aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Sorting/Permutation.v
blob: 301548a1dfc2ac55284fb0811d183d9ba201dd72 (plain)
1
2
3
4
5
6
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
37
38
39
40
41
42
43
44
45
46
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
223
224
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
252
253
254
255
256
257
258
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
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
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(*********************************************************************)
(** * List permutations as a composition of adjacent transpositions  *)
(*********************************************************************)

(* Adapted in May 2006 by Jean-Marc Notin from initial contents by
   Laurent Théry (Huffmann contribution, October 2003) *)

Require Import List Setoid Compare_dec Morphisms FinFun.
Import ListNotations. (* For notations [] and [a;b;c] *)
Set Implicit Arguments.
(* Set Universe Polymorphism. *)

Section Permutation.

Variable A:Type.

Inductive Permutation : list A -> list A -> Prop :=
| perm_nil: Permutation [] []
| perm_skip x l l' : Permutation l l' -> Permutation (x::l) (x::l')
| perm_swap x y l : Permutation (y::x::l) (x::y::l)
| perm_trans l l' l'' :
    Permutation l l' -> Permutation l' l'' -> Permutation l l''.

Local Hint Constructors Permutation.

(** Some facts about [Permutation] *)

Theorem Permutation_nil : forall (l : list A), Permutation [] l -> l = [].
Proof.
  intros l HF.
  remember (@nil A) as m in HF.
  induction HF; discriminate || auto.
Qed.

Theorem Permutation_nil_cons : forall (l : list A) (x : A),
 ~ Permutation nil (x::l).
Proof.
  intros l x HF.
  apply Permutation_nil in HF; discriminate.
Qed.

(** Permutation over lists is a equivalence relation *)

Theorem Permutation_refl : forall l : list A, Permutation l l.
Proof.
  induction l; constructor. exact IHl.
Qed.

Theorem Permutation_sym : forall l l' : list A,
 Permutation l l' -> Permutation l' l.
Proof.
  intros l l' Hperm; induction Hperm; auto.
  apply perm_trans with (l':=l'); assumption.
Qed.

Theorem Permutation_trans : forall l l' l'' : list A,
 Permutation l l' -> Permutation l' l'' -> Permutation l l''.
Proof.
  exact perm_trans.
Qed.

End Permutation.

Hint Resolve Permutation_refl perm_nil perm_skip.

(* These hints do not reduce the size of the problem to solve and they
   must be used with care to avoid combinatoric explosions *)

Local Hint Resolve perm_swap perm_trans.
Local Hint Resolve Permutation_sym Permutation_trans.

(* This provides reflexivity, symmetry and transitivity and rewriting
   on morphims to come *)

Instance Permutation_Equivalence A : Equivalence (@Permutation A) | 10 := {
  Equivalence_Reflexive := @Permutation_refl A ;
  Equivalence_Symmetric := @Permutation_sym A ;
  Equivalence_Transitive := @Permutation_trans A }.

Instance Permutation_cons A :
 Proper (Logic.eq ==> @Permutation A ==> @Permutation A) (@cons A) | 10.
Proof.
  repeat intro; subst; auto using perm_skip.
Qed.

Section Permutation_properties.

Variable A:Type.

Implicit Types a b : A.
Implicit Types l m : list A.

(** Compatibility with others operations on lists *)

Theorem Permutation_in : forall (l l' : list A) (x : A),
 Permutation l l' -> In x l -> In x l'.
Proof.
  intros l l' x Hperm; induction Hperm; simpl; tauto.
Qed.

Global Instance Permutation_in' :
 Proper (Logic.eq ==> @Permutation A ==> iff) (@In A) | 10.
Proof.
  repeat red; intros; subst; eauto using Permutation_in.
Qed.

Lemma Permutation_app_tail : forall (l l' tl : list A),
 Permutation l l' -> Permutation (l++tl) (l'++tl).
Proof.
  intros l l' tl Hperm; induction Hperm as [|x l l'|x y l|l l' l'']; simpl; auto.
  eapply Permutation_trans with (l':=l'++tl); trivial.
Qed.

