docs/htmldoc/mathcomp.ssreflect.path.html


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
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
609
610
611
612
613
614
615
616
617
618
619
620
621
622
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
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title>mathcomp.ssreflect.path</title>
</head>

<body>

<div id="page">

<div id="header">
</div>

<div id="main">

<h1 class="libtitle">Library mathcomp.ssreflect.path</h1>

<div class="code">
<span class="comment">(*&nbsp;(c)&nbsp;Copyright&nbsp;2006-2016&nbsp;Microsoft&nbsp;Corporation&nbsp;and&nbsp;Inria.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
&nbsp;Distributed&nbsp;under&nbsp;the&nbsp;terms&nbsp;of&nbsp;CeCILL-B.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*)</span><br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/>

<br/>
</div>

<div class="doc">
    The basic theory of paths over an eqType; this file is essentially a    
 complement to seq.v. Paths are non-empty sequences that obey a progression 
 relation. They are passed around in three parts: the head and tail of the  
 sequence, and a proof of (boolean) predicate asserting the progression.    
 This "exploded" view is rarely embarrassing, as the first two parameters   
 are usually inferred from the type of the third; on the contrary, it saves 
 the hassle of constantly constructing and destructing a dependent record.  
    We define similarly cycles, for which we allow the empty sequence,      
 which represents a non-rooted empty cycle; by contrast, the "empty" path   
 from a point x is the one-item sequence containing only x.                 
   We allow duplicates; uniqueness, if desired (as is the case for several  
 geometric constructions), must be asserted separately. We do provide       
 shorthand, but only for cycles, because the equational properties of       
 "path" and "uniq" are unfortunately  incompatible (esp. wrt "cat").        
    We define notations for the common cases of function paths, where the   
 progress relation is actually a function. In detail:                       
   path e x p == x :: p is an e-path [:: x_0; x_1; ... ; x_n], i.e., we     
                 e x_i x{i+1} for all i &lt; n. The path x :: p starts at x   
                 and ends at last x p.                                      
  fpath f x p == x :: p is an f-path, where f is a function, i.e., p is of  
                 the form [:: f x; f (f x); ... ]. This is just a notation   
                 for path (frel f) x p.                                     
   sorted e s == s is an e-sorted sequence: either s = [:: ], or s = x :: p  
                 is an e-path (this is oten used with e = leq or ltn).      
    cycle e c == c is an e-cycle: either c = [:: ], or c = x :: p with       
                 x :: (rcons p x) an e-path.                                
   fcycle f c == c is an f-cycle, for a function f.                         
 traject f x n == the f-path of size n starting at x                        
              := [:: x; f x; ...; iter n.-1 f x]                            
 looping f x n == the f-paths of size greater than n starting at x loop     
                 back, or, equivalently, traject f x n contains all         
                 iterates of f at x.                                        
 merge e s1 s2 == the e-sorted merge of sequences s1 and s2: this is always 
                 a permutation of s1 ++ s2, and is e-sorted when s1 and s2  
                 are and e is total.                                        
     sort e s == a permutation of the sequence s, that is e-sorted when e   
                 is total (computed by a merge sort with the merge function 
                 above).                                                    
   mem2 s x y == x, then y occur in the sequence (path) s; this is          
                 non-strict: mem2 s x x = (x \in s).                        
     next c x == the successor of the first occurrence of x in the sequence 
                 c (viewed as a cycle), or x if x \notin c.                 
     prev c x == the predecessor of the first occurrence of x in the        
                 sequence c (viewed as a cycle), or x if x \notin c.        
    arc c x y == the sub-arc of the sequece c (viewed as a cycle) starting  
                 at the first occurrence of x in c, and ending just before  
                 the next ocurrence of y (in cycle order); arc c x y        
                 returns an unspecified sub-arc of c if x and y do not both 
                 occur in c.                                                
  ucycle e c &lt;-&gt; ucycleb e c (ucycle e c is a Coercion target of type Prop) 
 ufcycle f c &lt;-&gt; c is a simple f-cycle, for a function f.                   
  shorten x p == the tail a duplicate-free subpath of x :: p with the same  
                 endpoints (x and last x p), obtained by removing all loops 
                 from x :: p.                                               
 rel_base e e' h b &lt;-&gt; the function h is a functor from relation e to       
                 relation e', EXCEPT at points whose image under h satisfy  
                 the "base" predicate b:                                    
                    e' (h x) (h y) = e x y UNLESS b (h x) holds             
                 This is the statement of the side condition of the path    
                 functorial mapping lemma map_path.                         
 fun_base f f' h b &lt;-&gt; the function h is a functor from function f to f',   
                 except at the preimage of predicate b under h.             
 We also provide three segmenting dependently-typed lemmas (splitP, splitPl 
 and splitPr) whose elimination split a path x0 :: p at an internal point x 
 as follows:                                                                
<ul class="doclist">
<li> splitP applies when x \in p; it replaces p with (rcons p1 x ++ p2), so  
    that x appears explicitly at the end of the left part. The elimination  
    of splitP will also simultaneously replace take (index x p) with p1 and 
    drop (index x p).+1 p with p2.                                          

