290 291 292 293 294

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
295
    | [] => Some l | x :: k => list_remove x l ≫= list_remove_list k
296 297
    end.
End contains_dec_help.
298

299 300 301 302 303
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').
304 305

(** Set operations on lists *)
306 307 308
Definition included {A} (l1 l2 : list A) := ∀ x, x ∈ l1 → x ∈ l2.
Infix "`included`" := included (at level 70) : C_scope.

309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331
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
332
      then list_difference l k else x :: list_difference l k
333
    end.
334
  Definition list_union (l k : list A) : list A := list_difference l k ++ k.
335 336 337 338 339
  Fixpoint list_intersection (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel (∈) x k
340
      then x :: list_intersection l k else list_intersection l k
341 342 343 344 345 346 347 348 349
    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.
350 351

(** * Basic tactics on lists *)
352
(** The tactic [discriminate_list] discharges a goal if it contains
353 354
a list equality involving [(::)] and [(++)] of two lists that have a different
length as one of its hypotheses. *)
355
Tactic Notation "discriminate_list" hyp(H) :=
356
  apply (f_equal length) in H;
357
  repeat (csimpl in H || rewrite app_length in H); exfalso; lia.
358 359
Tactic Notation "discriminate_list" :=
  match goal with H : @eq (list _) _ _ |- _ => discriminate_list H end.
360

361
(** The tactic [simplify_list_eq] simplifies hypotheses involving
362 363
equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies
lookups in singleton lists. *)
364
Lemma app_inj_1 {A} (l1 k1 l2 k2 : list A) :
365 366
  length l1 = length k1 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2.
Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed.
367
Lemma app_inj_2 {A} (l1 k1 l2 k2 : list A) :
368 369
  length l2 = length k2 → l1 ++ l2 = k1 ++ k2 → l1 = k1 ∧ l2 = k2.
Proof.
370
  intros ? Hl. apply app_inj_1; auto.
371 372
  apply (f_equal length) in Hl. rewrite !app_length in Hl. lia.
Qed.
373
Ltac simplify_list_eq :=
374
  repeat match goal with
375
  | _ => progress simplify_eq/=
376
  | H : _ ++ _ = _ ++ _ |- _ => first
377
    [ apply app_inv_head in H | apply app_inv_tail in H
378 379
    | apply app_inj_1 in H; [destruct H|done]
    | apply app_inj_2 in H; [destruct H|done] ]
Robbert Krebbers's avatar
Robbert Krebbers committed
380
  | H : [?x] !! ?i = Some ?y |- _ =>
381
    destruct i; [change (Some x = Some y) in H | discriminate]
382
  end.
383

384 385
(** * General theorems *)
Section general_properties.
Robbert Krebbers's avatar
Robbert Krebbers committed
386
Context {A : Type}.
387 388
Implicit Types x y z : A.
Implicit Types l k : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
389

390
Global Instance: Inj2 (=) (=) (=) (@cons A).
391
Proof. by injection 1. Qed.
392
Global Instance: ∀ k, Inj (=) (=) (k ++).
393
Proof. intros ???. apply app_inv_head. Qed.
394
Global Instance: ∀ k, Inj (=) (=) (++ k).
395
Proof. intros ???. apply app_inv_tail. Qed.
396
Global Instance: Assoc (=) (@app A).
397 398 399 400 401
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.
402

