list.v 153 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
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. *)
5
6
From Coq Require Export Permutation.
From prelude Require Export numbers base decidable option.
Robbert Krebbers's avatar
Robbert Krebbers committed
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

Arguments length {_} _.
Arguments cons {_} _ _.
Arguments app {_} _ _.
Arguments Permutation {_} _ _.
Arguments Forall_cons {_} _ _ _ _ _.

Notation tail := tl.
Notation take := firstn.
Notation drop := skipn.

Arguments take {_} !_ !_ /.
Arguments drop {_} !_ !_ /.

Notation "(::)" := cons (only parsing) : C_scope.
Notation "( x ::)" := (cons x) (only parsing) : C_scope.
Notation "(:: l )" := (λ x, cons x l) (only parsing) : C_scope.
Notation "(++)" := app (only parsing) : C_scope.
Notation "( l ++)" := (app l) (only parsing) : C_scope.
Notation "(++ k )" := (λ l, app l k) (only parsing) : C_scope.

Infix "≡ₚ" := Permutation (at level 70, no associativity) : C_scope.
Notation "(≡ₚ)" := Permutation (only parsing) : C_scope.
Notation "( x ≡ₚ)" := (Permutation x) (only parsing) : C_scope.
Notation "(≡ₚ x )" := (λ y, y  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.

Robbert Krebbers's avatar
Robbert Krebbers committed
37
38
39
Instance maybe_cons {A} : Maybe2 (@cons A) := λ l,
  match l with x :: l => Some (x,l) | _ => None end.

Robbert Krebbers's avatar
Robbert Krebbers committed
40
(** * Definitions *)
41
42
43
44
45
46
(** Setoid equality lifted to lists *)
Inductive list_equiv `{Equiv A} : Equiv (list A) :=
  | nil_equiv : []  []
  | cons_equiv x y l k : x  y  l  k  x :: l  y :: k.
Existing Instance list_equiv.

Robbert Krebbers's avatar
Robbert Krebbers committed
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
(** The operation [l !! i] gives the [i]th element of the list [l], or [None]
in case [i] is out of bounds. *)
Instance list_lookup {A} : Lookup nat A (list A) :=
  fix go i l {struct l} : option A := let _ : Lookup _ _ _ := @go in
  match l with
  | [] => None | x :: l => match i with 0 => Some x | S i => l !! i end
  end.

(** The operation [alter f i l] applies the function [f] to the [i]th element
of [l]. In case [i] is out of bounds, the list is returned unchanged. *)
Instance list_alter {A} : Alter nat A (list A) := λ f,
  fix go i l {struct l} :=
  match l with
  | [] => []
  | x :: l => match i with 0 => f x :: l | S i => x :: go i l end
  end.

(** The operation [<[i:=x]> l] overwrites the element at position [i] with the
value [x]. In case [i] is out of bounds, the list is returned unchanged. *)
Instance list_insert {A} : Insert nat A (list A) :=
  fix go i y l {struct l} := let _ : Insert _ _ _ := @go in
  match l with
  | [] => []
  | x :: l => match i with 0 => y :: l | S i => x :: <[i:=y]>l end
  end.
Fixpoint list_inserts {A} (i : nat) (k l : list A) : list A :=
  match k with
  | [] => l
  | y :: k => <[i:=y]>(list_inserts (S i) k l)
  end.

(** The operation [delete i l] removes the [i]th element of [l] and moves
all consecutive elements one position ahead. In case [i] is out of bounds,
the list is returned unchanged. *)
Instance list_delete {A} : Delete nat (list A) :=
  fix go (i : nat) (l : list A) {struct l} : list A :=
  match l with
  | [] => []
  | x :: l => match i with 0 => l | S i => x :: @delete _ _ go i l end
  end.

(** The function [option_list o] converts an element [Some x] into the
singleton list [[x]], and [None] into the empty list [[]]. *)
Definition option_list {A} : option A  list A := option_rect _ (λ x, [x]) [].
Definition list_singleton {A} (l : list A) : option A :=
  match l with [x] => Some x | _ => None end.

(** The function [filter P l] returns the list of elements of [l] that
satisfies [P]. The order remains unchanged. *)
Instance list_filter {A} : Filter A (list A) :=
  fix go P _ l := let _ : Filter _ _ := @go in
  match l with
  | [] => []
  | x :: l => if decide (P x) then x :: filter P l else filter P l
  end.

(** The function [list_find P l] returns the first index [i] whose element
satisfies the predicate [P]. *)
Definition list_find {A} P `{ x, Decision (P x)} : list A  option (nat * A) :=
  fix go l :=
  match l with
  | [] => None
  | x :: l => if decide (P x) then Some (0,x) else prod_map S id <$> go l
  end.

