list.v 138 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2014, 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
Require Export Permutation.
6
Require Export numbers base decidable option.
Robbert Krebbers's avatar
Robbert Krebbers committed
7

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

14 15 16
Notation tail := tl.
Notation take := firstn.
Notation drop := skipn.
17

18 19 20
Arguments take {_} !_ !_ /.
Arguments drop {_} !_ !_ /.

Robbert Krebbers's avatar
Robbert Krebbers committed
21 22 23 24 25 26 27
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.

28 29 30 31 32 33 34 35 36
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  x) (only parsing) : C_scope.
Notation "(≢ₚ)" := (λ x y, ¬x  y) (only parsing) : C_scope.
Notation "x ≢ₚ y":= (¬x  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.

37 38 39
(** * Definitions *)
(** The operation [l !! i] gives the [i]th element of the list [l], or [None]
in case [i] is out of bounds. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
40
Instance list_lookup {A} : Lookup nat A (list A) :=
41
  fix go i l {struct l} : option A := let _ : Lookup _ _ _ := @go in
42
  match l with
43
  | [] => None | x :: l => match i with 0 => Some x | S i => l !! i end
44
  end.
45 46 47

(** 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. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
48
Instance list_alter {A} (f : A  A) : AlterD nat A (list A) f :=
49
  fix go i l {struct l} := let _ : AlterD _ _ _ f := @go in
50 51
  match l with
  | [] => []
52
  | x :: l => match i with 0 => f x :: l | S i => x :: alter f i l end
53
  end.
54

55 56
(** 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. *)
57 58 59 60 61 62
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.
63

64 65 66
(** 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. *)
67 68
Instance list_delete {A} : Delete nat (list A) :=
  fix go (i : nat) (l : list A) {struct l} : list A :=
69 70
  match l with
  | [] => []
71
  | x :: l => match i with 0 => l | S i => x :: @delete _ _ go i l end
72
  end.
73 74 75

(** 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
76
Definition option_list {A} : option A  list A := option_rect _ (λ x, [x]) [].
77 78
Definition list_singleton {A} (l : list A) : option A :=
  match l with [x] => Some x | _ => None end.
Robbert Krebbers's avatar
Robbert Krebbers committed
79 80 81 82

(** 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) :=
83
  fix go P _ l := let _ : Filter _ _ := @go in
Robbert Krebbers's avatar
Robbert Krebbers committed
84 85
  match l with
  | [] => []
86
  | x :: l => if decide (P x) then x :: filter P l else filter P l
87 88 89 90 91 92 93
  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 :=
  fix go l :=
  match l with
94
  | [] => None | x :: l => if decide (P x) then Some 0 else S <$> go l
95
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
96 97 98 99

(** 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 :=
100
  match n with 0 => [] | S n => x :: replicate n x end.
Robbert Krebbers's avatar
Robbert Krebbers committed
101 102 103 104

(** The function [reverse l] returns the elements of [l] in reverse order. *)
Definition reverse {A} (l : list A) : list A := rev_append l [].

105 106 107 108
(** 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.
109

Robbert Krebbers's avatar
Robbert Krebbers committed
110 111 112 113 114 115
(** 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
116
  | x :: l => match n with 0 => [] | S n => x :: resize n y l end
Robbert Krebbers's avatar
Robbert Krebbers committed
117 118 119
  end.
Arguments resize {_} !_ _ !_.

120 121 122
(** 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. *)
123 124
Fixpoint reshape {A} (szs : list nat) (l : list A) : list (list A) :=
  match szs with
125
  | [] => [] | sz :: szs => take sz l :: reshape szs (drop sz l)
126 127
  end.

128
Definition sublist_lookup {A} (i n : nat) (l : list A) : option (list A) :=
129 130 131 132
  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.
133

134 135 136 137
(** 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 :=
138
  fix go a l := match l with [] => a | x :: l => go (f a x) l end.
139 140 141 142 143

(** The monadic operations. *)
Instance list_ret: MRet list := λ A x, x :: @nil A.
Instance list_fmap {A B} (f : A  B) : FMapD list f :=
  fix go (l : list A) :=
144
  match l with [] => [] | x :: l => f x :: @fmap _ _ _ f go l end.
145 146
Instance list_bind {A B} (f : A  list B) : MBindD list f :=
  fix go (l : list A) :=
147
  match l with [] => [] | x :: l => f x ++ @mbind _ _ _ f go l end.
148 149
Instance list_join: MJoin list :=
  fix go A (ls : list (list A)) : list A :=
150
  match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end.
151
Definition mapM `{MBind M, MRet M} {A B} (f : A  M B) : list A  M (list B) :=
152
  fix go l :=
153
  match l with [] => mret [] | x :: l => y  f x; k  go l; mret (y :: k) end.
154 155 156 157 158