Lemma Permutation_app_head : forall (l tl tl' : list A),
 Permutation tl tl' -> Permutation (l++tl) (l++tl').
Proof.
  intros l tl tl' Hperm; induction l;
   [trivial | repeat rewrite <- app_comm_cons; constructor; assumption].
Qed.

Theorem Permutation_app : forall (l m l' m' : list A),
 Permutation l l' -> Permutation m m' -> Permutation (l++m) (l'++m').
Proof.
  intros l m l' m' Hpermll' Hpermmm';
   induction Hpermll' as [|x l l'|x y l|l l' l''];
    repeat rewrite <- app_comm_cons; auto.
  apply Permutation_trans with (l' := (x :: y :: l ++ m));
   [idtac | repeat rewrite app_comm_cons; apply Permutation_app_head]; trivial.
  apply Permutation_trans with (l' := (l' ++ m')); try assumption.
  apply Permutation_app_tail; assumption.
Qed.

Global Instance Permutation_app' :
 Proper (@Permutation A ==> @Permutation A ==> @Permutation A) (@app A) | 10.
Proof.
  repeat intro; now apply Permutation_app.
Qed.

Lemma Permutation_add_inside : forall a (l l' tl tl' : list A),
  Permutation l l' -> Permutation tl tl' ->
  Permutation (l ++ a :: tl) (l' ++ a :: tl').
Proof.
  intros; apply Permutation_app; auto.
Qed.

Lemma Permutation_cons_append : forall (l : list A) x,
  Permutation (x :: l) (l ++ x :: nil).
Proof. induction l; intros; auto. simpl. rewrite <- IHl; auto. Qed.
Local Hint Resolve Permutation_cons_append.

Theorem Permutation_app_comm : forall (l l' : list A),
  Permutation (l ++ l') (l' ++ l).
Proof.
  induction l as [|x l]; simpl; intro l'.
  rewrite app_nil_r; trivial. rewrite IHl.
  rewrite app_comm_cons, Permutation_cons_append.
  now rewrite <- app_assoc.
Qed.
Local Hint Resolve Permutation_app_comm.

Theorem Permutation_cons_app : forall (l l1 l2:list A) a,
  Permutation l (l1 ++ l2) -> Permutation (a :: l) (l1 ++ a :: l2).
Proof.
  intros l l1 l2 a H. rewrite H.
  rewrite app_comm_cons, Permutation_cons_append.
  now rewrite <- app_assoc.
Qed.
Local Hint Resolve Permutation_cons_app.

Lemma Permutation_Add a l l' : Add a l l' -> Permutation (a::l) l'.
Proof.
 induction 1; simpl; trivial.
 rewrite perm_swap. now apply perm_skip.
Qed.

Theorem Permutation_middle : forall (l1 l2:list A) a,
  Permutation (a :: l1 ++ l2) (l1 ++ a :: l2).
Proof.
  auto.
Qed.
Local Hint Resolve Permutation_middle.

Theorem Permutation_rev : forall (l : list A), Permutation l (rev l).
Proof.
  induction l as [| x l]; simpl; trivial. now rewrite IHl at 1.
Qed.

Global Instance Permutation_rev' :
 Proper (@Permutation A ==> @Permutation A) (@rev A) | 10.
Proof.
  repeat intro; now rewrite <- 2 Permutation_rev.
Qed.

Theorem Permutation_length : forall (l l' : list A),
 Permutation l l' -> length l = length l'.
Proof.
  intros l l' Hperm; induction Hperm; simpl; auto. now transitivity (length l').
Qed.

Global Instance Permutation_length' :
 Proper (@Permutation A ==> Logic.eq) (@length A) | 10.
Proof.
  exact Permutation_length.
Qed.

