list.v 161 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
2
3
4
(* This file is distributed under the terms of the BSD license. *)
(** This file collects general purpose definitions and theorems on lists that
are not in the Coq standard library. *)
5
From Coq Require Export Permutation.
6
From stdpp Require Export numbers base option.
Robbert Krebbers's avatar
Robbert Krebbers committed
7

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

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

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

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

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

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

Robbert Krebbers's avatar
Robbert Krebbers committed
32
33
34
35
36
37
38
Notation "(::)" := cons (only parsing) : C_scope.
Notation "( x ::)" := (cons x) (only parsing) : C_scope.
Notation "(:: l )" := (λ x, cons x l) (only parsing) : C_scope.
Notation "(++)" := app (only parsing) : C_scope.
Notation "( l ++)" := (app l) (only parsing) : C_scope.
Notation "(++ k )" := (λ l, app l k) (only parsing) : C_scope.

39
40
41
42
43
44
45
46
47
Infix "≡ₚ" := Permutation (at level 70, no associativity) : C_scope.
Notation "(≡ₚ)" := Permutation (only parsing) : C_scope.
Notation "( x ≡ₚ)" := (Permutation x) (only parsing) : C_scope.
Notation "(≡ₚ x )" := (λ y, y  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
48
49
50
Instance maybe_cons {A} : Maybe2 (@cons A) := λ l,
  match l with x :: l => Some (x,l) | _ => None end.

51
(** * Definitions *)
52
53
54
55
56
57
(** Setoid equality lifted to lists *)
Inductive list_equiv `{Equiv A} : Equiv (list A) :=
  | nil_equiv : []  []
  | cons_equiv x y l k : x  y  l  k  x :: l  y :: k.
Existing Instance list_equiv.

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

(** The operation [alter f i l] applies the function [f] to the [i]th element
of [l]. In case [i] is out of bounds, the list is returned unchanged. *)
68
Instance list_alter {A} : Alter nat A (list A) := λ f,
69
  fix go i l {struct l} :=
70
71
  match l with
  | [] => []
72
  | x :: l => match i with 0 => f x :: l | S i => x :: go i l end
73
  end.
74

75
76
(** The operation [<[i:=x]> l] overwrites the element at position [i] with the
value [x]. In case [i] is out of bounds, the list is returned unchanged. *)
77
78
Instance list_insert {A} : Insert nat A (list A) :=
  fix go i y l {struct l} := let _ : Insert _ _ _ := @go in
79
80
81
82
  match l with
  | [] => []
  | x :: l => match i with 0 => y :: l | S i => x :: <[i:=y]>l end
  end.
83
84
85
86
87
Fixpoint list_inserts {A} (i : nat) (k l : list A) : list A :=
  match k with
  | [] => l
  | y :: k => <[i:=y]>(list_inserts (S i) k l)
  end.
88
Instance: Params (@list_inserts) 1.
89

90
91
92
(** The operation [delete i l] removes the [i]th element of [l] and moves
all consecutive elements one position ahead. In case [i] is out of bounds,
the list is returned unchanged. *)
93
94
Instance list_delete {A} : Delete nat (list A) :=
  fix go (i : nat) (l : list A) {struct l} : list A :=
95
96
  match l with
  | [] => []
97
  | x :: l => match i with 0 => l | S i => x :: @delete _ _ go i l end
98
  end.
99
100
101

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

(** The function [filter P l] returns the list of elements of [l] that
satisfies [P]. The order remains unchanged. *)
Instance list_filter {A} : Filter A (list A) :=
110
  fix go P _ l := let _ : Filter _ _ := @go in
Robbert Krebbers's avatar
Robbert Krebbers committed
111
112
  match l with
  | [] => []
113
  | x :: l => if decide (P x) then x :: filter P l else filter P l
114
115
116
117
  end.