</li>
<li> splitPl applies when x \in x0 :: p; it replaces p with p1 ++ p2 and     
    simulaneously generates an equation x = last x0 p.                      

</li>
<li> splitPr applies when x \in p; it replaces p with (p1 ++ x :: p2), so x  
    appears explicitly at the start of the right part.                      

</li>
</ul>
 The parts p1 and p2 are computed using index/take/drop in all cases, but   
 only splitP attemps to subsitute the explicit values. The substitution of  
 p can be deferred using the dependent equation generation feature of       
 ssreflect, e.g.: case/splitPr def_p: {1}p / x_in_p =&gt; [p1 p2] generates    
 the equation p = p1 ++ p2 instead of performing the substitution outright. 
   Similarly, eliminating the loop removal lemma shortenP simultaneously    
 replaces shorten e x p with a fresh constant p', and last x p with         
 last x p'.                                                                 
   Note that although all "path" functions actually operate on the          
 underlying sequence, we provide a series of lemmas that define their       
 interaction with thepath and cycle predicates, e.g., the cat_path equation 
 can be used to split the path predicate after splitting the underlying     
 sequence.                                                                   
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Paths"><span class="id" title="section">Paths</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="Paths.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Paths.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>).<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Paths.Path"><span class="id" title="section">Path</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="Paths.Path.x0_cycle"><span class="id" title="variable">x0_cycle</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#Paths.T"><span class="id" title="variable">T</span></a>) (<a name="Paths.Path.e"><span class="id" title="variable">e</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="path"><span class="id" title="definition">path</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.T"><span class="id" title="variable">T</span></a>) :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.Path.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <span class="id" title="var">y</span> <span class="id" title="var">p'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="cat_path"><span class="id" title="lemma">cat_path</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> : <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="rcons_path"><span class="id" title="lemma">rcons_path</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.Path.e"><span class="id" title="variable">e</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="pathP"><span class="id" title="lemma">pathP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> <span class="id" title="var">x0</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.Path.e"><span class="id" title="variable">e</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x0"><span class="id" title="variable">x0</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a>))<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="cycle"><span class="id" title="definition">cycle</span></a> <span class="id" title="var">p</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <span class="id" title="var">x</span> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <span class="id" title="var">p'</span> <span class="id" title="var">x</span>) <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="cycle_path"><span class="id" title="lemma">cycle_path</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.Path.x0_cycle"><span class="id" title="variable">x0_cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="rot_cycle"><span class="id" title="lemma">rot_cycle</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="rotr_cycle"><span class="id" title="lemma">rotr_cycle</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#Paths.Path"><span class="id" title="section">Path</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_path"><span class="id" title="lemma">eq_path</span></a> <span class="id" title="var">e</span> <span class="id" title="var">e'</span> : <a class="idref" href="mathcomp.ssreflect.path.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#86efabaf3ab961ac3d2bac35d2b3c35d"><span class="id" title="notation">=2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e'"><span class="id" title="variable">e'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#86efabaf3ab961ac3d2bac35d2b3c35d"><span class="id" title="notation">=2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e'"><span class="id" title="variable">e'</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_cycle"><span class="id" title="lemma">eq_cycle</span></a> <span class="id" title="var">e</span> <span class="id" title="var">e'</span> : <a class="idref" href="mathcomp.ssreflect.path.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#86efabaf3ab961ac3d2bac35d2b3c35d"><span class="id" title="notation">=2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e'"><span class="id" title="variable">e'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e'"><span class="id" title="variable">e'</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="sub_path"><span class="id" title="lemma">sub_path</span></a> <span class="id" title="var">e</span> <span class="id" title="var">e'</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#subrel"><span class="id" title="definition">subrel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e'"><span class="id" title="variable">e'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">p</span>, <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e'"><span class="id" title="variable">e'</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="rev_path"><span class="id" title="lemma">rev_path</span></a> <span class="id" title="var">e</span> <span class="id" title="var">x</span> <span class="id" title="var">p</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">z</span> ⇒ <a class="idref" href="mathcomp.ssreflect.path.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#Paths"><span class="id" title="section">Paths</span></a>.<br/>