403
Lemma app_nil l1 l2 : l1 ++ l2 = [] ↔ l1 = [] ∧ l2 = [].
404
Proof. split. apply app_eq_nil. by intros [-> ->]. Qed.
405 406
Lemma app_singleton l1 l2 x :
  l1 ++ l2 = [x] ↔ l1 = [] ∧ l2 = [x] ∨ l1 = [x] ∧ l2 = [].
407
Proof. split. apply app_eq_unit. by intros [[-> ->]|[-> ->]]. Qed.
408 409 410
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.
411
Proof.
412
  revert l2. induction l1 as [|x l1 IH]; intros [|y l2] H.
413 414 415
  - done.
  - discriminate (H 0).
  - discriminate (H 0).
416
  - f_equal; [by injection (H 0)|]. apply (IH _ $ λ i, H (S i)).
417
Qed.
418
Global Instance list_eq_dec {dec : ∀ x y, Decision (x = y)} : ∀ l k,
419
  Decision (l = k) := list_eq_dec dec.
420 421 422
Global Instance list_eq_nil_dec l : Decision (l = []).
Proof. by refine match l with [] => left _ | _ => right _ end. Defined.
Lemma list_singleton_reflect l :
423
  option_reflect (λ x, l = [x]) (length l ≠ 1) (maybe (λ x, [x]) l).
424 425 426 427
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.
428
Lemma nil_or_length_pos l : l = [] ∨ length l ≠ 0.
429
Proof. destruct l; simpl; auto with lia. Qed.
430
Lemma nil_length_inv l : length l = 0 → l = [].
431 432
Proof. by destruct l. Qed.
Lemma lookup_nil i : @nil A !! i = None.
433
Proof. by destruct i. Qed.
434
Lemma lookup_tail l i : tail l !! i = l !! S i.
435
Proof. by destruct l. Qed.
436
Lemma lookup_lt_Some l i x : l !! i = Some x → i < length l.
437
Proof. revert i. induction l; intros [|?] ?; naive_solver auto with arith. Qed.
438 439 440
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).
441
Proof. revert i. induction l; intros [|?] ?; naive_solver eauto with lia. Qed.
442 443 444 445 446 447 448 449
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.
450 451 452
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
453
Proof.
454
  intros <- Hlen Hl; apply list_eq; intros i. destruct (l2 !! i) as [x|] eqn:Hx.
455
  - destruct (lookup_lt_is_Some_2 l1 i) as [y Hy].
456 457
    { rewrite Hlen; eauto using lookup_lt_Some. }
    rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some.
458
  - by rewrite lookup_ge_None, Hlen, <-lookup_ge_None.
Robbert Krebbers's avatar
Robbert Krebbers committed
459
Qed.
460
Lemma lookup_app_l l1 l2 i : i < length l1 → (l1 ++ l2) !! i = l1 !! i.
461
Proof. revert i. induction l1; intros [|?]; naive_solver auto with lia. Qed.
462 463
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.
464
Lemma lookup_app_r l1 l2 i :
465
  length l1 ≤ i → (l1 ++ l2) !! i = l2 !! (i - length l1).
466 467 468 469 470 471
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.
472
  - revert i. induction l1 as [|y l1 IH]; intros [|i] ?;
473
      simplify_eq/=; auto with lia.
474
    destruct (IH i) as [?|[??]]; auto with lia.
475
  - intros [?|[??]]; auto using lookup_app_l_Some. by rewrite lookup_app_r.
476
Qed.
477 478 479
Lemma list_lookup_middle l1 l2 x n :
  n = length l1 → (l1 ++ x :: l2) !! n = Some x.
Proof. intros ->. by induction l1. Qed.
480

