list.v 141 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. *)
48
49
Instance list_alter {A} : Alter nat A (list A) := λ f,
  fix go i l {struct l} :=
50
51
  match l with
  | [] => []
52
  | x :: l => match i with 0 => f x :: l | S i => x :: go 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

(** The monadic operations. *)
Instance list_ret: MRet list := λ A x, x :: @nil A.
142
143
Instance list_fmap : FMap list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x :: go l end.
144
145
146
147
148
149
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.
150
151
Instance list_bind : MBind list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x ++ go l end.
152
153
Instance list_join: MJoin list :=
  fix go A (ls : list (list A)) : list A :=
154
  match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end.
155
Definition mapM `{MBind M, MRet M} {A B} (f : A  M B) : list A  M (list B) :=
156
  fix go l :=
157
  match l with [] => mret [] | x :: l => y  f x; k  go l; mret (y :: k) end.
158
159
160
161
162

(** 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) :=
163
  match l with [] => [] | x :: l => f n x :: go (S n) l end.
164
Definition imap {A B} (f : nat  A  B) : list A  list B := imap_go f 0.
165
166
167
168
169
170
171
172
173
174
175
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 {_ _} _ _ _ _ _.
176

177
178
179
180
181
182
183
(** 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.
184
185
186
187

(** The function [permutations l] yields all permutations of [l]. *)
Fixpoint interleave {A} (x : A) (l : list A) : list (list A) :=
  match l with
188
  | [] => [[x]]| y :: l => (x :: y :: l) :: ((y ::) <$> interleave x l)
189
190
  end.
Fixpoint permutations {A} (l : list A) : list (list A) :=
191
  match l with [] => [[]] | x :: l => permutations l = interleave x end.
192

193
194
(** 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. *)
195
196
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.
197
198
Infix "`suffix_of`" := suffix_of (at level 70) : C_scope.
Infix "`prefix_of`" := prefix_of (at level 70) : C_scope.
199
200
Hint Extern 0 (?x `prefix_of` ?y) => reflexivity.
Hint Extern 0 (?x `suffix_of` ?y) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
201

202
203
204
205
206
207
208
209
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 =>
210
      if decide_rel (=) x1 x2
211
      then prod_map id (x1 ::) (go l1 l2) else (x1 :: l1, x2 :: l2, [])
212
213
214
215
216
    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.
217
218
  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.
219
End prefix_suffix_ops.
Robbert Krebbers's avatar
Robbert Krebbers committed
220

221
(** A list [l1] is a sublist of [l2] if [l2] is obtained by removing elements
222
223
224
from [l1] without changing the order. *)
Inductive sublist {A} : relation (list A) :=
  | sublist_nil : sublist [] []
225
  | sublist_skip x l1 l2 : sublist l1 l2  sublist (x :: l1) (x :: l2)
226
  | sublist_cons x l1 l2 : sublist l1 l2  sublist l1 (x :: l2).
227
Infix "`sublist`" := sublist (at level 70) : C_scope.
228
Hint Extern 0 (?x `sublist` ?y) => reflexivity.
229
230

(** A list [l2] contains a list [l1] if [l2] is obtained by removing elements
231
from [l1] while possiblity changing the order. *)
232
233
234
235
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)
236
  | contains_cons x l1 l2 : contains l1 l2  contains l1 (x :: l2)
237
238
  | contains_trans l1 l2 l3 : contains l1 l2  contains l2 l3  contains l1 l3.
Infix "`contains`" := contains (at level 70) : C_scope.
239
Hint Extern 0 (?x `contains` ?y) => reflexivity.
240
241
242
243
244
245
246
247
248
249

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
250
    | [] => Some l | x :: k => list_remove x l = list_remove_list k
251
252
    end.
End contains_dec_help.
253

254
255
256
257
258
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').
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283