(** The function [replicate n x] generates a list with length [n] of elements
with value [x]. *)
Fixpoint replicate {A} (n : nat) (x : A) : list A :=
  match n with 0 => [] | S n => x :: replicate n x end.

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

(** The function [last l] returns the last element of the list [l], or [None]
if the list [l] is empty. *)
Fixpoint last {A} (l : list A) : option A :=
  match l with [] => None | [x] => Some x | _ :: l => last l end.

(** The function [resize n y l] takes the first [n] elements of [l] in case
[length l ≤ n], and otherwise appends elements with value [x] to [l] to obtain
a list of length [n]. *)
Fixpoint resize {A} (n : nat) (y : A) (l : list A) : list A :=
  match l with
  | [] => replicate n y
  | x :: l => match n with 0 => [] | S n => x :: resize n y l end
  end.
Arguments resize {_} !_ _ !_.

(** The function [reshape k l] transforms [l] into a list of lists whose sizes
are specified by [k]. In case [l] is too short, the resulting list will be
padded with empty lists. In case [l] is too long, it will be truncated. *)
Fixpoint reshape {A} (szs : list nat) (l : list A) : list (list A) :=
  match szs with
  | [] => [] | sz :: szs => take sz l :: reshape szs (drop sz l)
  end.

Definition sublist_lookup {A} (i n : nat) (l : list A) : option (list A) :=
  guard (i + n  length l); Some (take n (drop i l)).
Definition sublist_alter {A} (f : list A  list A)
    (i n : nat) (l : list A) : list A :=
  take i l ++ f (take n (drop i l)) ++ drop (i + n) l.

(** Functions to fold over a list. We redefine [foldl] with the arguments in
the same order as in Haskell. *)
Notation foldr := fold_right.
Definition foldl {A B} (f : A  B  A) : A  list B  A :=
  fix go a l := match l with [] => a | x :: l => go (f a x) l end.

(** The monadic operations. *)
Instance list_ret: MRet list := λ A x, x :: @nil A.
Instance list_fmap : FMap list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x :: go l end.
Instance list_omap : OMap list := λ A B f,
  fix go (l : list A) :=
  match l with
  | [] => []
  | x :: l => match f x with Some y => y :: go l | None => go l end
  end.
Instance list_bind : MBind list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x ++ go l end.
Instance list_join: MJoin list :=
  fix go A (ls : list (list A)) : list A :=
  match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end.
Definition mapM `{MBind M, MRet M} {A B} (f : A  M B) : list A  M (list B) :=
  fix go l :=
  match l with [] => mret [] | x :: l => y  f x; k  go l; mret (y :: k) end.

(** We define stronger variants of map and fold that allow the mapped
function to use the index of the elements. *)
Definition imap_go {A B} (f : nat  A  B) : nat  list A  list B :=
  fix go (n : nat) (l : list A) :=
  match l with [] => [] | x :: l => f n x :: go (S n) l end.
Definition imap {A B} (f : nat  A  B) : list A  list B := imap_go f 0.
Definition zipped_map {A B} (f : list A  list A  A  B) :
  list A  list A  list B := fix go l k :=
  match k with [] => [] | x :: k => f l k x :: go (x :: l) k end.

Definition imap2_go {A B C} (f : nat  A  B  C) :
    nat  list A  list B  list C:=
  fix go (n : nat) (l : list A) (k : list B) :=
  match l, k with
  | [], _ |_, [] => [] | x :: l, y :: k => f n x y :: go (S n) l k
  end.
Definition imap2 {A B C} (f : nat  A  B  C) :
  list A  list B  list C := imap2_go f 0.

Inductive zipped_Forall {A} (P : list A  list A  A  Prop) :
    list A  list A  Prop :=
  | zipped_Forall_nil l : zipped_Forall P l []
  | zipped_Forall_cons l k x :
     P l k x  zipped_Forall P (x :: l) k  zipped_Forall P l (x :: k).
Arguments zipped_Forall_nil {_ _} _.
Arguments zipped_Forall_cons {_ _} _ _ _ _ _.

(** The function [mask f βs l] applies the function [f] to elements in [l] at
positions that are [true] in [βs]. *)
Fixpoint mask {A} (f : A  A) (βs : list bool) (l : list A) : list A :=
  match βs, l with
  | β :: βs, x :: l => (if β then f x else x) :: mask f βs l
  | _, _ => l
  end.