(** 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) :=
159
  match l with [] => [] | x :: l => f n x :: go (S n) l end.
160
Definition imap {A B} (f : nat  A  B) : list A  list B := imap_go f 0.
161 162 163 164 165 166 167 168 169 170 171
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.

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 {_ _} _ _ _ _ _.
172

173 174 175 176 177 178 179
(** 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.
180 181 182 183

(** The function [permutations l] yields all permutations of [l]. *)
Fixpoint interleave {A} (x : A) (l : list A) : list (list A) :=
  match l with
184
  | [] => [[x]]| y :: l => (x :: y :: l) :: ((y ::) <$> interleave x l)
185 186
  end.
Fixpoint permutations {A} (l : list A) : list (list A) :=
187
  match l with [] => [[]] | x :: l => permutations l = interleave x end.
188

189 190
(** 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. *)
191 192
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.
193 194
Infix "`suffix_of`" := suffix_of (at level 70) : C_scope.
Infix "`prefix_of`" := prefix_of (at level 70) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
195

196 197 198 199 200 201 202 203
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 =>
204
      if decide_rel (=) x1 x2
205
      then prod_map id (x1 ::) (go l1 l2) else (x1 :: l1, x2 :: l2, [])
206 207 208 209 210
    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.
211 212
  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.
213
End prefix_suffix_ops.
Robbert Krebbers's avatar
Robbert Krebbers committed
214

215
(** A list [l1] is a sublist of [l2] if [l2] is obtained by removing elements
216 217 218
from [l1] without changing the order. *)
Inductive sublist {A} : relation (list A) :=
  | sublist_nil : sublist [] []
219
  | sublist_skip x l1 l2 : sublist l1 l2  sublist (x :: l1) (x :: l2)
220
  | sublist_cons x l1 l2 : sublist l1 l2  sublist l1 (x :: l2).
221 222 223
Infix "`sublist`" := sublist (at level 70) : C_scope.

(** A list [l2] contains a list [l1] if [l2] is obtained by removing elements
224
from [l1] while possiblity changing the order. *)
225 226 227 228
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)
229
  | contains_cons x l1 l2 : contains l1 l2  contains l1 (x :: l2)
230 231 232 233 234 235 236 237 238 239 240 241
  | contains_trans l1 l2 l3 : contains l1 l2  contains l2 l3  contains l1 l3.
Infix "`contains`" := contains (at level 70) : C_scope.

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
242
    | [] => Some l | x :: k => list_remove x l = list_remove_list k
243 244
    end.
End contains_dec_help.
245

246 247 248 249 250
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').
251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275

(** Set operations on lists *)
Section list_set.
  Context {A} {dec :  x y : A, Decision (x = y)}.
  Global Instance elem_of_list_dec {dec :  x y : A, Decision (x = y)}
    (x : A) :  l, Decision (x  l).
  Proof.
   refine (
    fix go l :=
    match l return Decision (x  l) with
    | [] => right _
    | y :: l => cast_if_or (decide (x = y)) (go l)
    end); clear go dec; subst; try (by constructor); abstract by inversion 1.
  Defined.
  Fixpoint remove_dups (l : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x l then remove_dups l else x :: remove_dups l
    end.
  Fixpoint list_difference (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x k
276
      then list_difference l k else x :: list_difference l k
277 278 279 280 281 282
    end.
  Fixpoint list_intersection (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x k
283
      then x :: list_intersection l k else list_intersection l k
284 285 286 287 288 289 290 291 292
    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.
293 294

(** * Basic tactics on lists *)
295 296 297
(** The tactic [discriminate_list_equality] discharges a goal if it contains
a list equality involving [(::)] and [(++)] of two lists that have a different
length as one of its hypotheses. *)
298 299
Tactic Notation "discriminate_list_equality" hyp(H) :=
  apply (f_equal length) in H;
300
  repeat (simpl in H || rewrite app_length in H); exfalso; lia.
301
Tactic Notation "discriminate_list_equality" :=
302 303 304
  match goal with
  | H : @eq (list _) _ _ |- _ => discriminate_list_equality H
  end.
305