<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="EqPath"><span class="id" title="section">EqPath</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="EqPath.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="EqPath.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="EqPath.x0_cycle"><span class="id" title="variable">x0_cycle</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>) (<a name="EqPath.e"><span class="id" title="variable">e</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">CoInductive</span> <a name="split"><span class="id" title="inductive">split</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/>
&nbsp;&nbsp;<a name="Split"><span class="id" title="constructor">Split</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> : <a class="idref" href="mathcomp.ssreflect.path.html#split"><span class="id" title="inductive">split</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="splitP"><span class="id" title="lemma">splitP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> (<span class="id" title="var">i</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#split"><span class="id" title="inductive">split</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>).<br/>

<br/>
<span class="id" title="keyword">CoInductive</span> <a name="splitl"><span class="id" title="inductive">splitl</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/>
&nbsp;&nbsp;<a name="Splitl"><span class="id" title="constructor">Splitl</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#splitl"><span class="id" title="inductive">splitl</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="splitPl"><span class="id" title="lemma">splitPl</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#splitl"><span class="id" title="inductive">splitl</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">CoInductive</span> <a name="splitr"><span class="id" title="inductive">splitr</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/>
&nbsp;&nbsp;<a name="Splitr"><span class="id" title="constructor">Splitr</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> : <a class="idref" href="mathcomp.ssreflect.path.html#splitr"><span class="id" title="inductive">splitr</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="splitPr"><span class="id" title="lemma">splitPr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#splitr"><span class="id" title="inductive">splitr</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="next_at"><span class="id" title="definition">next_at</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y0</span> <span class="id" title="var">y</span> <span class="id" title="var">p</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y0"><span class="id" title="variable">y0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><br/>
&nbsp;&nbsp;| <span class="id" title="var">y'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <span class="id" title="var">y'</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#next_at"><span class="id" title="definition">next_at</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y0"><span class="id" title="variable">y0</span></a> <span class="id" title="var">y'</span> <span class="id" title="var">p'</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="next"><span class="id" title="definition">next</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.path.html#next_at"><span class="id" title="definition">next_at</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <span class="id" title="var">y</span> <span class="id" title="var">p'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>.<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="prev_at"><span class="id" title="definition">prev_at</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y0</span> <span class="id" title="var">y</span> <span class="id" title="var">p</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a>     ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y0"><span class="id" title="variable">y0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><br/>
&nbsp;&nbsp;| <span class="id" title="var">y'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <span class="id" title="var">y'</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#prev_at"><span class="id" title="definition">prev_at</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y0"><span class="id" title="variable">y0</span></a> <span class="id" title="var">y'</span> <span class="id" title="var">p'</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="prev"><span class="id" title="definition">prev</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.path.html#prev_at"><span class="id" title="definition">prev_at</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <span class="id" title="var">y</span> <span class="id" title="var">p'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="next_nth"><span class="id" title="lemma">next_nth</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <span class="id" title="var">y</span> <span class="id" title="var">p'</span> (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="prev_nth"><span class="id" title="lemma">prev_nth</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">p'</span>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem_next"><span class="id" title="lemma">mem_next</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem_prev"><span class="id" title="lemma">mem_prev</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>

<br/>
</div>

<div class="doc">
 ucycleb is the boolean predicate, but ucycle is defined as a Prop 
 so that it can be used as a coercion target.  
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="ucycleb"><span class="id" title="definition">ucycleb</span></a> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="ucycle"><span class="id" title="definition">ucycle</span></a> <span class="id" title="var">p</span> : <span class="id" title="keyword">Prop</span> := <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
</div>