Theorem Permutation_ind_bis :
 forall P : list A -> list A -> Prop,
   P [] [] ->
   (forall x l l', Permutation l l' -> P l l' -> P (x :: l) (x :: l')) ->
   (forall x y l l', Permutation l l' -> P l l' -> P (y :: x :: l) (x :: y :: l')) ->
   (forall l l' l'', Permutation l l' -> P l l' -> Permutation l' l'' -> P l' l'' -> P l l'') ->
   forall l l', Permutation l l' -> P l l'.
Proof.
  intros P Hnil Hskip Hswap Htrans.
  induction 1; auto.
  apply Htrans with (x::y::l); auto.
  apply Hswap; auto.
  induction l; auto.
  apply Hskip; auto.
  apply Hskip; auto.
  induction l; auto.
  eauto.
Qed.

Theorem Permutation_nil_app_cons : forall (l l' : list A) (x : A),
 ~ Permutation nil (l++x::l').
Proof.
  intros l l' x HF.
  apply Permutation_nil in HF. destruct l; discriminate.
Qed.

Ltac InvAdd := repeat (match goal with
 | H: Add ?x _ (_ :: _) |- _ => inversion H; clear H; subst
 end).

Ltac finish_basic_perms H :=
  try constructor; try rewrite perm_swap; try constructor; trivial;
  (rewrite <- H; now apply Permutation_Add) ||
  (rewrite H; symmetry; now apply Permutation_Add).

Theorem Permutation_Add_inv a l1 l2 :
  Permutation l1 l2 -> forall l1' l2', Add a l1' l1 -> Add a l2' l2 ->
   Permutation l1' l2'.
Proof.
 revert l1 l2. refine (Permutation_ind_bis _ _ _ _ _).
 - (* nil *)
   inversion_clear 1.
 - (* skip *)
   intros x l1 l2 PE IH. intros. InvAdd; try finish_basic_perms PE.
   constructor. now apply IH.
 - (* swap *)
   intros x y l1 l2 PE IH. intros. InvAdd; try finish_basic_perms PE.
   rewrite perm_swap; do 2 constructor. now apply IH.
 - (* trans *)
   intros l1 l l2 PE IH PE' IH' l1' l2' AD1 AD2.
   assert (Ha : In a l). { rewrite <- PE. rewrite (Add_in AD1). simpl; auto. }
   destruct (Add_inv _ _ Ha) as (l',AD).
   transitivity l'; auto.
Qed.

Theorem Permutation_app_inv (l1 l2 l3 l4:list A) a :
  Permutation (l1++a::l2) (l3++a::l4) -> Permutation (l1++l2) (l3 ++ l4).
Proof.
 intros. eapply Permutation_Add_inv; eauto using Add_app.
Qed.

Theorem Permutation_cons_inv l l' a :
 Permutation (a::l) (a::l') -> Permutation l l'.
Proof.
  intro. eapply Permutation_Add_inv; eauto using Add_head.
Qed.

Theorem Permutation_cons_app_inv l l1 l2 a :
 Permutation (a :: l) (l1 ++ a :: l2) -> Permutation l (l1 ++ l2).
Proof.
  intro. eapply Permutation_Add_inv; eauto using Add_head, Add_app.
Qed.

Theorem Permutation_app_inv_l : forall l l1 l2,
 Permutation (l ++ l1) (l ++ l2) -> Permutation l1 l2.
Proof.
  induction l; simpl; auto.
  intros.
  apply IHl.
  apply Permutation_cons_inv with a; auto.
Qed.

Theorem Permutation_app_inv_r l l1 l2 :
 Permutation (l1 ++ l) (l2 ++ l) -> Permutation l1 l2.
Proof.
 rewrite 2 (Permutation_app_comm _ l). apply Permutation_app_inv_l.
Qed.