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

(** The predicate [suffix_of] holds if the first list is a suffix of the second.
The predicate [prefix_of] holds if the first list is a prefix of the second. *)
Definition suffix_of {A} : relation (list A) := λ l1 l2,  k, l2 = k ++ l1.
Definition prefix_of {A} : relation (list A) := λ l1 l2,  k, l2 = l1 ++ k.
Infix "`suffix_of`" := suffix_of (at level 70) : C_scope.
Infix "`prefix_of`" := prefix_of (at level 70) : C_scope.
223
224
Hint Extern 0 (_ `prefix_of` _) => reflexivity.
Hint Extern 0 (_ `suffix_of` _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251

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.
252
Hint Extern 0 (_ `sublist` _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
253
254
255
256
257
258
259
260
261
262

(** 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.
263
Hint Extern 0 (_ `contains` _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327

Section contains_dec_help.
  Context {A} {dec :  x y : A, Decision (x = y)}.
  Fixpoint list_remove (x : A) (l : list A) : option (list A) :=
    match l with
    | [] => None
    | y :: l => if decide (x = y) then Some l else (y ::) <$> list_remove x l
    end.
  Fixpoint list_remove_list (k : list A) (l : list A) : option (list A) :=
    match k with
    | [] => Some l | x :: k => list_remove x l = list_remove_list k
    end.
End contains_dec_help.

Inductive Forall3 {A B C} (P : A  B  C  Prop) :
     list A  list B  list C  Prop :=
  | Forall3_nil : Forall3 P [] [] []
  | Forall3_cons x y z l k k' :
     P x y z  Forall3 P l k k'  Forall3 P (x :: l) (y :: k) (z :: k').

(** Set operations on lists *)
Section list_set.
  Context {A} {dec :  x y : A, Decision (x = y)}.
  Global Instance elem_of_list_dec {dec :  x y : A, Decision (x = y)}
    (x : A) :  l, Decision (x  l).
  Proof.
   refine (
    fix go l :=
    match l return Decision (x  l) with
    | [] => right _
    | y :: l => cast_if_or (decide (x = y)) (go l)
    end); clear go dec; subst; try (by constructor); abstract by inversion 1.
  Defined.
  Fixpoint remove_dups (l : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x l then remove_dups l else x :: remove_dups l
    end.
  Fixpoint list_difference (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x k
      then list_difference l k else x :: list_difference l k
    end.
  Definition list_union (l k : list A) : list A := list_difference l k ++ k.
  Fixpoint list_intersection (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x k
      then x :: list_intersection l k else list_intersection l k
    end.
  Definition list_intersection_with (f : A  A  option A) :
    list A  list A  list A := fix go l k :=
    match l with
    | [] => []
    | x :: l => foldr (λ y,
        match f x y with None => id | Some z => (z ::) end) (go l k) k
    end.
End list_set.

(** * Basic tactics on lists *)
328
(** The tactic [discriminate_list] discharges a goal if it contains
Robbert Krebbers's avatar
Robbert Krebbers committed
329
330
a list equality involving [(::)] and [(++)] of two lists that have a different
length as one of its hypotheses. *)
331
Tactic Notation "discriminate_list" hyp(H) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
332
333
  apply (f_equal length) in H;
  repeat (csimpl in H || rewrite app_length in H); exfalso; lia.
334
335
Tactic Notation "discriminate_list" :=
  match goal with H : @eq (list _) _ _ |- _ => discriminate_list H end.
Robbert Krebbers's avatar
Robbert Krebbers committed
336

337
(** The tactic [simplify_list_eq] simplifies hypotheses involving
Robbert Krebbers's avatar
Robbert Krebbers committed
338
339
equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies
lookups in singleton lists. *)
340
Lemma app_inj_1 {A} (l1 k1 l2 k2 : list A) :
Robbert Krebbers's avatar
Robbert Krebbers committed
341
342
  length l1 = length k1  l1 ++ l2 = k1 ++ k2  l1 = k1  l2 = k2.
Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed.
343
Lemma app_inj_2 {A} (l1 k1 l2 k2 : list A) :
Robbert Krebbers's avatar
Robbert Krebbers committed
344
345
  length l2 = length k2  l1 ++ l2 = k1 ++ k2  l1 = k1  l2 = k2.
Proof.
346
  intros ? Hl. apply app_inj_1; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
347
348
  apply (f_equal length) in Hl. rewrite !app_length in Hl. lia.
Qed.
349
Ltac simplify_list_eq :=
Robbert Krebbers's avatar
Robbert Krebbers committed
350
  repeat match goal with
351
  | _ => progress simplify_eq/=
Robbert Krebbers's avatar
Robbert Krebbers committed
352
353
  | H : _ ++ _ = _ ++ _ |- _ => first
    [ apply app_inv_head in H | apply app_inv_tail in H
354
355
    | 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
356
357
358
359
360
361
362
363
364
365
  | 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.