<div class="doc">
 Projections, used for creating local lemmas.  
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a name="ucycle_cycle"><span class="id" title="lemma">ucycle_cycle</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#ucycle"><span class="id" title="definition">ucycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="ucycle_uniq"><span class="id" title="lemma">ucycle_uniq</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#ucycle"><span class="id" title="definition">ucycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="next_cycle"><span class="id" title="lemma">next_cycle</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="prev_cycle"><span class="id" title="lemma">prev_cycle</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="rot_ucycle"><span class="id" title="lemma">rot_ucycle</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#ucycle"><span class="id" title="definition">ucycle</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#ucycle"><span class="id" title="definition">ucycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="rotr_ucycle"><span class="id" title="lemma">rotr_ucycle</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#ucycle"><span class="id" title="definition">ucycle</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#ucycle"><span class="id" title="definition">ucycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
</div>

<div class="doc">
 The "appears no later" partial preorder defined by a path.  
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Definition</span> <a name="mem2"><span class="id" title="definition">mem2</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> := <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2l"><span class="id" title="lemma">mem2l</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2lf"><span class="id" title="lemma">mem2lf</span></a> {<span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>} : <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2r"><span class="id" title="lemma">mem2r</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2rf"><span class="id" title="lemma">mem2rf</span></a> {<span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>} : <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2_cat"><span class="id" title="lemma">mem2_cat</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2_splice"><span class="id" title="lemma">mem2_splice</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p3</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">p2</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p3"><span class="id" title="variable">p3</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p3"><span class="id" title="variable">p3</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2_splice1"><span class="id" title="lemma">mem2_splice1</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p3</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p3"><span class="id" title="variable">p3</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p3"><span class="id" title="variable">p3</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2_cons"><span class="id" title="lemma">mem2_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2_seq1"><span class="id" title="lemma">mem2_seq1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2_last"><span class="id" title="lemma">mem2_last</span></a> <span class="id" title="var">y0</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y0"><span class="id" title="variable">y0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2l_cat"><span class="id" title="lemma">mem2l_cat</span></a> {<span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="var">x</span>} : <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2r_cat"><span class="id" title="lemma">mem2r_cat</span></a> {<span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>} : <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2lr_splice"><span class="id" title="lemma">mem2lr_splice</span></a> {<span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="var">p3</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>} :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p3"><span class="id" title="variable">p3</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p3"><span class="id" title="variable">p3</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>.<br/>

<br/>
<span class="id" title="keyword">CoInductive</span> <a name="split2r"><span class="id" title="inductive">split2r</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/>
&nbsp;&nbsp;<a name="Split2r"><span class="id" title="constructor">Split2r</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#split2r"><span class="id" title="inductive">split2r</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="splitP2r"><span class="id" title="lemma">splitP2r</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#split2r"><span class="id" title="inductive">split2r</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="shorten"><span class="id" title="definition">shorten</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">then</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#shorten"><span class="id" title="definition">shorten</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">p'</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#shorten"><span class="id" title="definition">shorten</span></a> <span class="id" title="var">y</span> <span class="id" title="var">p'</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a>.<br/>

<br/>
<span class="id" title="keyword">CoInductive</span> <a name="shorten_spec"><span class="id" title="inductive">shorten_spec</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/>
&nbsp;&nbsp;&nbsp;<a name="ShortenSpec"><span class="id" title="constructor">ShortenSpec</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a> &amp; <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a>) &amp; <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#subpred"><span class="id" title="definition">subpred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#shorten_spec"><span class="id" title="inductive">shorten_spec</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="shortenP"><span class="id" title="lemma">shortenP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#shorten_spec"><span class="id" title="inductive">shorten_spec</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#shorten"><span class="id" title="definition">shorten</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#EqPath"><span class="id" title="section">EqPath</span></a>.<br/>

<br/>
</div>

<div class="doc">
 Ordered paths and sorting.  
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Section</span> <a name="SortSeq"><span class="id" title="section">SortSeq</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="SortSeq.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/>
<span class="id" title="keyword">Variable</span> <a name="SortSeq.leT"><span class="id" title="variable">leT</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sorted"><span class="id" title="definition">sorted</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="path_sorted"><span class="id" title="lemma">path_sorted</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="path_min_sorted"><span class="id" title="lemma">path_min_sorted</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Section</span> <a name="SortSeq.Transitive"><span class="id" title="section">Transitive</span></a>.<br/>