Lemma Permutation_length_1_inv: forall a l, Permutation [a] l -> l = [a].
Proof.
  intros a l H; remember [a] as m in H.
  induction H; try (injection Heqm as -> ->);
    discriminate || auto.
  apply Permutation_nil in H as ->; trivial.
Qed.

Lemma Permutation_length_1: forall a b, Permutation [a] [b] -> a = b.
Proof.
  intros a b H.
  apply Permutation_length_1_inv in H; injection H as ->; trivial.
Qed.

Lemma Permutation_length_2_inv :
  forall a1 a2 l, Permutation [a1;a2] l -> l = [a1;a2] \/ l = [a2;a1].
Proof.
  intros a1 a2 l H; remember [a1;a2] as m in H.
  revert a1 a2 Heqm.
  induction H; intros; try (injection Heqm as ? ?; subst);
    discriminate || (try tauto).
  apply Permutation_length_1_inv in H as ->; left; auto.
  apply IHPermutation1 in Heqm as [H1|H1]; apply IHPermutation2 in H1 as [];
    auto.
Qed.

Lemma Permutation_length_2 :
  forall a1 a2 b1 b2, Permutation [a1;a2] [b1;b2] ->
    a1 = b1 /\ a2 = b2 \/ a1 = b2 /\ a2 = b1.
Proof.
  intros a1 b1 a2 b2 H.
  apply Permutation_length_2_inv in H as [H|H]; injection H as -> ->; auto.
Qed.

Lemma NoDup_Permutation l l' : NoDup l -> NoDup l' ->
  (forall x:A, In x l <-> In x l') -> Permutation l l'.
Proof.
 intros N. revert l'. induction N as [|a l Hal Hl IH].
 - destruct l'; simpl; auto.
   intros Hl' H. exfalso. rewrite (H a); auto.
 - intros l' Hl' H.
   assert (Ha : In a l') by (apply H; simpl; auto).
   destruct (Add_inv _ _ Ha) as (l'' & AD).
   rewrite <- (Permutation_Add AD).
   apply perm_skip.
   apply IH; clear IH.
   * now apply (NoDup_Add AD).
   * split.
     + apply incl_Add_inv with a l'; trivial. intro. apply H.
     + intro Hx.
       assert (Hx' : In x (a::l)).
       { apply H. rewrite (Add_in AD). now right. }
       destruct Hx'; simpl; trivial. subst.
       rewrite (NoDup_Add AD) in Hl'. tauto.
Qed.

Lemma NoDup_Permutation_bis l l' : NoDup l -> NoDup l' ->
  length l' <= length l -> incl l l' -> Permutation l l'.
Proof.
 intros. apply NoDup_Permutation; auto.
 split; auto. apply NoDup_length_incl; trivial.
Qed.

Lemma Permutation_NoDup l l' : Permutation l l' -> NoDup l -> NoDup l'.
Proof.
 induction 1; auto.
 * inversion_clear 1; constructor; eauto using Permutation_in.
 * inversion_clear 1 as [|? ? H1 H2]. inversion_clear H2; simpl in *.
   constructor. simpl; intuition. constructor; intuition.
Qed.

Global Instance Permutation_NoDup' :
 Proper (@Permutation A ==> iff) (@NoDup A) | 10.
Proof.
  repeat red; eauto using Permutation_NoDup.
Qed.

End Permutation_properties.

Section Permutation_map.

Variable A B : Type.
Variable f : A -> B.

Lemma Permutation_map l l' :
  Permutation l l' -> Permutation (map f l) (map f l').
Proof.
 induction 1; simpl; eauto.
Qed.

Global Instance Permutation_map' :
  Proper (@Permutation A ==> @Permutation B) (map f) | 10.
Proof.
  exact Permutation_map.
Qed.

End Permutation_map.

Lemma nat_bijection_Permutation n f :
 bFun n f ->
 Injective f ->
 let l := seq 0 n in Permutation (map f l) l.
Proof.
 intros Hf BD.
 apply NoDup_Permutation_bis; auto using Injective_map_NoDup, seq_NoDup.
 * now rewrite map_length.
 * intros x. rewrite in_map_iff. intros (y & <- & Hy').
   rewrite in_seq in *. simpl in *.
   destruct Hy' as (_,Hy'). auto with arith.
Qed.