366
367
368
369
370
Section setoid.
  Context `{Equiv A} `{!Equivalence (() : relation A)}.
  Global Instance map_equivalence : Equivalence (() : relation (list A)).
  Proof.
    split.
371
372
373
    - intros l; induction l; constructor; auto.
    - induction 1; constructor; auto.
    - intros l1 l2 l3 Hl; revert l3.
374
      induction Hl; inversion_clear 1; constructor; try etrans; eauto.
375
376
377
378
379
380
381
382
383
  Qed.
  Global Instance cons_proper : Proper (() ==> () ==> ()) (@cons A).
  Proof. by constructor. Qed.
  Global Instance app_proper : Proper (() ==> () ==> ()) (@app A).
  Proof.
    induction 1 as [|x y l k ?? IH]; intros ?? Htl; simpl; auto.
    by apply cons_equiv, IH.
  Qed.
  Global Instance list_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (list A).
384
  Proof. induction 1; f_equal; fold_leibniz; auto. Qed.
385
386
End setoid.

387
Global Instance: Inj2 (=) (=) (=) (@cons A).
Robbert Krebbers's avatar
Robbert Krebbers committed
388
Proof. by injection 1. Qed.
389
Global Instance:  k, Inj (=) (=) (k ++).
Robbert Krebbers's avatar
Robbert Krebbers committed
390
Proof. intros ???. apply app_inv_head. Qed.
391
Global Instance:  k, Inj (=) (=) (++ k).
Robbert Krebbers's avatar
Robbert Krebbers committed
392
Proof. intros ???. apply app_inv_tail. Qed.
393
Global Instance: Assoc (=) (@app A).
Robbert Krebbers's avatar
Robbert Krebbers committed
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
Proof. intros ???. apply app_assoc. Qed.
Global Instance: LeftId (=) [] (@app A).
Proof. done. Qed.
Global Instance: RightId (=) [] (@app A).
Proof. intro. apply app_nil_r. Qed.

Lemma app_nil l1 l2 : l1 ++ l2 = []  l1 = []  l2 = [].
Proof. split. apply app_eq_nil. by intros [-> ->]. Qed.
Lemma app_singleton l1 l2 x :
  l1 ++ l2 = [x]  l1 = []  l2 = [x]  l1 = [x]  l2 = [].
Proof. split. apply app_eq_unit. by intros [[-> ->]|[-> ->]]. Qed.
Lemma cons_middle x l1 l2 : l1 ++ x :: l2 = l1 ++ [x] ++ l2.
Proof. done. Qed.
Lemma list_eq l1 l2 : ( i, l1 !! i = l2 !! i)  l1 = l2.
Proof.
  revert l2. induction l1; intros [|??] H.
410
411
412
413
  - done.
  - discriminate (H 0).
  - discriminate (H 0).
  - f_equal; [by injection (H 0)|]. apply (IHl1 _ $ λ i, H (S i)).
Robbert Krebbers's avatar
Robbert Krebbers committed
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
Qed.
Global Instance list_eq_dec {dec :  x y, Decision (x = y)} :  l k,
  Decision (l = k) := list_eq_dec dec.
Global Instance list_eq_nil_dec l : Decision (l = []).
Proof. by refine match l with [] => left _ | _ => right _ end. Defined.
Lemma list_singleton_reflect l :
  option_reflect (λ x, l = [x]) (length l  1) (list_singleton l).
Proof. by destruct l as [|? []]; constructor. Defined.