(** The function [list_find P l] returns the first index [i] whose element
satisfies the predicate [P]. *)
118
Definition list_find {A} P `{ x, Decision (P x)} : list A  option (nat * A) :=
119
120
  fix go l :=
  match l with
121
122
  | [] => None
  | x :: l => if decide (P x) then Some (0,x) else prod_map S id <$> go l
123
  end.
124
Instance: Params (@list_find) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
125
126
127
128

(** The function [replicate n x] generates a list with length [n] of elements
with value [x]. *)
Fixpoint replicate {A} (n : nat) (x : A) : list A :=
129
  match n with 0 => [] | S n => x :: replicate n x end.
130
Instance: Params (@replicate) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
131
132
133

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

136
137
138
139
(** The function [last l] returns the last element of the list [l], or [None]
if the list [l] is empty. *)
Fixpoint last {A} (l : list A) : option A :=
  match l with [] => None | [x] => Some x | _ :: l => last l end.
140
Instance: Params (@last) 1.
141

Robbert Krebbers's avatar
Robbert Krebbers committed
142
143
144
145
146
147
(** The function [resize n y l] takes the first [n] elements of [l] in case
[length l ≤ n], and otherwise appends elements with value [x] to [l] to obtain
a list of length [n]. *)
Fixpoint resize {A} (n : nat) (y : A) (l : list A) : list A :=
  match l with
  | [] => replicate n y
148
  | x :: l => match n with 0 => [] | S n => x :: resize n y l end
Robbert Krebbers's avatar
Robbert Krebbers committed
149
150
  end.
Arguments resize {_} !_ _ !_.
151
Instance: Params (@resize) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
152

153
154
155
(** The function [reshape k l] transforms [l] into a list of lists whose sizes
are specified by [k]. In case [l] is too short, the resulting list will be
padded with empty lists. In case [l] is too long, it will be truncated. *)
156
157
Fixpoint reshape {A} (szs : list nat) (l : list A) : list (list A) :=
  match szs with
158
  | [] => [] | sz :: szs => take sz l :: reshape szs (drop sz l)
159
  end.
160
Instance: Params (@reshape) 2.
161

162
Definition sublist_lookup {A} (i n : nat) (l : list A) : option (list A) :=
163
164
165
166
  guard (i + n  length l); Some (take n (drop i l)).
Definition sublist_alter {A} (f : list A  list A)
    (i n : nat) (l : list A) : list A :=
  take i l ++ f (take n (drop i l)) ++ drop (i + n) l.
167

168
169
170
171
(** Functions to fold over a list. We redefine [foldl] with the arguments in
the same order as in Haskell. *)
Notation foldr := fold_right.
Definition foldl {A B} (f : A  B  A) : A  list B  A :=
172
  fix go a l := match l with [] => a | x :: l => go (f a x) l end.
173
174
175

(** The monadic operations. *)
Instance list_ret: MRet list := λ A x, x :: @nil A.
176
177
Instance list_fmap : FMap list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x :: go l end.
178
179
180
181
182
183
Instance list_omap : OMap list := λ A B f,
  fix go (l : list A) :=
  match l with
  | [] => []
  | x :: l => match f x with Some y => y :: go l | None => go l end
  end.
184
185
Instance list_bind : MBind list := λ A B f,
  fix go (l : list A) := match l with [] => [] | x :: l => f x ++ go l end.
186
187
Instance list_join: MJoin list :=
  fix go A (ls : list (list A)) : list A :=
188
  match ls with [] => [] | l :: ls => l ++ @mjoin _ go _ ls end.
189
Definition mapM `{MBind M, MRet M} {A B} (f : A  M B) : list A  M (list B) :=
190
  fix go l :=
191
  match l with [] => mret [] | x :: l => y  f x; k  go l; mret (y :: k) end.
192
193
194
195
196

(** We define stronger variants of map and fold that allow the mapped
function to use the index of the elements. *)
Definition imap_go {A B} (f : nat  A  B) : nat  list A  list B :=
  fix go (n : nat) (l : list A) :=
197
  match l with [] => [] | x :: l => f n x :: go (S n) l end.
198
Definition imap {A B} (f : nat  A  B) : list A  list B := imap_go f 0.
199
200
Arguments imap : simpl never.