306 307 308
(** The tactic [simplify_list_equality] simplifies hypotheses involving
equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies
lookups in singleton lists. *)
309 310 311 312 313 314 315 316 317
Lemma app_injective_1 {A} (l1 k1 l2 k2 : list A) :
  length l1 = length k1  l1 ++ l2 = k1 ++ k2  l1 = k1  l2 = k2.
Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed.
Lemma app_injective_2 {A} (l1 k1 l2 k2 : list A) :
  length l2 = length k2  l1 ++ l2 = k1 ++ k2  l1 = k1  l2 = k2.
Proof.
  intros ? Hl. apply app_injective_1; auto.
  apply (f_equal length) in Hl. rewrite !app_length in Hl. lia.
Qed.
318 319 320
Ltac simplify_list_equality :=
  repeat match goal with
  | _ => progress simplify_equality
321
  | H : _ ++ _ = _ ++ _ |- _ => first
322 323 324
    [ apply app_inv_head in H | apply app_inv_tail in H
    | apply app_injective_1 in H; [destruct H|done]
    | apply app_injective_2 in H; [destruct H|done] ]
Robbert Krebbers's avatar
Robbert Krebbers committed
325
  | H : [?x] !! ?i = Some ?y |- _ =>
326 327 328
    destruct i; [change (Some x = Some y) in H | discriminate]
  end;
  try discriminate_list_equality.
329 330
Ltac simplify_list_equality' :=
  repeat (progress simpl in * || simplify_list_equality).
331

332 333
(** * General theorems *)
Section general_properties.
Robbert Krebbers's avatar
Robbert Krebbers committed
334
Context {A : Type}.
335 336
Implicit Types x y z : A.
Implicit Types l k : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
337

338 339 340
Global Instance: Injective2 (=) (=) (=) (@cons A).
Proof. by injection 1. Qed.
Global Instance:  k, Injective (=) (=) (k ++).
341
Proof. intros ???. apply app_inv_head. Qed.
342
Global Instance:  k, Injective (=) (=) (++ k).
343
Proof. intros ???. apply app_inv_tail. Qed.
344 345 346 347 348 349
Global Instance: Associative (=) (@app A).
Proof. intros ???. apply app_assoc. Qed.
Global Instance: LeftId (=) [] (@app A).
Proof. done. Qed.
Global Instance: RightId (=) [] (@app A).
Proof. intro. apply app_nil_r. Qed.
350

351
Lemma app_nil l1 l2 : l1 ++ l2 = []  l1 = []  l2 = [].
352
Proof. split. apply app_eq_nil. by intros [-> ->]. Qed.
353 354
Lemma app_singleton l1 l2 x :
  l1 ++ l2 = [x]  l1 = []  l2 = [x]  l1 = [x]  l2 = [].
355
Proof. split. apply app_eq_unit. by intros [[-> ->]|[-> ->]]. Qed.
356 357 358
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.
359 360
Proof.
  revert l2. induction l1; intros [|??] H.
361
  * done.
362 363
  * discriminate (H 0).
  * discriminate (H 0).
364
  * f_equal; [by injection (H 0)|]. apply (IHl1 _ $ λ i, H (S i)).
365
Qed.
366
Global Instance list_eq_dec {dec :  x y, Decision (x = y)} :  l k,
367
  Decision (l = k) := list_eq_dec dec.
368 369 370 371 372 373 374 375
Global Instance list_eq_nil_dec l : Decision (l = []).
Proof. by refine match l with [] => left _ | _ => right _ end. Defined.
Lemma list_singleton_reflect l :
  option_reflect (λ x, l = [x]) (length l  1) (list_singleton l).
Proof. by destruct l as [|? []]; constructor. Defined.