Definition nil_length : length (@nil A) = 0 := eq_refl.
Definition cons_length x l : length (x :: l) = S (length l) := eq_refl.
Lemma nil_or_length_pos l : l = []  length l  0.
Proof. destruct l; simpl; auto with lia. Qed.
Lemma nil_length_inv l : length l = 0  l = [].
Proof. by destruct l. Qed.
Lemma lookup_nil i : @nil A !! i = None.
Proof. by destruct i. Qed.
Lemma lookup_tail l i : tail l !! i = l !! S i.
Proof. by destruct l. Qed.
Lemma lookup_lt_Some l i x : l !! i = Some x  i < length l.
Proof.
435
  revert i. induction l; intros [|?] ?; simplify_eq/=; auto with arith.
Robbert Krebbers's avatar
Robbert Krebbers committed
436
437
438
439
440
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.
441
  revert i. induction l; intros [|?] ?; simplify_eq/=; eauto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
442
443
444
445
446
447
448
449
450
451
452
453
454
455
Qed.
Lemma lookup_lt_is_Some l i : is_Some (l !! i)  i < length l.
Proof. split; auto using lookup_lt_is_Some_1, lookup_lt_is_Some_2. Qed.
Lemma lookup_ge_None l i : l !! i = None  length l  i.
Proof. rewrite eq_None_not_Some, lookup_lt_is_Some. lia. Qed.
Lemma lookup_ge_None_1 l i : l !! i = None  length l  i.
Proof. by rewrite lookup_ge_None. Qed.
Lemma lookup_ge_None_2 l i : length l  i  l !! i = None.
Proof. by rewrite lookup_ge_None. Qed.
Lemma list_eq_same_length l1 l2 n :
  length l2 = n  length l1 = n 
  ( i x y, i < n  l1 !! i = Some x  l2 !! i = Some y  x = y)  l1 = l2.
Proof.
  intros <- Hlen Hl; apply list_eq; intros i. destruct (l2 !! i) as [x|] eqn:Hx.
456
  - destruct (lookup_lt_is_Some_2 l1 i) as [y Hy].
Robbert Krebbers's avatar
Robbert Krebbers committed
457
458
    { rewrite Hlen; eauto using lookup_lt_Some. }
    rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some.
459
  - by rewrite lookup_ge_None, Hlen, <-lookup_ge_None.
Robbert Krebbers's avatar
Robbert Krebbers committed
460
461
462
463
464
465
466
467
468
469
470
471
472
Qed.
Lemma lookup_app_l l1 l2 i : i < length l1  (l1 ++ l2) !! i = l1 !! i.
Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed.
Lemma lookup_app_l_Some l1 l2 i x : l1 !! i = Some x  (l1 ++ l2) !! i = Some x.
Proof. intros. rewrite lookup_app_l; eauto using lookup_lt_Some. Qed.
Lemma lookup_app_r l1 l2 i :
  length l1  i  (l1 ++ l2) !! i = l2 !! (i - length l1).
Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed.
Lemma lookup_app_Some l1 l2 i x :
  (l1 ++ l2) !! i = Some x 
    l1 !! i = Some x  length l1  i  l2 !! (i - length l1) = Some x.
Proof.
  split.
473
  - revert i. induction l1 as [|y l1 IH]; intros [|i] ?;
474
      simplify_eq/=; auto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
475
    destruct (IH i) as [?|[??]]; auto with lia.
476
  - intros [?|[??]]; auto using lookup_app_l_Some. by rewrite lookup_app_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
477
478
479
480
481
482
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.
483
Proof. by revert i; induction l; intros []; intros; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
484
Lemma alter_length f l i : length (alter f i l) = length l.
485
Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
486
Lemma insert_length l i x : length (<[i:=x]>l) = length l.
487
Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
488
489
490
491
492
493
494
Lemma list_lookup_alter f l i : alter f i l !! i = f <$> l !! i.
Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed.
Lemma list_lookup_alter_ne f l i j : i  j  alter f i l !! j = l !! j.
Proof.
  revert i j. induction l; [done|]. intros [][] ?; csimpl; auto with congruence.
Qed.
Lemma list_lookup_insert l i x : i < length l  <[i:=x]>l !! i = Some x.
495
Proof. revert i. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
496
497
498
499
500
501
502
503
504
505
Lemma list_lookup_insert_ne l i j x : i  j  <[i:=x]>l !! j = l !! j.
Proof.
  revert i j. induction l; [done|]. intros [] [] ?; simpl; auto with congruence.
Qed.
Lemma list_lookup_insert_Some l i x j y :
  <[i:=x]>l !! j = Some y 
    i = j  x = y  j < length l  i  j  l !! j = Some y.
Proof.
  destruct (decide (i = j)) as [->|];
    [split|rewrite list_lookup_insert_ne by done; tauto].
506
  - intros Hy. assert (j < length l).
Robbert Krebbers's avatar
Robbert Krebbers committed
507
508
    { rewrite <-(insert_length l j x); eauto using lookup_lt_Some. }
    rewrite list_lookup_insert in Hy by done; naive_solver.
509
  - intros [(?&?&?)|[??]]; rewrite ?list_lookup_insert; naive_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
510
511
512
Qed.
Lemma list_insert_commute l i j x y :
  i  j  <[i:=x]>(<[j:=y]>l) = <[j:=y]>(<[i:=x]>l).
513
Proof. revert i j. by induction l; intros [|?] [|?] ?; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
514
515
516
Lemma list_lookup_other l i x :
  length l  1  l !! i = Some x   j y, j  i  l !! j = Some y.
Proof.
517
  intros. destruct i, l as [|x0 [|x1 l]]; simplify_eq/=.
518
519
  - by exists 1, x1.
  - by exists 0, x0.
Robbert Krebbers's avatar
Robbert Krebbers committed
520
521
522
Qed.
Lemma alter_app_l f l1 l2 i :
  i < length l1  alter f i (l1 ++ l2) = alter f i l1 ++ l2.
523
Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
524
525
Lemma alter_app_r f l1 l2 i :
  alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2.
526
Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
527
528
529
530
531
532
533
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.
534
Proof. intros ?. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
535
536
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.
537
Proof. intros H ->. revert i H. induction k; intros [|?] ?; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
538
539
Lemma list_alter_compose f g l i :
  alter (f  g) i l = alter f i (alter g i l).
540
Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
541
542
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).
543
Proof. revert i j. induction l; intros [|?][|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
544
545
Lemma insert_app_l l1 l2 i x :
  i < length l1  <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2.
546
Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
547
Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2.
548
Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
549
550
551
552
553
554
555
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.
556
Proof. induction l1; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594