481
Lemma list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l.
482
Proof. by revert i; induction l; intros []; intros; f_equal/=. Qed.
483
Lemma alter_length f l i : length (alter f i l) = length l.
484
Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
485
Lemma insert_length l i x : length (<[i:=x]>l) = length l.
486
Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
487
Lemma list_lookup_alter f l i : alter f i l !! i = f <$> l !! i.
488
Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
489
Lemma list_lookup_alter_ne f l i j : i ≠ j → alter f i l !! j = l !! j.
490
Proof. revert i j. induction l; [done|]. intros [] []; naive_solver. Qed.
491
Lemma list_lookup_insert l i x : i < length l → <[i:=x]>l !! i = Some x.
492
Proof. revert i. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
493
Lemma list_lookup_insert_ne l i j x : i ≠ j → <[i:=x]>l !! j = l !! j.
494
Proof. revert i j. induction l; [done|]. intros [] []; naive_solver. Qed.
495 496 497 498 499 500
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].
501
  - intros Hy. assert (j < length l).
502 503
    { rewrite <-(insert_length l j x); eauto using lookup_lt_Some. }
    rewrite list_lookup_insert in Hy by done; naive_solver.
504
  - intros [(?&?&?)|[??]]; rewrite ?list_lookup_insert; naive_solver.
505 506 507
Qed.
Lemma list_insert_commute l i j x y :
  i ≠ j → <[i:=x]>(<[j:=y]>l) = <[j:=y]>(<[i:=x]>l).
508
Proof. revert i j. by induction l; intros [|?] [|?] ?; f_equal/=; auto. Qed.
509 510
Lemma list_lookup_other l i x :
  length l ≠ 1 → l !! i = Some x → ∃ j y, j ≠ i ∧ l !! j = Some y.
Robbert Krebbers's avatar
Robbert Krebbers committed
511
Proof.
512
  intros. destruct i, l as [|x0 [|x1 l]]; simplify_eq/=.
513 514
  - by exists 1, x1.
  - by exists 0, x0.
Robbert Krebbers's avatar
Robbert Krebbers committed
515
Qed.
516 517
Lemma alter_app_l f l1 l2 i :
  i < length l1 → alter f i (l1 ++ l2) = alter f i l1 ++ l2.
518
Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
519
Lemma alter_app_r f l1 l2 i :
520
  alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2.
521
Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
522 523
Lemma alter_app_r_alt f l1 l2 i :
  length l1 ≤ i → alter f i (l1 ++ l2) = l1 ++ alter f (i - length l1) l2.
524 525 526 527
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply alter_app_r.
Qed.
528
Lemma list_alter_id f l i : (∀ x, f x = x) → alter f i l = l.
529
Proof. intros ?. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
530 531
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.
532
Proof. intros H ->. revert i H. induction k; intros [|?] ?; f_equal/=; auto. Qed.
533 534
Lemma list_alter_compose f g l i :
  alter (f ∘ g) i l = alter f i (alter g i l).
535
Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
536 537
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).
538
Proof. revert i j. induction l; intros [|?][|?] ?; f_equal/=; auto with lia. Qed.
539 540
Lemma insert_app_l l1 l2 i x :
  i < length l1 → <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2.
541
Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
542
Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2.
543
Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
544 545
Lemma insert_app_r_alt l1 l2 i x :
  length l1 ≤ i → <[i:=x]>(l1 ++ l2) = l1 ++ <[i - length l1:=x]>l2.
546 547 548 549
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply insert_app_r.
Qed.
550
Lemma delete_middle l1 l2 x : delete (length l1) (l1 ++ x :: l2) = l1 ++ l2.
551
Proof. induction l1; f_equal/=; auto. Qed.
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
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.
590
  - intros Hy. assert (j < length l).
591 592
    { rewrite <-(inserts_length l i k); eauto using lookup_lt_Some. }
    rewrite list_lookup_inserts in Hy by lia. intuition lia.
593
  - intuition. by rewrite list_lookup_inserts by lia.
594 595 596 597 598 599 600 601
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.