201
202
203
204
Definition zipped_map {A B} (f : list A  list A  A  B) :
  list A  list A  list B := fix go l k :=
  match k with [] => [] | x :: k => f l k x :: go (x :: l) k end.

Robbert Krebbers's avatar
Robbert Krebbers committed
205
206
207
208
209
210
211
212
213
Definition imap2_go {A B C} (f : nat  A  B  C) :
    nat  list A  list B  list C:=
  fix go (n : nat) (l : list A) (k : list B) :=
  match l, k with
  | [], _ |_, [] => [] | x :: l, y :: k => f n x y :: go (S n) l k
  end.
Definition imap2 {A B C} (f : nat  A  B  C) :
  list A  list B  list C := imap2_go f 0.

214
215
216
217
218
219
220
Inductive zipped_Forall {A} (P : list A  list A  A  Prop) :
    list A  list A  Prop :=
  | zipped_Forall_nil l : zipped_Forall P l []
  | zipped_Forall_cons l k x :
     P l k x  zipped_Forall P (x :: l) k  zipped_Forall P l (x :: k).
Arguments zipped_Forall_nil {_ _} _.
Arguments zipped_Forall_cons {_ _} _ _ _ _ _.
221

222
223
224
225
226
227
228
(** The function [mask f βs l] applies the function [f] to elements in [l] at
positions that are [true] in [βs]. *)
Fixpoint mask {A} (f : A  A) (βs : list bool) (l : list A) : list A :=
  match βs, l with
  | β :: βs, x :: l => (if β then f x else x) :: mask f βs l
  | _, _ => l
  end.
229
230
231
232

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

238
239
(** The predicate [suffix_of] holds if the first list is a suffix of the second.
The predicate [prefix_of] holds if the first list is a prefix of the second. *)
240
241
Definition suffix_of {A} : relation (list A) := λ l1 l2,  k, l2 = k ++ l1.
Definition prefix_of {A} : relation (list A) := λ l1 l2,  k, l2 = l1 ++ k.
242
243
Infix "`suffix_of`" := suffix_of (at level 70) : C_scope.
Infix "`prefix_of`" := prefix_of (at level 70) : C_scope.
244
245
Hint Extern 0 (_ `prefix_of` _) => reflexivity.
Hint Extern 0 (_ `suffix_of` _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
246

247
Section prefix_suffix_ops.
248
249
  Context `{EqDecision A}.

250
251
252
253
254
255
  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 =>
256
      if decide_rel (=) x1 x2
257
      then prod_map id (x1 ::) (go l1 l2) else (x1 :: l1, x2 :: l2, [])
258
259
260
261
262
    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.
263
264
  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.
265
End prefix_suffix_ops.
Robbert Krebbers's avatar
Robbert Krebbers committed
266

267
(** A list [l1] is a sublist of [l2] if [l2] is obtained by removing elements
268
269
270
from [l1] without changing the order. *)
Inductive sublist {A} : relation (list A) :=
  | sublist_nil : sublist [] []
271
  | sublist_skip x l1 l2 : sublist l1 l2  sublist (x :: l1) (x :: l2)
272
  | sublist_cons x l1 l2 : sublist l1 l2  sublist l1 (x :: l2).
273
Infix "`sublist`" := sublist (at level 70) : C_scope.
274
Hint Extern 0 (_ `sublist` _) => reflexivity.
275
276

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

Section contains_dec_help.
288
  Context `{EqDecision A}.
289
290
291
292
293
294
295
  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
296
    | [] => Some l | x :: k => list_remove x l = list_remove_list k
297
298
    end.
End contains_dec_help.
299

300
301
302
303
304
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').
305

306
307
(** Set operations on lists *)
Instance list_subseteq {A} : SubsetEq (list A) := λ l1 l2,  x, x  l1  x  l2.
308