Definition nil_length : length (@nil A) = 0 := eq_refl.
Definition cons_length x l : length (x :: l) = S (length l) := eq_refl.
376
Lemma nil_or_length_pos l : l = []  length l  0.
377
Proof. destruct l; simpl; auto with lia. Qed.
378
Lemma nil_length_inv l : length l = 0  l = [].
379 380
Proof. by destruct l. Qed.
Lemma lookup_nil i : @nil A !! i = None.
381
Proof. by destruct i. Qed.
382
Lemma lookup_tail l i : tail l !! i = l !! S i.
383
Proof. by destruct l. Qed.
384 385
Lemma lookup_lt_Some l i x : l !! i = Some x  i < length l.
Proof.
386
  revert i. induction l; intros [|?] ?; simplify_equality'; auto with arith.
387 388 389 390 391
Qed.
Lemma lookup_lt_is_Some_1 l i : is_Some (l !! i)  i < length l.
Proof. intros [??]; eauto using lookup_lt_Some. Qed.
Lemma lookup_lt_is_Some_2 l i : i < length l  is_Some (l !! i).
Proof.
392
  revert i. induction l; intros [|?] ?; simplify_equality'; eauto with lia.
393 394 395 396 397 398 399 400 401 402
Qed.
Lemma lookup_lt_is_Some l i : is_Some (l !! i)  i < length l.
Proof. split; auto using lookup_lt_is_Some_1, lookup_lt_is_Some_2. Qed.
Lemma lookup_ge_None l i : l !! i = None  length l  i.
Proof. rewrite eq_None_not_Some, lookup_lt_is_Some. lia. Qed.
Lemma lookup_ge_None_1 l i : l !! i = None  length l  i.
Proof. by rewrite lookup_ge_None. Qed.
Lemma lookup_ge_None_2 l i : length l  i  l !! i = None.
Proof. by rewrite lookup_ge_None. Qed.
Lemma list_eq_length l1 l2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
403
  length l2 = length l1 
404
  ( i x y, l1 !! i = Some x  l2 !! i = Some y  x = y)  l1 = l2.
Robbert Krebbers's avatar
Robbert Krebbers committed
405
Proof.
406 407 408
  intros Hl ?; apply list_eq; intros i. destruct (l2 !! i) as [x|] eqn:Hx.
  * destruct (lookup_lt_is_Some_2 l1 i) as [y ?]; [|naive_solver].
    rewrite <-Hl. eauto using lookup_lt_Some.
409
  * by rewrite lookup_ge_None, <-Hl, <-lookup_ge_None.
Robbert Krebbers's avatar
Robbert Krebbers committed
410
Qed.
411
Lemma lookup_app_l l1 l2 i : i < length l1  (l1 ++ l2) !! i = l1 !! i.
412
Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed.
413 414
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.
415
Lemma lookup_app_r l1 l2 i : (l1 ++ l2) !! (length l1 + i) = l2 !! i.
416 417 418 419
Proof. revert i. induction l1; intros [|i]; simplify_equality'; auto. Qed.
Lemma lookup_app_r_alt l1 l2 i j :
  j = length l1  (l1 ++ l2) !! (j + i) = l2 !! i.
Proof. intros ->. by apply lookup_app_r. Qed.
420 421
Lemma lookup_app_r_Some l1 l2 i x :
  l2 !! i = Some x  (l1 ++ l2) !! (length l1 + i) = Some x.
422
Proof. by rewrite lookup_app_r. Qed.
423 424 425
Lemma lookup_app_minus_r l1 l2 i :
  length l1  i  (l1 ++ l2) !! i = l2 !! (i - length l1).
Proof. intros. rewrite <-(lookup_app_r l1 l2). f_equal. lia. Qed.
426 427
Lemma lookup_app_inv l1 l2 i x :
  (l1 ++ l2) !! i = Some x  l1 !! i = Some x  l2 !! (i - length l1) = Some x.
428
Proof. revert i. induction l1; intros [|i] ?; simplify_equality'; auto. Qed.
429 430 431
Lemma list_lookup_middle l1 l2 x n :
  n = length l1  (l1 ++ x :: l2) !! n = Some x.
Proof. intros ->. by induction l1. Qed.
432