602
(** ** Properties of the [elem_of] predicate *)
603
Lemma not_elem_of_nil x : x ∉ [].
604
Proof. by inversion 1. Qed.
605
Lemma elem_of_nil x : x ∈ [] ↔ False.
606
Proof. intuition. by destruct (not_elem_of_nil x). Qed.
607
Lemma elem_of_nil_inv l : (∀ x, x ∉ l) → l = [].
608
Proof. destruct l. done. by edestruct 1; constructor. Qed.
609 610
Lemma elem_of_not_nil x l : x ∈ l → l ≠ [].
Proof. intros ? ->. by apply (elem_of_nil x). Qed.
611
Lemma elem_of_cons l x y : x ∈ y :: l ↔ x = y ∨ x ∈ l.
Robbert Krebbers's avatar
Robbert Krebbers committed
612
Proof. by split; [inversion 1; subst|intros [->|?]]; constructor. Qed.
613
Lemma not_elem_of_cons l x y : x ∉ y :: l ↔ x ≠ y ∧ x ∉ l.
Robbert Krebbers's avatar
Robbert Krebbers committed
614
Proof. rewrite elem_of_cons. tauto. Qed.
615
Lemma elem_of_app l1 l2 x : x ∈ l1 ++ l2 ↔ x ∈ l1 ∨ x ∈ l2.
616
Proof.
617
  induction l1.
618 619
  - split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x).
  - simpl. rewrite !elem_of_cons, IHl1. tauto.
620
Qed.
621
Lemma not_elem_of_app l1 l2 x : x ∉ l1 ++ l2 ↔ x ∉ l1 ∧ x ∉ l2.
Robbert Krebbers's avatar
Robbert Krebbers committed
622
Proof. rewrite elem_of_app. tauto. Qed.
623
Lemma elem_of_list_singleton x y : x ∈ [y] ↔ x = y.
624
Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
625
Global Instance elem_of_list_permutation_proper x : Proper ((≡ₚ) ==> iff) (x ∈).
626
Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed.
627
Lemma elem_of_list_split l x : x ∈ l → ∃ l1 l2, l = l1 ++ x :: l2.
628
Proof.
629
  induction 1 as [x l|x y l ? [l1 [l2 ->]]]; [by eexists [], l|].
630
  by exists (y :: l1), l2.
631
Qed.
632
Lemma elem_of_list_lookup_1 l x : x ∈ l → ∃ i, l !! i = Some x.
633
Proof.
634 635
  induction 1 as [|???? IH]; [by exists 0 |].
  destruct IH as [i ?]; auto. by exists (S i).
636
Qed.
637
Lemma elem_of_list_lookup_2 l i x : l !! i = Some x → x ∈ l.
638
Proof.
639
  revert i. induction l; intros [|i] ?; simplify_eq/=; constructor; eauto.
640
Qed.
641 642
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.
643 644 645 646
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.
647
  - induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst;
648
      setoid_rewrite elem_of_cons; naive_solver.
649
  - intros (x&Hx&?). by induction Hx; csimpl; repeat case_match;
650
      simplify_eq; try constructor; auto.
651
Qed.
652

653
(** ** Properties of the [NoDup] predicate *)
654 655
Lemma NoDup_nil : NoDup (@nil A) ↔ True.
Proof. split; constructor. Qed.
656
Lemma NoDup_cons x l : NoDup (x :: l) ↔ x ∉ l ∧ NoDup l.
657
Proof. split. by inversion 1. intros [??]. by constructor. Qed.
658
Lemma NoDup_cons_11 x l : NoDup (x :: l) → x ∉ l.
659
Proof. rewrite NoDup_cons. by intros [??]. Qed.
660
Lemma NoDup_cons_12 x l : NoDup (x :: l) → NoDup l.
661
Proof. rewrite NoDup_cons. by intros [??]. Qed.
662
Lemma NoDup_singleton x : NoDup [x].
663
Proof. constructor. apply not_elem_of_nil. constructor. Qed.
664
Lemma NoDup_app l k : NoDup (l ++ k) ↔ NoDup l ∧ (∀ x, x ∈ l → x ∉ k) ∧ NoDup k.
Robbert Krebbers's avatar
Robbert Krebbers committed
665
Proof.
666
  induction l; simpl.
667 668
  - rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver.
  - rewrite !NoDup_cons.
Robbert Krebbers's avatar
Robbert Krebbers committed
669
    setoid_rewrite elem_of_cons. setoid_rewrite elem_of_app. naive_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
670
Qed.
671
Global Instance NoDup_proper: Proper ((≡ₚ) ==> iff) (@NoDup A).
672 673
Proof.
  induction 1 as [|x l k Hlk IH | |].
674 675 676 677
  - by rewrite !NoDup_nil.
  - by rewrite !NoDup_cons, IH, Hlk.
  - rewrite !NoDup_cons, !elem_of_cons. intuition.
  - intuition.
678
Qed.
679 680
Lemma NoDup_lookup l i j x :
  NoDup l → l !! i = Some x → l !! j = Some x → i = j.
681 682
Proof.
  intros Hl. revert i j. induction Hl as [|x' l Hx Hl IH].
683 684
  { intros; simplify_eq. }
  intros [|i] [|j] ??; simplify_eq/=; eauto with f_equal;
685 686
    exfalso; eauto using elem_of_list_lookup_2.
Qed.
687 688
Lemma NoDup_alt l :
  NoDup l ↔ ∀ i j x, l !! i = Some x → l !! j = Some x → i = j.
689
Proof.
690 691
  split; eauto using NoDup_lookup.
  induction l as [|x l IH]; intros Hl; constructor.
692
  - rewrite elem_of_list_lookup. intros [i ?].
693
    by feed pose proof (Hl (S i) 0 x); auto.
694
  - apply IH. intros i j x' ??. by apply (inj S), (Hl (S i) (S j) x').
695
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
696