<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="SortSeq.Transitive.leT_tr"><span class="id" title="variable">leT_tr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="subseq_order_path"><span class="id" title="lemma">subseq_order_path</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="order_path_min"><span class="id" title="lemma">order_path_min</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="subseq_sorted"><span class="id" title="lemma">subseq_sorted</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sorted_filter"><span class="id" title="lemma">sorted_filter</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="sorted_uniq"><span class="id" title="lemma">sorted_uniq</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#irreflexive"><span class="id" title="definition">irreflexive</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">s</span>, <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_sorted"><span class="id" title="lemma">eq_sorted</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#antisymmetric"><span class="id" title="definition">antisymmetric</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>, <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_sorted_irr"><span class="id" title="lemma">eq_sorted_irr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#irreflexive"><span class="id" title="definition">irreflexive</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>, <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.Transitive"><span class="id" title="section">Transitive</span></a>.<br/>

<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="SortSeq.leT_total"><span class="id" title="variable">leT_total</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#total"><span class="id" title="definition">total</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a>.<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="merge"><span class="id" title="definition">merge</span></a> <span class="id" title="var">s1</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x1</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span> <span class="id" title="keyword">then</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <span class="id" title="var">merge_s1</span> <span class="id" title="var">s2</span> :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">s2'</span> <span class="id" title="keyword">then</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <span class="id" title="var">x2</span> <span class="id" title="var">x1</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#merge_s1"><span class="id" title="variable">merge_s1</span></a> <span class="id" title="var">s2'</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <span class="id" title="var">x1</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#merge_s1"><span class="id" title="variable">merge_s1</span></a><br/>
&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="merge_path"><span class="id" title="lemma">merge_path</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="merge_sorted"><span class="id" title="lemma">merge_sorted</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="perm_merge"><span class="id" title="lemma">perm_merge</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem_merge"><span class="id" title="lemma">mem_merge</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="size_merge"><span class="id" title="lemma">size_merge</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="merge_uniq"><span class="id" title="lemma">merge_uniq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="merge_sort_push"><span class="id" title="definition">merge_sort_push</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">ss</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.path.html#ss"><span class="id" title="variable">ss</span></a> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">ss'</span> | <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">ss'</span> ⇒ <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">ss'</span><br/>
&nbsp;&nbsp;| <span class="id" title="var">s2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">ss'</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#merge_sort_push"><span class="id" title="definition">merge_sort_push</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <span class="id" title="var">s2</span>) <span class="id" title="var">ss'</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="merge_sort_pop"><span class="id" title="definition">merge_sort_pop</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">ss</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#ss"><span class="id" title="variable">ss</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">s2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">ss'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.path.html#merge_sort_pop"><span class="id" title="definition">merge_sort_pop</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#merge"><span class="id" title="definition">merge</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <span class="id" title="var">s2</span>) <span class="id" title="var">ss'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a>.<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="merge_sort_rec"><span class="id" title="definition">merge_sort_rec</span></a> <span class="id" title="var">ss</span> <span class="id" title="var">s</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="mathcomp.ssreflect.seq.html#fcbb95c5c5fd941c49e06d09a4f5a316"><span class="id" title="notation">[::</span></a> <span class="id" title="var">x1</span><a class="idref" href="mathcomp.ssreflect.seq.html#fcbb95c5c5fd941c49e06d09a4f5a316"><span class="id" title="notation">,</span></a> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#fcbb95c5c5fd941c49e06d09a4f5a316"><span class="id" title="notation">&amp;</span></a> <span class="id" title="var">s'</span><a class="idref" href="mathcomp.ssreflect.seq.html#fcbb95c5c5fd941c49e06d09a4f5a316"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">then</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">s1</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">[::</span></a> <span class="id" title="var">x1</span><a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">;</span></a> <span class="id" title="var">x2</span><a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">[::</span></a> <span class="id" title="var">x2</span><a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">;</span></a> <span class="id" title="var">x1</span><a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">]</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#merge_sort_rec"><span class="id" title="definition">merge_sort_rec</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#merge_sort_push"><span class="id" title="definition">merge_sort_push</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#ss"><span class="id" title="variable">ss</span></a>) <span class="id" title="var">s'</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.path.html#merge_sort_pop"><span class="id" title="definition">merge_sort_pop</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#ss"><span class="id" title="variable">ss</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sort"><span class="id" title="definition">sort</span></a> := <a class="idref" href="mathcomp.ssreflect.path.html#merge_sort_rec"><span class="id" title="definition">merge_sort_rec</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sort_sorted"><span class="id" title="lemma">sort_sorted</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="perm_sort"><span class="id" title="lemma">perm_sort</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem_sort"><span class="id" title="lemma">mem_sort</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.path.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="size_sort"><span class="id" title="lemma">size_sort</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="sort_uniq"><span class="id" title="lemma">sort_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="perm_sortP"><span class="id" title="lemma">perm_sortP</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#antisymmetric"><span class="id" title="definition">antisymmetric</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sort"><span class="id" title="definition">sort</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s2"><span class="id" title="variable">s2</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#SortSeq"><span class="id" title="section">SortSeq</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="rev_sorted"><span class="id" title="lemma">rev_sorted</span></a> (<span class="id" title="var">T</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<span class="id" title="var">leT</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">s</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#leT"><span class="id" title="variable">leT</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.ssreflect.path.html#leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="ltn_sorted_uniq_leq"><span class="id" title="lemma">ltn_sorted_uniq_leq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ltn"><span class="id" title="definition">ltn</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#leq"><span class="id" title="definition">leq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="iota_sorted"><span class="id" title="lemma">iota_sorted</span></a> <span class="id" title="var">i</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#leq"><span class="id" title="definition">leq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="iota_ltn_sorted"><span class="id" title="lemma">iota_ltn_sorted</span></a> <span class="id" title="var">i</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.path.html#sorted"><span class="id" title="definition">sorted</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ltn"><span class="id" title="definition">ltn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>).<br/>
 