433
Lemma alter_length f l i : length (alter f i l) = length l.
434
Proof. revert i. by induction l; intros [|?]; f_equal'. Qed.
435
Lemma insert_length l i x : length (<[i:=x]>l) = length l.
436
Proof. revert i. by induction l; intros [|?]; f_equal'. Qed.
437
Lemma list_lookup_alter f l i : alter f i l !! i = f <$> l !! i.
438
Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
439
Lemma list_lookup_alter_ne f l i j : i  j  alter f i l !! j = l !! j.
440
Proof.
441
  revert i j. induction l; [done|]. intros [] [] ?; simpl; auto with congruence.
442
Qed.
443
Lemma list_lookup_insert l i x : i < length l  <[i:=x]>l !! i = Some x.
444 445
Proof. revert i. induction l; intros [|?] ?; f_equal'; auto with lia. Qed.
Lemma list_lookup_insert_ne l i j x : i  j  <[i:=x]>l !! j = l !! j.
446
Proof.
447
  revert i j. induction l; [done|]. intros [] [] ?; simpl; auto with congruence.
448
Qed.
449 450
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
451
Proof.
452
  intros. destruct i, l as [|x0 [|x1 l]]; simplify_equality'.
Robbert Krebbers's avatar
Robbert Krebbers committed
453 454 455
  * by exists 1 x1.
  * by exists 0 x0.
Qed.
456 457
Lemma alter_app_l f l1 l2 i :
  i < length l1  alter f i (l1 ++ l2) = alter f i l1 ++ l2.
458
Proof. revert i. induction l1; intros [|?] ?; f_equal'; auto with lia. Qed.
459
Lemma alter_app_r f l1 l2 i :
460
  alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2.
461
Proof. revert i. induction l1; intros [|?]; f_equal'; auto. Qed.
462 463
Lemma alter_app_r_alt f l1 l2 i :
  length l1  i  alter f i (l1 ++ l2) = l1 ++ alter f (i - length l1) l2.
464 465 466 467
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply alter_app_r.
Qed.
468 469 470
Lemma list_alter_ext f g l k i :
  ( x, l !! i = Some x  f x = g x)  l = k  alter f i l = alter g i k.
Proof. intros H ->. revert i H. induction k; intros [|?] ?; f_equal'; auto. Qed.
471 472
Lemma list_alter_compose f g l i :
  alter (f  g) i l = alter f i (alter g i l).
473
Proof. revert i. induction l; intros [|?]; f_equal'; auto. Qed.
474 475
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).
476
Proof. revert i j. induction l; intros [|?][|?] ?; f_equal'; auto with lia. Qed.
477 478
Lemma insert_app_l l1 l2 i x :
  i < length l1  <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2.
479
Proof. revert i. induction l1; intros [|?] ?; f_equal'; auto with lia. Qed.
480
Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2.
481
Proof. revert i. induction l1; intros [|?]; f_equal'; auto. Qed.
482 483
Lemma insert_app_r_alt l1 l2 i x :
  length l1  i  <[i:=x]>(l1 ++ l2) = l1 ++ <[i - length l1:=x]>l2.
484 485 486 487
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply insert_app_r.
Qed.
488
Lemma delete_middle l1 l2 x : delete (length l1) (l1 ++ x :: l2) = l1 ++ l2.
489
Proof. induction l1; f_equal'; auto. Qed.
490

491
(** ** Properties of the [elem_of] predicate *)
492
Lemma not_elem_of_nil x : x  [].
493
Proof. by inversion 1. Qed.
494
Lemma elem_of_nil x : x  []  False.
495
Proof. intuition. by destruct (not_elem_of_nil x). Qed.
496
Lemma elem_of_nil_inv l : ( x, x  l)  l = [].
497
Proof. destruct l. done. by edestruct 1; constructor. Qed.
498
Lemma elem_of_cons l x y : x  y :: l  x = y  x  l.
499
Proof. split; [inversion 1; subst|intros [->|?]]; constructor (done). Qed.
500
Lemma not_elem_of_cons l x y : x  y :: l  x  y  x  l.
Robbert Krebbers's avatar
Robbert Krebbers committed
501
Proof. rewrite elem_of_cons. tauto. Qed.
502
Lemma elem_of_app l1 l2 x : x  l1 ++ l2  x  l1  x  l2.
503
Proof.
504
  induction l1.
505
  * split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x).
506
  * simpl. rewrite !elem_of_cons, IHl1. tauto.
507
Qed.
508
Lemma not_elem_of_app l1 l2 x : x  l1 ++ l2  x  l1  x  l2.
Robbert Krebbers's avatar
Robbert Krebbers committed
509
Proof. rewrite elem_of_app. tauto. Qed.
510
Lemma elem_of_list_singleton x y : x  [y]  x = y.
511
Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
512
Global Instance elem_of_list_permutation_proper x : Proper (() ==> iff) (x ).
513
Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed.
514
Lemma elem_of_list_split l x : x  l   l1 l2, l = l1 ++ x :: l2.
515
Proof.
516 517
  induction 1 as [x l|x y l ? [l1 [l2 ->]]]; [by eexists [], l|].
  by exists (y :: l1) l2.
518
Qed.
519
Lemma elem_of_list_lookup_1 l x : x  l   i, l !! i = Some x.
520
Proof.
521 522
  induction 1 as [|???? IH]; [by exists 0 |].
  destruct IH as [i ?]; auto. by exists (S i).
523
Qed.
524
Lemma elem_of_list_lookup_2 l i x : l !! i = Some x  x  l.
525
Proof.
526
  revert i. induction l; intros [|i] ?; simplify_equality'; constructor; eauto.
527
Qed.
528 529 530 531 532 533
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.