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

(** * Basic tactics on lists *)
351
(** The tactic [discriminate_list] discharges a goal if it contains
352
353
a list equality involving [(::)] and [(++)] of two lists that have a different
length as one of its hypotheses. *)
354
Tactic Notation "discriminate_list" hyp(H) :=
355
  apply (f_equal length) in H;
356
  repeat (csimpl in H || rewrite app_length in H); exfalso; lia.
357
358
Tactic Notation "discriminate_list" :=
  match goal with H : @eq (list _) _ _ |- _ => discriminate_list H end.
359

360
(** The tactic [simplify_list_eq] simplifies hypotheses involving
361
362
equalities on lists using injectivity of [(::)] and [(++)]. Also, it simplifies
lookups in singleton lists. *)
363
Lemma app_inj_1 {A} (l1 k1 l2 k2 : list A) :
364
365
  length l1 = length k1  l1 ++ l2 = k1 ++ k2  l1 = k1  l2 = k2.
Proof. revert k1. induction l1; intros [|??]; naive_solver. Qed.
366
Lemma app_inj_2 {A} (l1 k1 l2 k2 : list A) :
367
368
  length l2 = length k2  l1 ++ l2 = k1 ++ k2  l1 = k1  l2 = k2.
Proof.
369
  intros ? Hl. apply app_inj_1; auto.
370
371
  apply (f_equal length) in Hl. rewrite !app_length in Hl. lia.
Qed.
372
Ltac simplify_list_eq :=
373
  repeat match goal with
374
  | _ => progress simplify_eq/=
375
  | H : _ ++ _ = _ ++ _ |- _ => first
376
    [ apply app_inv_head in H | apply app_inv_tail in H
377
378
    | 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
379
  | H : [?x] !! ?i = Some ?y |- _ =>
380
    destruct i; [change (Some x = Some y) in H | discriminate]
381
  end.
382

383
384
(** * General theorems *)
Section general_properties.
Robbert Krebbers's avatar
Robbert Krebbers committed
385
Context {A : Type}.
386
387
Implicit Types x y z : A.
Implicit Types l k : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
388

389
Global Instance: Inj2 (=) (=) (=) (@cons A).
390
Proof. by injection 1. Qed.
391
Global Instance:  k, Inj (=) (=) (k ++).
392
Proof. intros ???. apply app_inv_head. Qed.
393
Global Instance:  k, Inj (=) (=) (++ k).
394
Proof. intros ???. apply app_inv_tail. Qed.
395
Global Instance: Assoc (=) (@app A).
396
397
398
399
400
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.
401