(** 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
284
      then list_difference l k else x :: list_difference l k
285
    end.
286
  Definition list_union (l k : list A) : list A := list_difference l k ++ k.
287
288
289
290
291
  Fixpoint list_intersection (l k : list A) : list A :=
    match l with
    | [] => []
    | x :: l =>
      if decide_rel () x k
292
      then x :: list_intersection l k else list_intersection l k
293
294
295
296
297
298
299
300
301
    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.
302
303

(** * Basic tactics on lists *)
304
305
306
(** 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. *)
307
308
Tactic Notation "discriminate_list_equality" hyp(H) :=
  apply (f_equal length) in H;
309
  repeat (simpl in H || rewrite app_length in H); exfalso; lia.
310
Tactic Notation "discriminate_list_equality" :=
311
312
313
  match goal with
  | H : @eq (list _) _ _ |- _ => discriminate_list_equality H
  end.
314

315
316
317
(** The tactic [simplify_list_equality] simplifies hypotheses involving
equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies
lookups in singleton lists. *)
318
319
320
321
322
323
324
325
326
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.
327
328
329
Ltac simplify_list_equality :=
  repeat match goal with
  | _ => progress simplify_equality
330
  | H : _ ++ _ = _ ++ _ |- _ => first
331
332
333
    [ 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
334
  | H : [?x] !! ?i = Some ?y |- _ =>
335
336
337
    destruct i; [change (Some x = Some y) in H | discriminate]
  end;
  try discriminate_list_equality.
338
339
Ltac simplify_list_equality' :=
  repeat (progress simpl in * || simplify_list_equality).
340

341
342
(** * General theorems *)
Section general_properties.
Robbert Krebbers's avatar
Robbert Krebbers committed
343
Context {A : Type}.
344
345
Implicit Types x y z : A.
Implicit Types l k : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
346

347
348
349
Global Instance: Injective2 (=) (=) (=) (@cons A).
Proof. by injection 1. Qed.
Global Instance:  k, Injective (=) (=) (k ++).
350
Proof. intros ???. apply app_inv_head. Qed.
351
Global Instance:  k, Injective (=) (=) (++ k).
352
Proof. intros ???. apply app_inv_tail. Qed.
353
354
355
356
357
358
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.
359

360
Lemma app_nil l1 l2 : l1 ++ l2 = []  l1 = []  l2 = [].
361
Proof. split. apply app_eq_nil. by intros [-> ->]. Qed.
362
363
Lemma app_singleton l1 l2 x :
  l1 ++ l2 = [x]  l1 = []  l2 = [x]  l1 = [x]  l2 = [].
364
Proof. split. apply app_eq_unit. by intros [[-> ->]|[-> ->]]. Qed.
365
366
367
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.
368
369
Proof.
  revert l2. induction l1; intros [|??] H.
370
  * done.
371
372
  * discriminate (H 0).
  * discriminate (H 0).
373
  * f_equal; [by injection (H 0)|]. apply (IHl1 _ $ λ i, H (S i)).
374
Qed.
375
Global Instance list_eq_dec {dec :  x y, Decision (x = y)} :  l k,
376
  Decision (l = k) := list_eq_dec dec.
377
378
379
380
381
382
383
384
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.
385
Lemma nil_or_length_pos l : l = []  length l  0.
386
Proof. destruct l; simpl; auto with lia. Qed.
387
Lemma nil_length_inv l : length l = 0  l = [].
388
389
Proof. by destruct l. Qed.
Lemma lookup_nil i : @nil A !! i = None.
390
Proof. by destruct i. Qed.
391
Lemma lookup_tail l i : tail l !! i = l !! S i.
392
Proof. by destruct l. Qed.
393
394
Lemma lookup_lt_Some l i x : l !! i = Some x  i < length l.
Proof.
395
  revert i. induction l; intros [|?] ?; simplify_equality'; auto with arith.
396
397
398
399
400
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.
401
  revert i. induction l; intros [|?] ?; simplify_equality'; eauto with lia.
402
403
404
405
406
407
408
409
410
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.
411
412
413
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
414
Proof.
415
416
417
418
419
  intros <- Hlen Hl; apply list_eq; intros i. destruct (l2 !! i) as [x|] eqn:Hx.
  * destruct (lookup_lt_is_Some_2 l1 i) as [y Hy].
    { rewrite Hlen; eauto using lookup_lt_Some. }
    rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some.
  * by rewrite lookup_ge_None, Hlen, <-lookup_ge_None.
Robbert Krebbers's avatar
Robbert Krebbers committed
420
Qed.
421
Lemma lookup_app_l l1 l2 i : i < length l1  (l1 ++ l2) !! i = l1 !! i.
422
Proof. revert i. induction l1; intros [|?]; simpl; auto with lia. Qed.
423
424
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.
425
Lemma lookup_app_r l1 l2 i : (l1 ++ l2) !! (length l1 + i) = l2 !! i.
426
427
428
429
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.
430
431
Lemma lookup_app_r_Some l1 l2 i x :
  l2 !! i = Some x  (l1 ++ l2) !! (length l1 + i) = Some x.
432
Proof. by rewrite lookup_app_r. Qed.
433
434
435
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.
436
437
Lemma lookup_app_inv l1 l2 i x :
  (l1 ++ l2) !! i = Some x  l1 !! i = Some x  l2 !! (i - length l1) = Some x.
438
Proof. revert i. induction l1; intros [|i] ?; simplify_equality'; auto. Qed.
439
440
441
Lemma list_lookup_middle l1 l2 x n :
  n = length l1  (l1 ++ x :: l2) !! n = Some x.
Proof. intros ->. by induction l1. Qed.
442