Section Permutation_alt.
Variable A:Type.
Implicit Type a : A.
Implicit Type l : list A.

(** Alternative characterization of permutation
    via [nth_error] and [nth] *)

Let adapt f n :=
 let m := f (S n) in if le_lt_dec m (f 0) then m else pred m.

Let adapt_injective f : Injective f -> Injective (adapt f).
Proof.
 unfold adapt. intros Hf x y EQ.
 destruct le_lt_dec as [LE|LT]; destruct le_lt_dec as [LE'|LT'].
 - now apply eq_add_S, Hf.
 - apply Lt.le_lt_or_eq in LE.
   destruct LE as [LT|EQ']; [|now apply Hf in EQ'].
   unfold lt in LT. rewrite EQ in LT.
   rewrite <- (Lt.S_pred _ _ LT') in LT.
   elim (Lt.lt_not_le _ _ LT' LT).
 - apply Lt.le_lt_or_eq in LE'.
   destruct LE' as [LT'|EQ']; [|now apply Hf in EQ'].
   unfold lt in LT'. rewrite <- EQ in LT'.
   rewrite <- (Lt.S_pred _ _ LT) in LT'.
   elim (Lt.lt_not_le _ _ LT LT').
 - apply eq_add_S, Hf.
   now rewrite (Lt.S_pred _ _ LT), (Lt.S_pred _ _ LT'), EQ.
Qed.

Let adapt_ok a l1 l2 f : Injective f -> length l1 = f 0 ->
 forall n, nth_error (l1++a::l2) (f (S n)) = nth_error (l1++l2) (adapt f n).
Proof.
 unfold adapt. intros Hf E n.
 destruct le_lt_dec as [LE|LT].
 - apply Lt.le_lt_or_eq in LE.
   destruct LE as [LT|EQ]; [|now apply Hf in EQ].
   rewrite <- E in LT.
   rewrite 2 nth_error_app1; auto.
 - rewrite (Lt.S_pred _ _ LT) at 1.
   rewrite <- E, (Lt.S_pred _ _ LT) in LT.
   rewrite 2 nth_error_app2; auto with arith.
   rewrite <- Minus.minus_Sn_m; auto with arith.
Qed.

Lemma Permutation_nth_error l l' :
 Permutation l l' <->
  (length l = length l' /\
   exists f:nat->nat,
    Injective f /\ forall n, nth_error l' n = nth_error l (f n)).
Proof.
 split.
 { intros P.
   split; [now apply Permutation_length|].
   induction P.
   - exists (fun n => n).
     split; try red; auto.
   - destruct IHP as (f & Hf & Hf').
     exists (fun n => match n with O => O | S n => S (f n) end).
     split; try red.
     * intros [|y] [|z]; simpl; now auto.
     * intros [|n]; simpl; auto.
   - exists (fun n => match n with 0 => 1 | 1 => 0 | n => n end).
     split; try red.
     * intros [|[|z]] [|[|t]]; simpl; now auto.
     * intros [|[|n]]; simpl; auto.
   - destruct IHP1 as (f & Hf & Hf').
     destruct IHP2 as (g & Hg & Hg').
     exists (fun n => f (g n)).
     split; try red.
     * auto.
     * intros n. rewrite <- Hf'; auto. }
 { revert l. induction l'.
   - intros [|l] (E & _); now auto.
   - intros l (E & f & Hf & Hf').
     simpl in E.
     assert (Ha : nth_error l (f 0) = Some a)
      by (symmetry; apply (Hf' 0)).
     destruct (nth_error_split l (f 0) Ha) as (l1 & l2 & L12 & L1).
     rewrite L12. rewrite <- Permutation_middle. constructor.
     apply IHl'; split; [|exists (adapt f); split].
     * revert E. rewrite L12, !app_length. simpl.
       rewrite <- plus_n_Sm. now injection 1.
     * now apply adapt_injective.
     * intro n. rewrite <- (adapt_ok a), <- L12; auto.
       apply (Hf' (S n)). }
Qed.