(** ** Set operations on lists *)
Section list_set.
  Context {dec :  x y, Decision (x = y)}.
534
  Lemma elem_of_list_difference l k x : x  list_difference l k  x  l  x  k.
535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563
  Proof.
    split; induction l; simpl; try case_decide;
      rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence.
  Qed.
  Lemma list_difference_nodup l k : NoDup l  NoDup (list_difference l k).
  Proof.
    induction 1; simpl; try case_decide.
    * constructor.
    * done.
    * constructor. rewrite elem_of_list_difference; intuition. done.
  Qed.
  Lemma elem_of_list_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 list_intersection_nodup l k : NoDup l  NoDup (list_intersection l k).
  Proof.
    induction 1; simpl; try case_decide.
    * constructor.
    * constructor. rewrite elem_of_list_intersection; intuition. done.
    * done.
  Qed.
  Lemma elem_of_list_intersection_with f l k x :
    x  list_intersection_with f l k   x1 x2,
      x1  l  x2  k  f x1 x2 = Some x.
  Proof.
    split.
564 565 566 567 568 569 570 571
    * induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
      intros Hx. setoid_rewrite elem_of_cons.
      cut (( x2, x2  k  f x1 x2 = Some x)
         x  list_intersection_with f l k); [naive_solver|].
      clear IH. revert Hx. generalize (list_intersection_with f l k).
      induction k; simpl; [by auto|].
      case_match; setoid_rewrite elem_of_cons; naive_solver.
    * intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
572 573 574 575 576
      + 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).
577
        induction k; simpl; intros; [done|].
578 579 580
        case_match; simpl; rewrite ?elem_of_cons; auto.
  Qed.
End list_set.
Robbert Krebbers's avatar
Robbert Krebbers committed
581

582
(** ** Properties of the [NoDup] predicate *)
583 584
Lemma NoDup_nil : NoDup (@nil A)  True.
Proof. split; constructor. Qed.
585
Lemma NoDup_cons x l : NoDup (x :: l)  x  l  NoDup l.
586
Proof. split. by inversion 1. intros [??]. by constructor. Qed.
587
Lemma NoDup_cons_11 x l : NoDup (x :: l)  x  l.
588
Proof. rewrite NoDup_cons. by intros [??]. Qed.
589
Lemma NoDup_cons_12 x l : NoDup (x :: l)  NoDup l.
590
Proof. rewrite NoDup_cons. by intros [??]. Qed.
591
Lemma NoDup_singleton x : NoDup [x].
592
Proof. constructor. apply not_elem_of_nil. constructor. Qed.
593
Lemma NoDup_app l k : NoDup (l ++ k)  NoDup l  ( x, x  l  x  k)  NoDup k.
Robbert Krebbers's avatar
Robbert Krebbers committed
594
Proof.
595
  induction l; simpl.
596
  * rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver.
597
  * rewrite !NoDup_cons.
Robbert Krebbers's avatar
Robbert Krebbers committed
598
    setoid_rewrite elem_of_cons. setoid_rewrite elem_of_app. naive_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
599
Qed.
600
Global Instance NoDup_proper: Proper (() ==> iff) (@NoDup A).
601 602 603 604 605 606 607
Proof.
  induction 1 as [|x l k Hlk IH | |].
  * by rewrite !NoDup_nil.
  * by rewrite !NoDup_cons, IH, Hlk.
  * rewrite !NoDup_cons, !elem_of_cons. intuition.
  * intuition.
Qed.
608 609
Lemma NoDup_lookup l i j x :
  NoDup l  l !! i = Some x  l !! j = Some x  i = j.
610 611 612 613 614 615
Proof.
  intros Hl. revert i j. induction Hl as [|x' l Hx Hl IH].
  { intros; simplify_equality. }
  intros [|i] [|j] ??; simplify_equality'; eauto with f_equal;
    exfalso; eauto using elem_of_list_lookup_2.
Qed.
616 617
Lemma NoDup_alt l :
  NoDup l   i j x, l !! i = Some x  l !! j = Some x  i = j.
618
Proof.
619 620 621 622 623
  split; eauto using NoDup_lookup.
  induction l as [|x l IH]; intros Hl; constructor.
  * rewrite elem_of_list_lookup. intros [i ?].
    by feed pose proof (Hl (S i) 0 x); auto.
  * apply IH. intros i j x' ??. by apply (injective S), (Hl (S i) (S j) x').
624
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
625