<br/>
</div>

<div class="doc">
 Function trajectories.  
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Notation</span> <a name="fpath"><span class="id" title="abbreviation">fpath</span></a> <span class="id" title="var">f</span> := (<a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#coerced_frel"><span class="id" title="abbreviation">coerced_frel</span></a> <span class="id" title="var">f</span>)).<br/>
<span class="id" title="keyword">Notation</span> <a name="fcycle"><span class="id" title="abbreviation">fcycle</span></a> <span class="id" title="var">f</span> := (<a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#coerced_frel"><span class="id" title="abbreviation">coerced_frel</span></a> <span class="id" title="var">f</span>)).<br/>
<span class="id" title="keyword">Notation</span> <a name="ufcycle"><span class="id" title="abbreviation">ufcycle</span></a> <span class="id" title="var">f</span> := (<a class="idref" href="mathcomp.ssreflect.path.html#ucycle"><span class="id" title="definition">ucycle</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#coerced_frel"><span class="id" title="abbreviation">coerced_frel</span></a> <span class="id" title="var">f</span>)).<br/>

<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Trajectory"><span class="id" title="section">Trajectory</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="Trajectory.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Trajectory.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="traject"><span class="id" title="definition">traject</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#Trajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <span class="id" title="var">n'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="trajectS"><span class="id" title="lemma">trajectS</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#Trajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="trajectSr"><span class="id" title="lemma">trajectSr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>) (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Trajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="last_traject"><span class="id" title="lemma">last_traject</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#Trajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Trajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="traject_iteri"><span class="id" title="lemma">traject_iteri</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iteri"><span class="id" title="definition">iteri</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Trajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="size_traject"><span class="id" title="lemma">size_traject</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="nth_traject"><span class="id" title="lemma">nth_traject</span></a> <span class="id" title="var">i</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Trajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#Trajectory"><span class="id" title="section">Trajectory</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="EqTrajectory"><span class="id" title="section">EqTrajectory</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="EqTrajectory.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="EqTrajectory.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_fpath"><span class="id" title="lemma">eq_fpath</span></a> <span class="id" title="var">f'</span> : <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#f'"><span class="id" title="variable">f'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#fpath"><span class="id" title="abbreviation">fpath</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#86efabaf3ab961ac3d2bac35d2b3c35d"><span class="id" title="notation">=2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#fpath"><span class="id" title="abbreviation">fpath</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#f'"><span class="id" title="variable">f'</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_fcycle"><span class="id" title="lemma">eq_fcycle</span></a> <span class="id" title="var">f'</span> : <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#f'"><span class="id" title="variable">f'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#fcycle"><span class="id" title="abbreviation">fcycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#fcycle"><span class="id" title="abbreviation">fcycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#f'"><span class="id" title="variable">f'</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="fpathP"><span class="id" title="lemma">fpathP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#fpath"><span class="id" title="abbreviation">fpath</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="fpath_traject"><span class="id" title="lemma">fpath_traject</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.path.html#fpath"><span class="id" title="abbreviation">fpath</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Definition</span> <a name="looping"><span class="id" title="definition">looping</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="loopingP"><span class="id" title="lemma">loopingP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">m</span>, <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#looping"><span class="id" title="definition">looping</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="trajectP"><span class="id" title="lemma">trajectP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> <span class="id" title="var">y</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="looping_uniq"><span class="id" title="lemma">looping_uniq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#traject"><span class="id" title="definition">traject</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#looping"><span class="id" title="definition">looping</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#n"><span class="id" title="variable">n</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#EqTrajectory"><span class="id" title="section">EqTrajectory</span></a>.<br/>