697 698 699 700 701 702
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
703
    | x :: l =>
704 705 706 707 708 709 710 711
      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
712
    end.
713
  Lemma elem_of_remove_dups l x : x ∈ remove_dups l ↔ x ∈ l.
714 715
  Proof.
    split; induction l; simpl; repeat case_decide;
716
      rewrite ?elem_of_cons; intuition (simplify_eq; auto).
717
  Qed.
718
  Lemma NoDup_remove_dups l : NoDup (remove_dups l).
719 720 721 722
  Proof.
    induction l; simpl; repeat case_decide; try constructor; auto.
    by rewrite elem_of_remove_dups.
  Qed.
723
End no_dup_dec.
724

725 726 727 728 729 730 731 732 733 734 735
(** ** 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.
736 737 738
    - constructor.
    - done.
    - constructor. rewrite elem_of_list_difference; intuition. done.
739 740 741 742 743 744 745 746 747
  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.
748 749 750
    - by apply NoDup_list_difference.
    - intro. rewrite elem_of_list_difference. intuition.
    - done.
751 752 753 754 755 756 757 758 759 760
  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.
761 762 763
    - constructor.
    - constructor. rewrite elem_of_list_intersection; intuition. done.
    - done.
764 765 766 767 768 769
  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.
770
    - induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
771 772 773 774 775 776
      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.
777
    - intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
778 779 780 781 782 783 784 785 786 787
      + 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.

788
(** ** Properties of the [filter] function *)
789 790 791 792 793 794 795
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.
796
  Lemma NoDup_filter l : NoDup l → NoDup (filter P l).
797 798 799 800 801
  Proof.
    unfold filter. induction 1; simpl; repeat case_decide;
      rewrite ?NoDup_nil, ?NoDup_cons, ?elem_of_list_filter; tauto.
  Qed.
End filter.
Robbert Krebbers's avatar
Robbert Krebbers committed
802

803 804 805
(** ** Properties of the [find] function *)
Section find.
  Context (P : A → Prop) `{∀ x, Decision (P x)}.
806 807
  Lemma list_find_Some l i x :
    list_find P l = Some (i,x) → l !! i = Some x ∧ P x.
808
  Proof.
809 810 811
    revert i; induction l; intros [] ?; repeat first
      [ match goal with x : prod _ _ |- _ => destruct x end
      | simplify_option_eq ]; eauto.
812
  Qed.
813
  Lemma list_find_elem_of l x : x ∈ l → P x → is_Some (list_find P l).
814
  Proof.
815
    induction 1 as [|x y l ? IH]; intros; simplify_option_eq; eauto.
816
    by destruct IH as [[i x'] ->]; [|exists (S i, x')].
817 818 819
  Qed.
End find.

820
(** ** Properties of the [reverse] function *)
821 822
Lemma reverse_nil : reverse [] = @nil A.
Proof. done. Qed.
823
Lemma reverse_singleton x : reverse [x] = [x].
824
Proof. done. Qed.
825
Lemma reverse_cons l x : reverse (x :: l) = reverse l ++ [x].
826
Proof. unfold reverse. by rewrite <-!rev_alt. Qed.
827
Lemma reverse_snoc l x : reverse (l ++ [x]) = x :: reverse l.
828
Proof. unfold reverse. by rewrite <-!rev_alt, rev_unit. Qed.
829
Lemma reverse_app l1 l2 : reverse (l1 ++ l2) = reverse l2 ++ reverse l1.
830
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_app_distr. Qed.
831
Lemma reverse_length l : length (reverse l) = length l.
832
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_length. Qed.
833
Lemma reverse_involutive l : reverse (reverse l) = l.
834
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_involutive. Qed.
835 836 837 838 839 840 841 842 843 844
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.
845
Global Instance: Inj (=) (=) (@reverse A).
846 847 848 849
Proof.
  intros l1 l2 Hl.
  by rewrite <-(reverse_involutive l1), <-(reverse_involutive l2), Hl.
Qed.
850 851 852 853 854 855 856 857
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.
858