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.
595
  - intros Hy. assert (j < length l).
Robbert Krebbers's avatar
Robbert Krebbers committed
596
597
    { rewrite <-(inserts_length l i k); eauto using lookup_lt_Some. }
    rewrite list_lookup_inserts in Hy by lia. intuition lia.
598
  - intuition. by rewrite list_lookup_inserts by lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
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.
623
624
  - split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x).
  - simpl. rewrite !elem_of_cons, IHl1. tauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
625
626
627
628
629
630
631
632
633
634
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 (() ==> 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|].
635
  by exists (y :: l1), l2.
Robbert Krebbers's avatar
Robbert Krebbers committed
636
637
638
639
640
641
642
643
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.
644
  revert i. induction l; intros [|i] ?; simplify_eq/=; constructor; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
645
646
647
648
649
650
651
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.
652
  - induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst;
Robbert Krebbers's avatar
Robbert Krebbers committed
653
      setoid_rewrite elem_of_cons; naive_solver.
654
  - intros (x&Hx&?). by induction Hx; csimpl; repeat case_match;
655
      simplify_eq; try constructor; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
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.
672
673
  - rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver.
  - rewrite !NoDup_cons.
Robbert Krebbers's avatar
Robbert Krebbers committed
674
675
676
677
678
    setoid_rewrite elem_of_cons. setoid_rewrite elem_of_app. naive_solver.
Qed.
Global Instance NoDup_proper: Proper (() ==> iff) (@NoDup A).
Proof.
  induction 1 as [|x l k Hlk IH | |].
679
680
681
682
  - by rewrite !NoDup_nil.
  - by rewrite !NoDup_cons, IH, Hlk.
  - rewrite !NoDup_cons, !elem_of_cons. intuition.
  - intuition.
Robbert Krebbers's avatar
Robbert Krebbers committed
683
684
685
686
687
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].
688
689
  { intros; simplify_eq. }
  intros [|i] [|j] ??; simplify_eq/=; eauto with f_equal;
Robbert Krebbers's avatar
Robbert Krebbers committed
690
691
692
693
694
695
696
    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.
697
  - rewrite elem_of_list_lookup. intros [i ?].
Robbert Krebbers's avatar
Robbert Krebbers committed
698
    by feed pose proof (Hl (S i) 0 x); auto.
699
  - apply IH. intros i j x' ??. by apply (inj S), (Hl (S i) (S j) x').
Robbert Krebbers's avatar
Robbert Krebbers committed
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
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;
721
      rewrite ?elem_of_cons; intuition (simplify_eq; auto).
Robbert Krebbers's avatar
Robbert Krebbers committed
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
  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.
741
742
743
    - constructor.
    - done.
    - constructor. rewrite elem_of_list_difference; intuition. done.
Robbert Krebbers's avatar
Robbert Krebbers committed
744
745
746
747
748
749
750
751
752
  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.
753
754
755
    - by apply NoDup_list_difference.
    - intro. rewrite elem_of_list_difference. intuition.
    - done.
Robbert Krebbers's avatar
Robbert Krebbers committed
756
757
758
759
760
761
762
763
764
765
  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.
766
767
768
    - constructor.
    - constructor. rewrite elem_of_list_intersection; intuition. done.
    - done.
Robbert Krebbers's avatar
Robbert Krebbers committed
769
770
771
772
773
774
  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.
775
    - induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
Robbert Krebbers's avatar
Robbert Krebbers committed
776
777
778
779
780
781
      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.
782
    - intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
      + generalize (list_intersection_with f l k).
        induction Hx2; simpl; [by rewrite Hx; left |].
        case_match; simpl; try setoid_rewrite elem_of_cons; auto.
      + generalize (IH Hx). clear Hx IH Hx2.
        generalize (list_intersection_with f l k).
        induction k; simpl; intros; [done|].
        case_match; simpl; rewrite ?elem_of_cons; auto.
  Qed.
End list_set.