443
Lemma alter_length f l i : length (alter f i l) = length l.
444
Proof. revert i. by induction l; intros [|?]; f_equal'. Qed.
445
Lemma insert_length l i x : length (<[i:=x]>l) = length l.
446
Proof. revert i. by induction l; intros [|?]; f_equal'. Qed.
447
Lemma list_lookup_alter f l i : alter f i l !! i = f <$> l !! i.
448
Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
449
Lemma list_lookup_alter_ne f l i j : i  j  alter f i l !! j = l !! j.
450
Proof.
451
  revert i j. induction l; [done|]. intros [][] ?; csimpl; auto with congruence.
452
Qed.
453
Lemma list_lookup_insert l i x : i < length l  <[i:=x]>l !! i = Some x.
454
455
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.
456
Proof.
457
  revert i j. induction l; [done|]. intros [] [] ?; simpl; auto with congruence.
458
Qed.
459
460
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
461
Proof.
462
  intros. destruct i, l as [|x0 [|x1 l]]; simplify_equality'.
Robbert Krebbers's avatar
Robbert Krebbers committed
463
464
465
  * by exists 1 x1.
  * by exists 0 x0.
Qed.
466
467
Lemma alter_app_l f l1 l2 i :
  i < length l1  alter f i (l1 ++ l2) = alter f i l1 ++ l2.
468
Proof. revert i. induction l1; intros [|?] ?; f_equal'; auto with lia. Qed.
469
Lemma alter_app_r f l1 l2 i :
470
  alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2.
471
Proof. revert i. induction l1; intros [|?]; f_equal'; auto. Qed.
472
473
Lemma alter_app_r_alt f l1 l2 i :
  length l1  i  alter f i (l1 ++ l2) = l1 ++ alter f (i - length l1) l2.
474
475
476
477
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply alter_app_r.
Qed.
478
479
480
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.
481
482
Lemma list_alter_compose f g l i :
  alter (f  g) i l = alter f i (alter g i l).
483
Proof. revert i. induction l; intros [|?]; f_equal'; auto. Qed.
484
485
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).
486
Proof. revert i j. induction l; intros [|?][|?] ?; f_equal'; auto with lia. Qed.
487
488
Lemma insert_app_l l1 l2 i x :
  i < length l1  <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2.
489
Proof. revert i. induction l1; intros [|?] ?; f_equal'; auto with lia. Qed.
490
Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2.
491
Proof. revert i. induction l1; intros [|?]; f_equal'; auto. Qed.
492
493
Lemma insert_app_r_alt l1 l2 i x :
  length l1  i  <[i:=x]>(l1 ++ l2) = l1 ++ <[i - length l1:=x]>l2.
494
495
496
497
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply insert_app_r.
Qed.
498
Lemma delete_middle l1 l2 x : delete (length l1) (l1 ++ x :: l2) = l1 ++ l2.
499
Proof. induction l1; f_equal'; auto. Qed.
500