<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="UniqCycle"><span class="id" title="section">UniqCycle</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="UniqCycle.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="UniqCycle.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="UniqCycle.e"><span class="id" title="variable">e</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>) (<a name="UniqCycle.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="UniqCycle.Up"><span class="id" title="variable">Up</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="prev_next"><span class="id" title="lemma">prev_next</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="next_prev"><span class="id" title="lemma">next_prev</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="cycle_next"><span class="id" title="lemma">cycle_next</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#fcycle"><span class="id" title="abbreviation">fcycle</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="cycle_prev"><span class="id" title="lemma">cycle_prev</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> ⇒ <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="cycle_from_next"><span class="id" title="lemma">cycle_from_next</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="cycle_from_prev"><span class="id" title="lemma">cycle_from_prev</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.e"><span class="id" title="variable">e</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#cycle"><span class="id" title="definition">cycle</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="next_rot"><span class="id" title="lemma">next_rot</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="prev_rot"><span class="id" title="lemma">prev_rot</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle.p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycle"><span class="id" title="section">UniqCycle</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="UniqRotrCycle"><span class="id" title="section">UniqRotrCycle</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="UniqRotrCycle.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="UniqRotrCycle.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="UniqRotrCycle.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="UniqRotrCycle.Up"><span class="id" title="variable">Up</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqRotrCycle.p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="next_rotr"><span class="id" title="lemma">next_rotr</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqRotrCycle.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqRotrCycle.p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqRotrCycle.p"><span class="id" title="variable">p</span></a>.  <br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="prev_rotr"><span class="id" title="lemma">prev_rotr</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqRotrCycle.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqRotrCycle.p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqRotrCycle.p"><span class="id" title="variable">p</span></a>.  <br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#UniqRotrCycle"><span class="id" title="section">UniqRotrCycle</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="UniqCycleRev"><span class="id" title="section">UniqCycleRev</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="UniqCycleRev.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycleRev.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="prev_rev"><span class="id" title="lemma">prev_rev</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="next_rev"><span class="id" title="lemma">next_rev</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#UniqCycleRev"><span class="id" title="section">UniqCycleRev</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="MapPath"><span class="id" title="section">MapPath</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="MapPath.T"><span class="id" title="variable">T</span></a> <a name="MapPath.T'"><span class="id" title="variable">T'</span></a> : <span class="id" title="keyword">Type</span>) (<a name="MapPath.h"><span class="id" title="variable">h</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#T'"><span class="id" title="variable">T'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>) (<a name="MapPath.e"><span class="id" title="variable">e</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>) (<a name="MapPath.e'"><span class="id" title="variable">e'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T'"><span class="id" title="variable">T'</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="rel_base"><span class="id" title="definition">rel_base</span></a> (<span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapPath.T"><span class="id" title="variable">T</span></a>) :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <span class="id" title="var">x'</span> <span class="id" title="var">y'</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#b"><span class="id" title="variable">b</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#MapPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x'"><span class="id" title="variable">x'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapPath.e"><span class="id" title="variable">e</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#MapPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x'"><span class="id" title="variable">x'</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#MapPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y'"><span class="id" title="variable">y'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapPath.e'"><span class="id" title="variable">e'</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x'"><span class="id" title="variable">x'</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y'"><span class="id" title="variable">y'</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="map_path"><span class="id" title="lemma">map_path</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x'</span> <span class="id" title="var">p'</span> (<span class="id" title="var">Bb</span> : <a class="idref" href="mathcomp.ssreflect.path.html#rel_base"><span class="id" title="definition">rel_base</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#b"><span class="id" title="variable">b</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#preim"><span class="id" title="definition">preim</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#b"><span class="id" title="variable">b</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x'"><span class="id" title="variable">x'</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapPath.e"><span class="id" title="variable">e</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#MapPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x'"><span class="id" title="variable">x'</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapPath.e'"><span class="id" title="variable">e'</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x'"><span class="id" title="variable">x'</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#MapPath"><span class="id" title="section">MapPath</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="MapEqPath"><span class="id" title="section">MapEqPath</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="MapEqPath.T"><span class="id" title="variable">T</span></a> <a name="MapEqPath.T'"><span class="id" title="variable">T'</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="MapEqPath.h"><span class="id" title="variable">h</span></a> : <a class="idref" href="mathcomp.ssreflect.path.html#T'"><span class="id" title="variable">T'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>) (<a name="MapEqPath.e"><span class="id" title="variable">e</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>) (<a name="MapEqPath.e'"><span class="id" title="variable">e'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T'"><span class="id" title="variable">T'</span></a>).<br/>