402
Lemma app_nil l1 l2 : l1 ++ l2 = []  l1 = []  l2 = [].
403
Proof. split. apply app_eq_nil. by intros [-> ->]. Qed.
404
405
Lemma app_singleton l1 l2 x :
  l1 ++ l2 = [x]  l1 = []  l2 = [x]  l1 = [x]  l2 = [].
406
Proof. split. apply app_eq_unit. by intros [[-> ->]|[-> ->]]. Qed.
407
408
409
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.
410
Proof.
411
  revert l2. induction l1 as [|x l1 IH]; intros [|y l2] H.
412
413
414
  - done.
  - discriminate (H 0).
  - discriminate (H 0).
415
  - f_equal; [by injection (H 0)|]. apply (IH _ $ λ i, H (S i)).
416
Qed.
417
418
Global Instance list_eq_dec {dec : EqDecision A} : EqDecision (list A) :=
  list_eq_dec dec.
419
420
421
Global Instance list_eq_nil_dec l : Decision (l = []).
Proof. by refine match l with [] => left _ | _ => right _ end. Defined.
Lemma list_singleton_reflect l :
422
  option_reflect (λ x, l = [x]) (length l  1) (maybe (λ x, [x]) l).
423
424
425
426
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.
427
Lemma nil_or_length_pos l : l = []  length l  0.
428
Proof. destruct l; simpl; auto with lia. Qed.
429
Lemma nil_length_inv l : length l = 0  l = [].
430
431
Proof. by destruct l. Qed.
Lemma lookup_nil i : @nil A !! i = None.
432
Proof. by destruct i. Qed.
433
Lemma lookup_tail l i : tail l !! i = l !! S i.
434
Proof. by destruct l. Qed.
435
Lemma lookup_lt_Some l i x : l !! i = Some x  i < length l.
436
Proof. revert i. induction l; intros [|?] ?; naive_solver auto with arith. Qed.
437
438
439
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).
440
Proof. revert i. induction l; intros [|?] ?; naive_solver eauto with lia. Qed.
441
442
443
444
445
446
447
448
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.
449
450
451
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
452
Proof.
453
  intros <- Hlen Hl; apply list_eq; intros i. destruct (l2 !! i) as [x|] eqn:Hx.
454
  - destruct (lookup_lt_is_Some_2 l1 i) as [y Hy].
455
456
    { rewrite Hlen; eauto using lookup_lt_Some. }
    rewrite Hy; f_equal; apply (Hl i); eauto using lookup_lt_Some.
457
  - by rewrite lookup_ge_None, Hlen, <-lookup_ge_None.
Robbert Krebbers's avatar
Robbert Krebbers committed
458
Qed.
459
Lemma lookup_app_l l1 l2 i : i < length l1  (l1 ++ l2) !! i = l1 !! i.
460
Proof. revert i. induction l1; intros [|?]; naive_solver auto with lia. Qed.
461
462
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.
463
Lemma lookup_app_r l1 l2 i :
464
  length l1  i  (l1 ++ l2) !! i = l2 !! (i - length l1).
465
466
467
468
469
470
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.
471
  - revert i. induction l1 as [|y l1 IH]; intros [|i] ?;
472
      simplify_eq/=; auto with lia.
473
    destruct (IH i) as [?|[??]]; auto with lia.
474
  - intros [?|[??]]; auto using lookup_app_l_Some. by rewrite lookup_app_r.
475
Qed.
476
477
478
Lemma list_lookup_middle l1 l2 x n :
  n = length l1  (l1 ++ x :: l2) !! n = Some x.
Proof. intros ->. by induction l1. Qed.
479

Ralf Jung's avatar
Ralf Jung committed
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
Lemma nth_lookup_or_length l i d :
  {l !! i = Some (nth i l d)} + {(length l  i)%nat}.
Proof.
  revert i; induction l; intros i.
  - right. simpl. omega.
  - destruct i; simpl.
    + left. done.
    + destruct (IHl i) as [->|]; [by left|].
      right. omega.
Qed.

Lemma nth_lookup l i d x :
  l !! i = Some x  nth i l d = x.
Proof.
  revert i; induction l; intros i; [done|].
  destruct i; simpl.
  - congruence.
  - apply IHl.
Qed.