501
(** ** Properties of the [elem_of] predicate *)
502
Lemma not_elem_of_nil x : x  [].
503
Proof. by inversion 1. Qed.
504
Lemma elem_of_nil x : x  []  False.
505
Proof. intuition. by destruct (not_elem_of_nil x). Qed.
506
Lemma elem_of_nil_inv l : ( x, x  l)  l = [].
507
Proof. destruct l. done. by edestruct 1; constructor. Qed.
508
509
Lemma elem_of_not_nil x l : x  l  l  [].
Proof. intros ? ->. by apply (elem_of_nil x). Qed.
510
Lemma elem_of_cons l x y : x  y :: l  x = y  x  l.
511
Proof. split; [inversion 1; subst|intros [->|?]]; constructor (done). Qed.
512
Lemma not_elem_of_cons l x y : x  y :: l  x  y  x  l.
Robbert Krebbers's avatar
Robbert Krebbers committed
513
Proof. rewrite elem_of_cons. tauto. Qed.
514
Lemma elem_of_app l1 l2 x : x  l1 ++ l2  x  l1  x  l2.
515
Proof.
516
  induction l1.
517
  * split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x).
518
  * simpl. rewrite !elem_of_cons, IHl1. tauto.
519
Qed.
520
Lemma not_elem_of_app l1 l2 x : x  l1 ++ l2  x  l1  x  l2.
Robbert Krebbers's avatar
Robbert Krebbers committed
521
Proof. rewrite elem_of_app. tauto. Qed.
522
Lemma elem_of_list_singleton x y : x  [y]  x = y.
523
Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
524
Global Instance elem_of_list_permutation_proper x : Proper (() ==> iff) (x ).
525
Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed.
526
Lemma elem_of_list_split l x : x  l   l1 l2, l = l1 ++ x :: l2.
527
Proof.
528
529
  induction 1 as [x l|x y l ? [l1 [l2 ->]]]; [by eexists [], l|].
  by exists (y :: l1) l2.
530
Qed.
531
Lemma elem_of_list_lookup_1 l x : x  l   i, l !! i = Some x.
532
Proof.
533
534
  induction 1 as [|???? IH]; [by exists 0 |].
  destruct IH as [i ?]; auto. by exists (S i).
535
Qed.
536
Lemma elem_of_list_lookup_2 l i x : l !! i = Some x  x  l.
537
Proof.
538
  revert i. induction l; intros [|i] ?; simplify_equality'; constructor; eauto.
539
Qed.
540
541
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.
542
543
544
545
546
547
548
549
550
Lemma elem_of_list_omap {B} (f : A  option B) l (y : B) :
  y  omap f l   x, x  l  f x = Some y.
Proof.
  split.
  * induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst;
      setoid_rewrite elem_of_cons; naive_solver.
  * intros (x&Hx&?). induction Hx; csimpl; repeat case_match;
      simplify_equality; auto; constructor (by auto).
Qed.
551