<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="MapEqPath.Ih"><span class="id" title="variable">Ih</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mem2_map"><span class="id" title="lemma">mem2_map</span></a> <span class="id" title="var">x'</span> <span class="id" title="var">y'</span> <span class="id" title="var">p'</span> : <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x'"><span class="id" title="variable">x'</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y'"><span class="id" title="variable">y'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#mem2"><span class="id" title="definition">mem2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p'"><span class="id" title="variable">p'</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x'"><span class="id" title="variable">x'</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y'"><span class="id" title="variable">y'</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="next_map"><span class="id" title="lemma">next_map</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#next"><span class="id" title="definition">next</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="prev_map"><span class="id" title="lemma">prev_map</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath.h"><span class="id" title="variable">h</span></a> (<a class="idref" href="mathcomp.ssreflect.path.html#prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#MapEqPath"><span class="id" title="section">MapEqPath</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="fun_base"><span class="id" title="definition">fun_base</span></a> (<span class="id" title="var">T</span> <span class="id" title="var">T'</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<span class="id" title="var">h</span> : <a class="idref" href="mathcomp.ssreflect.path.html#T'"><span class="id" title="variable">T'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">f</span> <span class="id" title="var">f'</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#rel_base"><span class="id" title="definition">rel_base</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#h"><span class="id" title="variable">h</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#frel"><span class="id" title="definition">frel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#f"><span class="id" title="variable">f</span></a>) (<a class="idref" href="mathcomp.ssreflect.eqtype.html#frel"><span class="id" title="definition">frel</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#f'"><span class="id" title="variable">f'</span></a>).<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="CycleArc"><span class="id" title="section">CycleArc</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="CycleArc.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#CycleArc.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="arc"><span class="id" title="definition">arc</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> := <span class="id" title="keyword">let</span> <span class="id" title="var">px</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#px"><span class="id" title="variable">px</span></a>) <a class="idref" href="mathcomp.ssreflect.path.html#px"><span class="id" title="variable">px</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="arc_rot"><span class="id" title="lemma">arc_rot</span></a> <span class="id" title="var">i</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#arc"><span class="id" title="definition">arc</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#86efabaf3ab961ac3d2bac35d2b3c35d"><span class="id" title="notation">=2</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#arc"><span class="id" title="definition">arc</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="left_arc"><span class="id" title="lemma">left_arc</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> (<span class="id" title="var">p</span> := <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#arc"><span class="id" title="definition">arc</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="right_arc"><span class="id" title="lemma">right_arc</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> (<span class="id" title="var">p</span> := <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#arc"><span class="id" title="definition">arc</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a>.<br/>

<br/>
<span class="id" title="keyword">CoInductive</span> <a name="rot_to_arc_spec"><span class="id" title="inductive">rot_to_arc_spec</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a name="RotToArcSpec"><span class="id" title="constructor">RotToArcSpec</span></a> <span class="id" title="var">i</span> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#arc"><span class="id" title="definition">arc</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&amp; <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#arc"><span class="id" title="definition">arc</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&amp; <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#2ac9001c05ad5bd2f6d5f68e59f48fbb"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p2"><span class="id" title="variable">p2</span></a> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.path.html#rot_to_arc_spec"><span class="id" title="inductive">rot_to_arc_spec</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="rot_to_arc"><span class="id" title="lemma">rot_to_arc</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#rot_to_arc_spec"><span class="id" title="inductive">rot_to_arc_spec</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#y"><span class="id" title="variable">y</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#CycleArc"><span class="id" title="section">CycleArc</span></a>.<br/>

<br/>

<br/>
</div>
</div>

<div id="footer">
<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
</div>

</div>

</body>
</html>