Lemma Permutation_nth_error_bis l l' :
 Permutation l l' <->
  exists f:nat->nat,
    Injective f /\
    bFun (length l) f /\
    (forall n, nth_error l' n = nth_error l (f n)).
Proof.
 rewrite Permutation_nth_error; split.
 - intros (E & f & Hf & Hf').
   exists f. do 2 (split; trivial).
   intros n Hn.
   destruct (Lt.le_or_lt (length l) (f n)) as [LE|LT]; trivial.
   rewrite <- nth_error_None, <- Hf', nth_error_None, <- E in LE.
   elim (Lt.lt_not_le _ _ Hn LE).
 - intros (f & Hf & Hf2 & Hf3); split; [|exists f; auto].
   assert (H : length l' <= length l') by auto with arith.
   rewrite <- nth_error_None, Hf3, nth_error_None in H.
   destruct (Lt.le_or_lt (length l) (length l')) as [LE|LT];
    [|apply Hf2 in LT; elim (Lt.lt_not_le _ _ LT H)].
   apply Lt.le_lt_or_eq in LE. destruct LE as [LT|EQ]; trivial.
   rewrite <- nth_error_Some, Hf3, nth_error_Some in LT.
   assert (Hf' : bInjective (length l) f).
   { intros x y _ _ E. now apply Hf. }
   rewrite (bInjective_bSurjective Hf2) in Hf'.
   destruct (Hf' _ LT) as (y & Hy & Hy').
   apply Hf in Hy'. subst y. elim (Lt.lt_irrefl _ Hy).
Qed.

Lemma Permutation_nth l l' d :
 Permutation l l' <->
  (let n := length l in
   length l' = n /\
   exists f:nat->nat,
    bFun n f /\
    bInjective n f /\
    (forall x, x < n -> nth x l' d = nth (f x) l d)).
Proof.
 split.
 - intros H.
   assert (E := Permutation_length H).
   split; auto.
   apply Permutation_nth_error_bis in H.
   destruct H as (f & Hf & Hf2 & Hf3).
   exists f. split; [|split]; auto.
   intros x y _ _ Hxy. now apply Hf.
   intros n Hn. rewrite <- 2 nth_default_eq. unfold nth_default.
    now rewrite Hf3.
 - intros (E & f & Hf1 & Hf2 & Hf3).
   rewrite Permutation_nth_error.
   split; auto.
   exists (fun n => if le_lt_dec (length l) n then n else f n).
   split.
   * intros x y.
     destruct le_lt_dec as [LE|LT];
      destruct le_lt_dec as [LE'|LT']; auto.
     + apply Hf1 in LT'. intros ->.
       elim (Lt.lt_irrefl (f y)). eapply Lt.lt_le_trans; eauto.
     + apply Hf1 in LT. intros <-.
       elim (Lt.lt_irrefl (f x)). eapply Lt.lt_le_trans; eauto.
   * intros n.
     destruct le_lt_dec as [LE|LT].
     + assert (LE' : length l' <= n) by (now rewrite E).
       rewrite <- nth_error_None in LE, LE'. congruence.
     + assert (LT' : n < length l') by (now rewrite E).
       specialize (Hf3 n LT). rewrite <- 2 nth_default_eq in Hf3.
       unfold nth_default in Hf3.
       apply Hf1 in LT.
       rewrite <- nth_error_Some in LT, LT'.
       do 2 destruct nth_error; congruence.
Qed.

End Permutation_alt.

(* begin hide *)
Notation Permutation_app_swap := Permutation_app_comm (only parsing).
(* end hide *)