552
(** ** Properties of the [NoDup] predicate *)
553
554
Lemma NoDup_nil : NoDup (@nil A)  True.
Proof. split; constructor. Qed.
555
Lemma NoDup_cons x l : NoDup (x :: l)  x  l  NoDup l.
556
Proof. split. by inversion 1. intros [??]. by constructor. Qed.
557
Lemma NoDup_cons_11 x l : NoDup (x :: l)  x  l.
558
Proof. rewrite NoDup_cons. by intros [??]. Qed.
559
Lemma NoDup_cons_12 x l : NoDup (x :: l)  NoDup l.
560
Proof. rewrite NoDup_cons. by intros [??]. Qed.
561
Lemma NoDup_singleton x : NoDup [x].
562
Proof. constructor. apply not_elem_of_nil. constructor. Qed.
563
Lemma NoDup_app l k : NoDup (l ++ k)  NoDup l  ( x, x  l  x  k)  NoDup k.
Robbert Krebbers's avatar
Robbert Krebbers committed
564
Proof.
565
  induction l; simpl.
566
  * rewrite NoDup_nil. setoid_rewrite elem_of_nil. naive_solver.
567
  * rewrite !NoDup_cons.
Robbert Krebbers's avatar
Robbert Krebbers committed
568
    setoid_rewrite elem_of_cons. setoid_rewrite elem_of_app. naive_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
569
Qed.
570
Global Instance NoDup_proper: Proper (() ==> iff) (@NoDup A).
571
572
573
574
575
576
577
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.
578
579
Lemma NoDup_lookup l i j x :
  NoDup l  l !! i = Some x  l !! j = Some x  i = j.
580
581
582
583
584
585
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.
586
587
Lemma NoDup_alt l :
  NoDup l   i j x, l !! i = Some x  l !! j = Some x  i = j.
588
Proof.
589
590
591
592
593
  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').
594
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
595

596
597
598
599
600
601
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
602
    | x :: l =>
603
604
605
606
607
608
609
610
      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
611
    end.
612
  Lemma elem_of_remove_dups l x : x  remove_dups l  x  l.
613
614
615
616
  Proof.
    split; induction l; simpl; repeat case_decide;
      rewrite ?elem_of_cons; intuition (simplify_equality; auto).
  Qed.
617
  Lemma NoDup_remove_dups l : NoDup (remove_dups l).
618
619
620
621
  Proof.
    induction l; simpl; repeat case_decide; try constructor; auto.
    by rewrite elem_of_remove_dups.
  Qed.
622
End no_dup_dec.
623

624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
(** ** Set operations on lists *)
Section list_set.
  Context {dec :  x y, Decision (x = y)}.
  Lemma elem_of_list_difference l k x : x  list_difference l k  x  l  x  k.
  Proof.
    split; induction l; simpl; try case_decide;
      rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence.
  Qed.
  Lemma NoDup_list_difference l k : NoDup l  NoDup (list_difference l k).
  Proof.
    induction 1; simpl; try case_decide.
    * constructor.
    * done.
    * constructor. rewrite elem_of_list_difference; intuition. done.
  Qed.
  Lemma elem_of_list_union l k x : x  list_union l k  x  l  x  k.
  Proof.
    unfold list_union. rewrite elem_of_app, elem_of_list_difference.
    intuition. case (decide (x  k)); intuition.
  Qed.
  Lemma NoDup_list_union l k : NoDup l  NoDup k  NoDup (list_union l k).
  Proof.
    intros. apply NoDup_app. repeat split.
    * by apply NoDup_list_difference.
    * intro. rewrite elem_of_list_difference. intuition.
    * done.
  Qed.
  Lemma elem_of_list_intersection l k x :
    x  list_intersection l k  x  l  x  k.
  Proof.
    split; induction l; simpl; repeat case_decide;
      rewrite ?elem_of_nil, ?elem_of_cons; intuition congruence.
  Qed.
  Lemma NoDup_list_intersection l k : NoDup l  NoDup (list_intersection l k).
  Proof.
    induction 1; simpl; try case_decide.
    * constructor.
    * constructor. rewrite elem_of_list_intersection; intuition. done.
    * done.
  Qed.
  Lemma elem_of_list_intersection_with f l k x :
    x  list_intersection_with f l k   x1 x2,
      x1  l  x2  k  f x1 x2 = Some x.
  Proof.
    split.
    * induction l as [|x1 l IH]; simpl; [by rewrite elem_of_nil|].
      intros Hx. setoid_rewrite elem_of_cons.
      cut (( x2, x2  k  f x1 x2 = Some x)
         x  list_intersection_with f l k); [naive_solver|].
      clear IH. revert Hx. generalize (list_intersection_with f l k).
      induction k; simpl; [by auto|].
      case_match; setoid_rewrite elem_of_cons; naive_solver.
    * intros (x1&x2&Hx1&Hx2&Hx). induction Hx1 as [x1|x1 ? l ? IH]; simpl.
      + generalize (list_intersection_with f l k).
        induction Hx2; simpl; [by rewrite Hx; left |].
        case_match; simpl; try setoid_rewrite elem_of_cons; auto.
      + generalize (IH Hx). clear Hx IH Hx2.
        generalize (list_intersection_with f l k).
        induction k; simpl; intros; [done|].
        case_match; simpl; rewrite ?elem_of_cons; auto.
  Qed.
End list_set.