500
Lemma list_insert_alter l i x : <[i:=x]>l = alter (λ _, x) i l.
501
Proof. by revert i; induction l; intros []; intros; f_equal/=. Qed.
502
Lemma alter_length f l i : length (alter f i l) = length l.
503
Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
504
Lemma insert_length l i x : length (<[i:=x]>l) = length l.
505
Proof. revert i. by induction l; intros [|?]; f_equal/=. Qed.
506
Lemma list_lookup_alter f l i : alter f i l !! i = f <$> l !! i.
507
Proof. revert i. induction l. done. intros [|i]. done. apply (IHl i). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
508
Lemma list_lookup_alter_ne f l i j : i  j  alter f i l !! j = l !! j.
509
Proof. revert i j. induction l; [done|]. intros [] []; naive_solver. Qed.
510
Lemma list_lookup_insert l i x : i < length l  <[i:=x]>l !! i = Some x.
511
Proof. revert i. induction l; intros [|?] ?; f_equal/=; auto with lia. Qed.
512
Lemma list_lookup_insert_ne l i j x : i  j  <[i:=x]>l !! j = l !! j.
513
Proof. revert i j. induction l; [done|]. intros [] []; naive_solver. Qed.
514
515
516
517
518
519
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].
520
  - intros Hy. assert (j < length l).
521
522
    { rewrite <-(insert_length l j x); eauto using lookup_lt_Some. }
    rewrite list_lookup_insert in Hy by done; naive_solver.
523
  - intros [(?&?&?)|[??]]; rewrite ?list_lookup_insert; naive_solver.
524
525
526
Qed.
Lemma list_insert_commute l i j x y :
  i  j  <[i:=x]>(<[j:=y]>l) = <[j:=y]>(<[i:=x]>l).
527
Proof. revert i j. by induction l; intros [|?] [|?] ?; f_equal/=; auto. Qed.
528
529
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
530
Proof.
531
  intros. destruct i, l as [|x0 [|x1 l]]; simplify_eq/=.
532
533
  - by exists 1, x1.
  - by exists 0, x0.
Robbert Krebbers's avatar
Robbert Krebbers committed
534
Qed.
535
536
Lemma alter_app_l f l1 l2 i :
  i < length l1  alter f i (l1 ++ l2) = alter f i l1 ++ l2.
537
Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
538
Lemma alter_app_r f l1 l2 i :
539
  alter f (length l1 + i) (l1 ++ l2) = l1 ++ alter f i l2.
540
Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
541
542
Lemma alter_app_r_alt f l1 l2 i :
  length l1  i  alter f i (l1 ++ l2) = l1 ++ alter f (i - length l1) l2.
543
544
545
546
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply alter_app_r.
Qed.
547
Lemma list_alter_id f l i : ( x, f x = x)  alter f i l = l.
548
Proof. intros ?. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
549
550
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.
551
Proof. intros H ->. revert i H. induction k; intros [|?] ?; f_equal/=; auto. Qed.
552
553
Lemma list_alter_compose f g l i :
  alter (f  g) i l = alter f i (alter g i l).
554
Proof. revert i. induction l; intros [|?]; f_equal/=; auto. Qed.
555
556
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).
557
Proof. revert i j. induction l; intros [|?][|?] ?; f_equal/=; auto with lia. Qed.
558
559
Lemma insert_app_l l1 l2 i x :
  i < length l1  <[i:=x]>(l1 ++ l2) = <[i:=x]>l1 ++ l2.
560
Proof. revert i. induction l1; intros [|?] ?; f_equal/=; auto with lia. Qed.
561
Lemma insert_app_r l1 l2 i x : <[length l1+i:=x]>(l1 ++ l2) = l1 ++ <[i:=x]>l2.
562
Proof. revert i. induction l1; intros [|?]; f_equal/=; auto. Qed.
563
564
Lemma insert_app_r_alt l1 l2 i x :
  length l1  i  <[i:=x]>(l1 ++ l2) = l1 ++ <[i - length l1:=x]>l2.
565
566
567
568
Proof.
  intros. assert (i = length l1 + (i - length l1)) as Hi by lia.
  rewrite Hi at 1. by apply insert_app_r.
Qed.
569
Lemma delete_middle l1 l2 x : delete (length l1) (l1 ++ x :: l2) = l1 ++ l2.
570
Proof. induction l1; f_equal/=; auto. Qed.
571

572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
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.
609
  - intros Hy. assert (j < length l).
610
611
    { rewrite <-(inserts_length l i k); eauto using lookup_lt_Some. }
    rewrite list_lookup_inserts in Hy by lia. intuition lia.
612
  - intuition. by rewrite list_lookup_inserts by lia.
613
614
615
616
617
618
619
620
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.