(** ** Properties of the [filter] function *)
Section filter.
  Context (P : A  Prop) `{ x, Decision (P x)}.
  Lemma elem_of_list_filter l x : x  filter P l  P x  x  l.
  Proof.
    unfold filter. induction l; simpl; repeat case_decide;
       rewrite ?elem_of_nil, ?elem_of_cons; naive_solver.
  Qed.
  Lemma NoDup_filter l : NoDup l  NoDup (filter P l).
  Proof.
    unfold filter. induction 1; simpl; repeat case_decide;
      rewrite ?NoDup_nil, ?NoDup_cons, ?elem_of_list_filter; tauto.
  Qed.
End filter.

(** ** Properties of the [find] function *)
Section find.
  Context (P : A  Prop) `{ x, Decision (P x)}.
  Lemma list_find_Some l i x :
    list_find P l = Some (i,x)  l !! i = Some x  P x.
  Proof.
    revert i; induction l; intros [] ?;
      repeat (match goal with x : prod _ _ |- _ => destruct x end
816
              || simplify_option_eq); eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
817
818
819
  Qed.
  Lemma list_find_elem_of l x : x  l  P x  is_Some (list_find P l).
  Proof.
820
    induction 1 as [|x y l ? IH]; intros; simplify_option_eq; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
    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.
850
Global Instance: Inj (=) (=) (@reverse A).
Robbert Krebbers's avatar
Robbert Krebbers committed
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
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.
877
  revert i x. induction l; intros [|?] ??; simplify_eq/=; f_equal; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
878
879
880
881
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.
882
Proof. induction l; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
883
884
885
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.
886
Proof. intros ->. by rewrite <-(assoc_L (++)), take_app. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
887
Lemma take_app_le l k n : n  length l  take n (l ++ k) = take n l.
888
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
889
890
Lemma take_plus_app l k n m :
  length l = n  take (n + m) (l ++ k) = l ++ take m k.
891
Proof. intros <-. induction l; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
892
893
Lemma take_app_ge l k n :
  length l  n  take n (l ++ k) = l ++ take (n - length l) k.
894
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
895
Lemma take_ge l n : length l  n  take n l = l.
896
Proof. revert n. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
897
Lemma take_take l n m : take n (take m l) = take (min n m) l.
898
Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed.
899
Lemma take_idemp l n : take n (take n l) = take n l.
Robbert Krebbers's avatar
Robbert Krebbers committed
900
901
Proof. by rewrite take_take, Min.min_idempotent. Qed.
Lemma take_length l n : length (take n l) = min n (length l).
902
Proof. revert n. induction l; intros [|?]; f_equal/=; done. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
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).
918
919
  - by rewrite !lookup_take_ge.
  - by rewrite !lookup_take, !list_lookup_alter_ne by lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
920
921
922
923
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).
924
925
  - by rewrite !lookup_take_ge.
  - by rewrite !lookup_take, !list_lookup_insert_ne by lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
926
927
928
929
930
931
932
933
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.
934
Proof. revert n. by induction l; intros [|i]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
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.
950
Proof. intros ->. by rewrite <-(assoc_L (++)), drop_app. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
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.
973
Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
974
Lemma take_take_drop l n m : take n l ++ take m (drop n l) = take (n + m) l.
975
Proof. revert n m. induction l; intros [|?] [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
976
977
978
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 [|?] [|?] ?;
979
    f_equal/=; auto using take_drop with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
980
981
982
983
984
985
986
987
988
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.
989
990
  - revert i. induction n; intros [|?]; naive_solver auto with lia.
  - intros [-> Hi]. revert i Hi.
Robbert Krebbers's avatar
Robbert Krebbers committed
991
992
993
994
995
996
997
998
999
1000
    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.
1001
1002
  - 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's avatar
Robbert Krebbers committed
1003
1004
Qed.
Lemma insert_replicate x n i : <[i:=x]>(replicate n x) = replicate n x.
1005
Proof. revert i. induction n; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1006
1007
1008
1009
1010
1011
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.
1012
Proof. induction n; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1013
Lemma take_replicate n m x : take n (replicate m x) = replicate (min n m) x.
1014
Proof. revert m. by induction n; intros [|?]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1015
1016
1017
Lemma take_replicate_plus n m x : take n (replicate (n + m) x) = replicate n x.
Proof. by rewrite take_replicate, min_l by lia. Qed.
Lemma drop_replicate n m x : drop n (replicate m x) = replicate (m - n) x.
1018
Proof. revert m. by induction n; intros [|?]; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1019
1020
1021
1022
1023
1024
Lemma drop_replicate_plus n m x : drop n (replicate (n + m) x) = replicate m x.
Proof. rewrite drop_replicate. f_equal. lia. Qed.
Lemma replicate_as_elem_of x n l :
  replicate n x = l  length l = n   y, y  l  y = x.
Proof.
  split; [intros <-; eauto using elem_of_replicate_inv, replicate_length|].
1025
  intros [<- Hl]. symmetry. induction l as [|y l IH]; f_equal/=.
1026
1027
  - apply Hl. by left.
  - apply IH. intros ??. apply Hl. by right.
Robbert Krebbers's avatar
Robbert Krebbers committed
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
Qed.
Lemma reverse_replicate n x : reverse (replicate n x) = replicate n x.
Proof.
  symmetry. apply replicate_as_elem_of.
  rewrite reverse_length, replicate_length. split; auto.
  intros y. rewrite elem_of_reverse. by apply elem_of_replicate_inv.
Qed.
Lemma replicate_false βs n : length βs = n  replicate n false =.>* βs.
Proof. intros <-. by induction βs; simpl; constructor. Qed.