626 627 628 629 630 631
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
632
    | x :: l =>
633 634 635 636 637 638 639 640
      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
641
    end.
642
  Lemma elem_of_remove_dups l x : x  remove_dups l  x  l.
643 644 645 646 647 648 649 650 651
  Proof.
    split; induction l; simpl; repeat case_decide;
      rewrite ?elem_of_cons; intuition (simplify_equality; auto).
  Qed.
  Lemma remove_dups_nodup l : NoDup (remove_dups l).
  Proof.
    induction l; simpl; repeat case_decide; try constructor; auto.
    by rewrite elem_of_remove_dups.
  Qed.
652
End no_dup_dec.
653

654
(** ** Properties of the [filter] function *)
655 656 657 658 659 660 661 662 663 664 665 666 667
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 filter_nodup 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.
Robbert Krebbers's avatar
Robbert Krebbers committed
668

669 670 671
(** ** Properties of the [find] function *)
Section find.
  Context (P : A  Prop) `{ x, Decision (P x)}.
672 673
  Lemma list_find_Some l i :
    list_find P l = Some i   x, l !! i = Some x  P x.
674
  Proof.
675
    revert i. induction l; intros [] ?; simplify_option_equality; eauto.
676 677 678
  Qed.
  Lemma list_find_elem_of l x : x  l  P x   i, list_find P l = Some i.
  Proof.
679 680
    induction 1 as [|x y l ? IH]; intros; simplify_option_equality; eauto.
    by destruct IH as [i ->]; [|exists (S i)].
681 682 683 684 685 686 687
  Qed.
End find.

Section find_eq.
  Context `{ x y, Decision (x = y)}.
  Lemma list_find_eq_Some l i x : list_find (x =) l = Some i  l !! i = Some x.
  Proof.
688 689
    intros.
    destruct (list_find_Some (x =) l i) as (?&?&?); auto with congruence.
690 691 692 693 694
  Qed.
  Lemma list_find_eq_elem_of l x : x  l   i, list_find (x=) l = Some i.
  Proof. eauto using list_find_elem_of. Qed.
End find_eq.

695
(** ** Properties of the [reverse] function *)
696 697
Lemma reverse_nil : reverse [] = @nil A.
Proof. done. Qed.
698
Lemma reverse_singleton x : reverse [x] = [x].
699
Proof. done. Qed.
700
Lemma reverse_cons l x : reverse (x :: l) = reverse l ++ [x].
701
Proof. unfold reverse. by rewrite <-!rev_alt. Qed.
702
Lemma reverse_snoc l x : reverse (l ++ [x]) = x :: reverse l.
703
Proof. unfold reverse. by rewrite <-!rev_alt, rev_unit. Qed.
704
Lemma reverse_app l1 l2 : reverse (l1 ++ l2) = reverse l2 ++ reverse l1.
705
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_app_distr. Qed.
706
Lemma reverse_length l : length (reverse l) = length l.
707
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_length. Qed.
708
Lemma reverse_involutive l : reverse (reverse l) = l.
709
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_involutive. Qed.
710 711 712 713 714 715 716 717 718 719 720 721 722 723 724
Lemma elem_of_reverse_2 x l : x  l  x  reverse l.
Proof.
  induction 1; rewrite reverse_cons, elem_of_app,
    ?elem_of_list_singleton; intuition.
Qed.
Lemma elem_of_reverse x l : x  reverse l  x  l.
Proof.
  split; auto using elem_of_reverse_2.
  intros. rewrite <-(reverse_involutive l). by apply elem_of_reverse_2.
Qed.
Global Instance: Injective (=) (=) (@reverse A).
Proof.
  intros l1 l2 Hl.
  by rewrite <-(reverse_involutive l1), <-(reverse_involutive l2), Hl.
Qed.