621
(** ** Properties of the [elem_of] predicate *)
622
Lemma not_elem_of_nil x : x  [].
623
Proof. by inversion 1. Qed.
624
Lemma elem_of_nil x : x  []  False.
625
Proof. intuition. by destruct (not_elem_of_nil x). Qed.
626
Lemma elem_of_nil_inv l : ( x, x  l)  l = [].
627
Proof. destruct l. done. by edestruct 1; constructor. Qed.
628
629
Lemma elem_of_not_nil x l : x  l  l  [].
Proof. intros ? ->. by apply (elem_of_nil x). Qed.
630
Lemma elem_of_cons l x y : x  y :: l  x = y  x  l.
Robbert Krebbers's avatar
Robbert Krebbers committed
631
Proof. by split; [inversion 1; subst|intros [->|?]]; constructor. Qed.
632
Lemma not_elem_of_cons l x y : x  y :: l  x  y  x  l.
Robbert Krebbers's avatar
Robbert Krebbers committed
633
Proof. rewrite elem_of_cons. tauto. Qed.
634
Lemma elem_of_app l1 l2 x : x  l1 ++ l2  x  l1  x  l2.
635
Proof.
636
  induction l1.
637
638
  - split; [by right|]. intros [Hx|]; [|done]. by destruct (elem_of_nil x).
  - simpl. rewrite !elem_of_cons, IHl1. tauto.
639
Qed.
640
Lemma not_elem_of_app l1 l2 x : x  l1 ++ l2  x  l1  x  l2.
Robbert Krebbers's avatar
Robbert Krebbers committed
641
Proof. rewrite elem_of_app. tauto. Qed.
642
Lemma elem_of_list_singleton x y : x  [y]  x = y.
643
Proof. rewrite elem_of_cons, elem_of_nil. tauto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
644
Global Instance elem_of_list_permutation_proper x : Proper (() ==> iff) (x ).
645
Proof. induction 1; rewrite ?elem_of_nil, ?elem_of_cons; intuition. Qed.
646
Lemma elem_of_list_split l x : x  l   l1 l2, l = l1 ++ x :: l2.
647
Proof.
648
  induction 1 as [x l|x y l ? [l1 [l2 ->]]]; [by eexists [], l|].
649
  by exists (y :: l1), l2.
650
Qed.
651
Lemma elem_of_list_lookup_1 l x : x  l   i, l !! i = Some x.
652
Proof.
653
654
  induction 1 as [|???? IH]; [by exists 0 |].
  destruct IH as [i ?]; auto. by exists (S i).
655
Qed.
656
Lemma elem_of_list_lookup_2 l i x : l !! i = Some x  x  l.
657
Proof.
658
  revert i. induction l; intros [|i] ?; simplify_eq/=; constructor; eauto.
659
Qed.
660
661
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.
662
663
664
665
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.
666
  - induction l as [|x l]; csimpl; repeat case_match; inversion 1; subst;
667
      setoid_rewrite elem_of_cons; naive_solver.
668
  - intros (x&Hx&?). by induction Hx; csimpl; repeat case_match;
669
      simplify_eq; try constructor; auto.
670
Qed.
671

672
(** ** Properties of the [NoDup] predicate *)
673
674
Lemma NoDup_nil : NoDup (@nil A)  True.
Proof. split; constructor. Qed.
675
Lemma NoDup_cons x l : NoDup (x :: l)  x  l  NoDup l.
676
Proof. split. by inversion 1. intros [??]. by constructor. Qed.
677
Lemma NoDup_cons_11 x l : NoDup (x :: l)  x  l.
678
Proof. rewrite NoDup_cons. by intros [??]. Qed.
679
Lemma NoDup_cons_12 x l : NoDup (x :: l)  NoDup l.
680
Proof. rewrite NoDup_cons. by intros [??]. Qed.
681
Lemma NoDup_singleton x : NoDup [x].
682
Proof. constructor. apply not_elem_of_nil. constructor. Qed.
683
Lemma NoDup_app l k : NoDup (l ++ k)  NoDup l  ( x, x  l  x  k)  NoDup k.
Robbert Krebbers's avatar
Robbert Krebbers committed
684
Proof.