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diff --git a/docs/htmldoc/mathcomp.ssreflect.path.html b/docs/htmldoc/mathcomp.ssreflect.path.html deleted file mode 100644 index 39eba95..0000000 --- a/docs/htmldoc/mathcomp.ssreflect.path.html +++ /dev/null @@ -1,789 +0,0 @@ -<!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">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> - Distributed under the terms of CeCILL-B. *)</span><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 < 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 <-> ucycleb e c (ucycle e c is a Coercion target of type Prop) - ufcycle f c <-> 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 <-> 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 <-> 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 => [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/V8.9.0/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/V8.9.0/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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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="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/V8.9.0/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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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#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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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#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/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/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#00fe0eaf5e6949f0a31725357afa4bba"><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#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#407cde5b61fbf27196d1a7c5a475e083"><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/> - (<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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#c717e24548d7d4d1086535addc681262"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#c717e24548d7d4d1086535addc681262"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#c717e24548d7d4d1086535addc681262"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><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/V8.9.0/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/V8.9.0/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">Variant</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <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#93e0a78b945d3f9f22195c004c67aa36"><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/> - <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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#bda89d73ec4a8f23ae92b565ffb5aaa6"><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">Variant</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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">Variant</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <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#93e0a78b945d3f9f22195c004c67aa36"><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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/> - <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/> - | <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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#df45e8c2e8370fd4f0f7c4fdaf208180"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/> - | <span class="id" title="var">y'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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#df45e8c2e8370fd4f0f7c4fdaf208180"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">y'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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/> - | <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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#df45e8c2e8370fd4f0f7c4fdaf208180"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/> - | <span class="id" title="var">y'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <span class="id" title="var">y'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a><br/> - <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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a><br/> - <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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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/> -<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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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/> -</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/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/> - <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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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/> - <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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#93e0a78b945d3f9f22195c004c67aa36"><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#93e0a78b945d3f9f22195c004c67aa36"><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/> - <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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#93e0a78b945d3f9f22195c004c67aa36"><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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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#df45e8c2e8370fd4f0f7c4fdaf208180"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#506674b18256ef8f50efed43fa1dfd7d"><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#506674b18256ef8f50efed43fa1dfd7d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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#df45e8c2e8370fd4f0f7c4fdaf208180"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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#df45e8c2e8370fd4f0f7c4fdaf208180"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/> - <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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#93e0a78b945d3f9f22195c004c67aa36"><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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#93e0a78b945d3f9f22195c004c67aa36"><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">Variant</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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#407cde5b61fbf27196d1a7c5a475e083"><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#93e0a78b945d3f9f22195c004c67aa36"><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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">p'</span> <span class="id" title="keyword">then</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><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/> - <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variant</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <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> & <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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#subpred"><span class="id" title="definition">subpred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/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/V8.9.0/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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span> <span class="id" title="keyword">then</span><br/> - <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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s2'</span> <span class="id" title="keyword">then</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <span class="id" title="var">x1</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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/> - <a class="idref" href="mathcomp.ssreflect.path.html#merge_s1"><span class="id" title="variable">merge_s1</span></a><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#93e0a78b945d3f9f22195c004c67aa36"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#93e0a78b945d3f9f22195c004c67aa36"><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/> - <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/> - | <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">ss'</span> | <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><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#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">ss'</span><br/> - | <span class="id" title="var">s2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">ss'</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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#ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">[::</span></a> <span class="id" title="var">x1</span><a class="idref" href="mathcomp.ssreflect.seq.html#ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">,</span></a> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">&</span></a> <span class="id" title="var">s'</span><a class="idref" href="mathcomp.ssreflect.seq.html#ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">then</span><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">s1</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><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/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">[::</span></a> <span class="id" title="var">x1</span><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <span class="id" title="var">x2</span><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">[::</span></a> <span class="id" title="var">x2</span><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <span class="id" title="var">x1</span><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">]</span></a> <span class="id" title="tactic">in</span><br/> - <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/> - <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#0a934e621391740b862762275992e626"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><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.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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#bda89d73ec4a8f23ae92b565ffb5aaa6"><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#407cde5b61fbf27196d1a7c5a475e083"><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#0a934e621391740b862762275992e626"><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#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#407cde5b61fbf27196d1a7c5a475e083"><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#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#c717e24548d7d4d1086535addc681262"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><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#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</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/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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/V8.9.0/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/V8.9.0/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/V8.9.0/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/V8.9.0/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#df45e8c2e8370fd4f0f7c4fdaf208180"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/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/V8.9.0/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/V8.9.0/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/> - <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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/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/V8.9.0/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/V8.9.0/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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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/V8.9.0/stdlib//Coq.ssr.ssrfun.html#c717e24548d7d4d1086535addc681262"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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#407cde5b61fbf27196d1a7c5a475e083"><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#93e0a78b945d3f9f22195c004c67aa36"><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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#407cde5b61fbf27196d1a7c5a475e083"><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#407cde5b61fbf27196d1a7c5a475e083"><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#93e0a78b945d3f9f22195c004c67aa36"><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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#407cde5b61fbf27196d1a7c5a475e083"><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">Variant</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/> - <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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/> - & <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#407cde5b61fbf27196d1a7c5a475e083"><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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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/> - & <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/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#407cde5b61fbf27196d1a7c5a475e083"><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#93e0a78b945d3f9f22195c004c67aa36"><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#407cde5b61fbf27196d1a7c5a475e083"><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/> - <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/> - <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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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#c385a484ee9d1b4e0615924561a9b75e"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><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/> -<span class="id" title="keyword">Section</span> <a name="Monotonicity"><span class="id" title="section">Monotonicity</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Monotonicity.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="Monotonicity.r"><span class="id" title="variable">r</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/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="Monotonicity.r_trans"><span class="id" title="variable">r_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Monotonicity.r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sorted_lt_nth"><span class="id" title="lemma">sorted_lt_nth</span></a> <span class="id" title="var">x0</span> (<span class="id" title="var">s</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#Monotonicity.T"><span class="id" title="variable">T</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#Monotonicity.r"><span class="id" title="variable">r</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</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#00fe0eaf5e6949f0a31725357afa4bba"><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><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">homo</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#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">:</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><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#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Monotonicity.r"><span class="id" title="variable">r</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#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltn_index"><span class="id" title="lemma">ltn_index</span></a> (<span class="id" title="var">s</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#Monotonicity.T"><span class="id" title="variable">T</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#Monotonicity.r"><span class="id" title="variable">r</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><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">y</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#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</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#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Monotonicity.r"><span class="id" title="variable">r</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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Monotonicity.r_refl"><span class="id" title="variable">r_refl</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflexive"><span class="id" title="definition">reflexive</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Monotonicity.r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sorted_le_nth"><span class="id" title="lemma">sorted_le_nth</span></a> <span class="id" title="var">x0</span> (<span class="id" title="var">s</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#Monotonicity.T"><span class="id" title="variable">T</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#Monotonicity.r"><span class="id" title="variable">r</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</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#00fe0eaf5e6949f0a31725357afa4bba"><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><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">homo</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#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">:</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><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#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Monotonicity.r"><span class="id" title="variable">r</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#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#7f2550512fc22f98c0de9a527bb1f752"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="leq_index"><span class="id" title="lemma">leq_index</span></a> (<span class="id" title="var">s</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#Monotonicity.T"><span class="id" title="variable">T</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#Monotonicity.r"><span class="id" title="variable">r</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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><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">y</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#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</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#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#Monotonicity.r"><span class="id" title="variable">r</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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.path.html#Monotonicity"><span class="id" title="section">Monotonicity</span></a>.<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>
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