859 860 861
(** ** Properties of the [last] function *)
Lemma last_snoc x l : last (l ++ [x]) = Some x.
Proof. induction l as [|? []]; simpl; auto. Qed.
862 863 864 865
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.
866

867 868 869 870 871
(** ** 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.
872
  revert i x. induction l; intros [|?] ??; simplify_eq/=; f_equal; auto.
873
Qed.
874
Lemma take_nil n : take n (@nil A) = [].
Robbert Krebbers's avatar
Robbert Krebbers committed
875
Proof. by destruct n. Qed.
876
Lemma take_app l k : take (length l) (l ++ k) = l.
877
Proof. induction l; f_equal/=; auto. Qed.
878
Lemma take_app_alt l k n : n = length l → take n (l ++ k) = l.
879 880
Proof. intros ->. by apply take_app. Qed.
Lemma take_app3_alt l1 l2 l3 n : n = length l1 → take n ((l1 ++ l2) ++ l3) = l1.
881
Proof. intros ->. by rewrite <-(assoc_L (++)), take_app. Qed.
882
Lemma take_app_le l k n : n ≤ length l → take n (l ++ k) = take n l.
883
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
884 885
Lemma take_plus_app l k n m :
  length l = n → take (n + m) (l ++ k) = l ++ take m k.
886
Proof. intros <-. induction l; f_equal/=; auto. Qed.
887 888
Lemma take_app_ge l k n :
  length l ≤ n → take n (l ++ k) = l ++ take (n - length l) k.
889
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
890
Lemma take_ge l n : length l ≤ n → take n l = l.
891
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
892
Lemma take_take l n m : take n (take m l) = take (min n m) l.
893
Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed.
894
Lemma take_idemp l n : take n (take n l) = take n l.
Robbert Krebbers's avatar
Robbert Krebbers committed
895
Proof. by rewrite take_take, Min.min_idempotent. Qed.
896
Lemma take_length l n : length (take n l) = min n (length l).
897
Proof. revert n. induction l; intros [|?]; f_equal/=; done. Qed.
898
Lemma take_length_le l n : n ≤ length l → length (take n l) = n.
Robbert Krebbers's avatar
Robbert Krebbers committed
899
Proof. rewrite take_length. apply Min.min_l. Qed.
900 901
Lemma take_length_ge l n : length l ≤ n → length (take n l) = length l.
Proof. rewrite take_length. apply Min.min_r. Qed.
902
Lemma take_drop_commute l n m : take n (drop m l) = drop m (take (m + n) l).
Robbert Krebbers's avatar
Robbert Krebbers committed
903
Proof.
904
  revert n m. induction l; intros [|?][|?]; simpl; auto using take_nil with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
905
Qed.
906 907
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.
908
Lemma lookup_take_ge l n i : n ≤ i → take n l !! i = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
909
Proof. revert n i. induction l; intros [|?] [|?] ?; simpl; auto with lia. Qed.
910
Lemma take_alter f l n i : n ≤ i → take n (alter f i l) = take n l.
Robbert Krebbers's avatar
Robbert Krebbers committed
911 912
Proof.
  intros. apply list_eq. intros j. destruct (le_lt_dec n j).
913 914
  - by rewrite !lookup_take_ge.
  - by rewrite !lookup_take, !list_lookup_alter_ne by lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
915
Qed.
916
Lemma take_insert l n i x : n ≤ i → take n (<[i:=x]>l) = take n l.
917 918
Proof.
  intros. apply list_eq. intros j. destruct (le_lt_dec n j).
919 920
  - by rewrite !lookup_take_ge.
  - by rewrite !lookup_take, !list_lookup_insert_ne by lia.
921
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
922

923
(** ** Properties of the [drop] function *)
924 925
Lemma drop_0 l : drop 0 l = l.
Proof. done. Qed.
926
Lemma drop_nil n : drop n (@nil A) = [].
Robbert Krebbers's avatar
Robbert Krebbers committed
927
Proof. by destruct n. Qed.
928
Lemma drop_length l n : length (drop n l) = length l - n.
929
Proof. revert n. by induction l; intros [|i]; f_equal/=. Qed.
930
Lemma drop_ge l n : length l ≤ n → drop n l = [].