687
(** ** Properties of the [filter] function *)
688
689
690
691
692
693
694
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.
695
  Lemma NoDup_filter l : NoDup l  NoDup (filter P l).
696
697
698
699
700
  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
701

702
703
704
(** ** Properties of the [find] function *)
Section find.
  Context (P : A  Prop) `{ x, Decision (P x)}.
705
706
  Lemma list_find_Some l i :
    list_find P l = Some i   x, l !! i = Some x  P x.
707
  Proof.
708
    revert i. induction l; intros [] ?; simplify_option_equality; eauto.
709
710
711
  Qed.
  Lemma list_find_elem_of l x : x  l  P x   i, list_find P l = Some i.
  Proof.
712
713
    induction 1 as [|x y l ? IH]; intros; simplify_option_equality; eauto.
    by destruct IH as [i ->]; [|exists (S i)].
714
715
716
717
718
719
720
  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.
721
722
    intros.
    destruct (list_find_Some (x =) l i) as (?&?&?); auto with congruence.
723
724
725
726
727
  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.

728
(** ** Properties of the [reverse] function *)
729
730
Lemma reverse_nil : reverse [] = @nil A.
Proof. done. Qed.
731
Lemma reverse_singleton x : reverse [x] = [x].
732
Proof. done. Qed.
733
Lemma reverse_cons l x : reverse (x :: l) = reverse l ++ [x].
734
Proof. unfold reverse. by rewrite <-!rev_alt. Qed.
735
Lemma reverse_snoc l x : reverse (l ++ [x]) = x :: reverse l.
736
Proof. unfold reverse. by rewrite <-!rev_alt, rev_unit. Qed.
737
Lemma reverse_app l1 l2 : reverse (l1 ++ l2) = reverse l2 ++ reverse l1.
738
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_app_distr. Qed.
739
Lemma reverse_length l : length (reverse l) = length l.
740
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_length. Qed.
741
Lemma reverse_involutive l : reverse (reverse l) = l.
742
Proof. unfold reverse. rewrite <-!rev_alt. apply rev_involutive. Qed.
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
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.
758

759
760
761
(** ** Properties of the [last] function *)
Lemma last_snoc x l : last (l ++ [x]) = Some x.
Proof. induction l as [|? []]; simpl; auto. Qed.
762
763
764
765
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.
766

767
768
769
770
771
772
773
(** ** Properties of the [take] function *)
Definition take_drop i l : take i l ++ drop i l = l := firstn_skipn i l.
Lemma take_drop_middle l i x :
  l !! i = Some x  take i l ++ x :: drop (S i) l = l.
Proof.
  revert i x. induction l; intros [|?] ??; simplify_equality'; f_equal; auto.