(** ** Properties of the [resize] function *)
Lemma resize_spec l n x : resize n x l = take n l ++ replicate (n - length l) x.
1040
Proof. revert n. induction l; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1041
1042
1043
Lemma resize_0 l x : resize 0 x l = [].
Proof. by destruct l. Qed.
Lemma resize_nil n x : resize n x [] = replicate n x.
1044
Proof. rewrite resize_spec. rewrite take_nil. f_equal/=. lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
Lemma resize_ge l n x :
  length l  n  resize n x l = l ++ replicate (n - length l) x.
Proof. intros. by rewrite resize_spec, take_ge. Qed.
Lemma resize_le l n x : n  length l  resize n x l = take n l.
Proof.
  intros. rewrite resize_spec, (proj2 (Nat.sub_0_le _ _)) by done.
  simpl. by rewrite (right_id_L [] (++)).
Qed.
Lemma resize_all l x : resize (length l) x l = l.
Proof. intros. by rewrite resize_le, take_ge. Qed.
Lemma resize_all_alt l n x : n = length l  resize n x l = l.
Proof. intros ->. by rewrite resize_all. Qed.
Lemma resize_plus l n m x :
  resize (n + m) x l = resize n x l ++ resize m x (drop n l).
Proof.
1060
  revert n m. induction l; intros [|?] [|?]; f_equal/=; auto.
1061
1062
  - by rewrite Nat.add_0_r, (right_id_L [] (++)).
  - by rewrite replicate_plus.
Robbert Krebbers's avatar
Robbert Krebbers committed
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
Qed.
Lemma resize_plus_eq l n m x :
  length l = n  resize (n + m) x l = l ++ replicate m x.
Proof. intros <-. by rewrite resize_plus, resize_all, drop_all, resize_nil. Qed.
Lemma resize_app_le l1 l2 n x :
  n  length l1  resize n x (l1 ++ l2) = resize n x l1.
Proof.
  intros. by rewrite !resize_le, take_app_le by (rewrite ?app_length; lia).
Qed.
Lemma resize_app l1 l2 n x : n = length l1  resize n x (l1 ++ l2) = l1.
Proof. intros ->. by rewrite resize_app_le, resize_all. Qed.
Lemma resize_app_ge l1 l2 n x :
  length l1  n  resize n x (l1 ++ l2) = l1 ++ resize (n - length l1) x l2.
Proof.
1077
  intros. rewrite !resize_spec, take_app_ge, (assoc_L (++)) by done.
Robbert Krebbers's avatar
Robbert Krebbers committed
1078
1079
1080
1081
1082
  do 2 f_equal. rewrite app_length. lia.
Qed.
Lemma resize_length l n x : length (resize n x l) = n.
Proof. rewrite resize_spec, app_length, replicate_length, take_length. lia. Qed.
Lemma resize_replicate x n m : resize n x (replicate m x) = replicate n x.
1083
Proof. revert m. induction n; intros [|?]; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
1084
1085
1086
Lemma resize_resize l n m x : n  m  resize n x (resize m x l) = resize n x l.
Proof.
  revert n m. induction l; simpl.
1087
  - intros. by rewrite !resize_nil, resize_replicate.
1088
  - intros [|?] [|?] ?; f_equal/=; auto with lia.