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| author | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
|---|---|---|
| committer | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
| commit | 6b59540a2460633df4e3d8347cb4dfe2fb3a3afb (patch) | |
| tree | 1239c1d5553d51a7d73f2f8b465f6a23178ff8a0 /docs/htmldoc/mathcomp.algebra.interval.html | |
| parent | dd82aaeae7e9478efc178ce8430986649555b032 (diff) | |
removing everything but index which redirects to the new page
Diffstat (limited to 'docs/htmldoc/mathcomp.algebra.interval.html')
| -rw-r--r-- | docs/htmldoc/mathcomp.algebra.interval.html | 661 |
1 files changed, 0 insertions, 661 deletions
diff --git a/docs/htmldoc/mathcomp.algebra.interval.html b/docs/htmldoc/mathcomp.algebra.interval.html deleted file mode 100644 index e0e16ed..0000000 --- a/docs/htmldoc/mathcomp.algebra.interval.html +++ /dev/null @@ -1,661 +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.algebra.interval</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.algebra.interval</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"> - This file provide support for intervals in numerical and real domains. - The datatype (interval R) gives a formal characterization of an - interval, as the pair of its right and left bounds. - interval R == the type of formal intervals on R. - x \in i == when i is a formal interval on a numeric domain, - \in can be used to test membership. - itvP x_in_i == where x_in_i has type x \in i, if i is ground, - gives a set of rewrite rules that x_in_i imply. - x <= y ?< if c == x is smaller than y, and strictly if c is true - -<div class="paragraph"> </div> - - We provide a set of notations to write intervals (see below) - ` [a, b], ` ]a, b], ..., ` ]-oo, a], ..., ` ]-oo, +oo[ - We also provide the lemma subitvP which computes the inequalities one - needs to prove when trying to prove the inclusion of intervals. - -<div class="paragraph"> </div> - - Remark that we cannot implement a boolean comparison test for intervals - on an arbitrary numeric domains, for this problem might be undecidable. - Note also that type (interval R) may contain several inhabitants coding - for the same interval. However, this pathological issues do nor arise - when R is a real domain: we could provide a specific theory for this - important case. - -<div class="paragraph"> </div> - - See also ``Formal proofs in real algebraic geometry: from ordered fields - to quantifier elimination'', LMCS journal, 2012 - by Cyril Cohen and Assia Mahboubi - -<div class="paragraph"> </div> - - And "Formalized algebraic numbers: construction and first-order theory" - Cyril Cohen, PhD, 2012, section 4.3. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> - -<br/> -<span class="id" title="keyword">Local Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">GRing.Theory</span> <span class="id" title="var">Num.Theory</span>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="LersifPo"><span class="id" title="section">LersifPo</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="LersifPo.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Exports.numDomainType"><span class="id" title="abbreviation">numDomainType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="lersif"><span class="id" title="definition">lersif</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#LersifPo.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">b</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.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subr_lersifr0"><span class="id" title="lemma">subr_lersifr0</span></a> <span class="id" title="var">b</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#LersifPo.R"><span class="id" title="variable">R</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.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="subr_lersif0r"><span class="id" title="lemma">subr_lersif0r</span></a> <span class="id" title="var">b</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#LersifPo.R"><span class="id" title="variable">R</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>0 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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">Definition</span> <a name="subr_lersif0"><span class="id" title="definition">subr_lersif0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#subr_lersifr0"><span class="id" title="lemma">subr_lersifr0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#subr_lersif0r"><span class="id" title="lemma">subr_lersif0r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_trans"><span class="id" title="lemma">lersif_trans</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif01"><span class="id" title="lemma">lersif01</span></a> <span class="id" title="var">b</span> : 0 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_anti"><span class="id" title="lemma">lersif_anti</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> - <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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</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">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#b2"><span class="id" title="variable">b2</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</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.algebra.interval.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.algebra.interval.html#y"><span class="id" title="variable">y</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersifxx"><span class="id" title="lemma">lersifxx</span></a> <span class="id" title="var">x</span> <span class="id" title="var">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersifNF"><span class="id" title="lemma">lersifNF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</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.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="lersifS"><span class="id" title="lemma">lersifS</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersifT"><span class="id" title="lemma">lersifT</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <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> <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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersifF"><span class="id" title="lemma">lersifF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</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> <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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.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>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_oppl"><span class="id" title="lemma">lersif_oppl</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="lersif_oppr"><span class="id" title="lemma">lersif_oppr</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="lersif_0oppr"><span class="id" title="lemma">lersif_0oppr</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="lersif_oppr0"><span class="id" title="lemma">lersif_oppr0</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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>0 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="lersif_opp2"><span class="id" title="lemma">lersif_opp2</span></a> <span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#a8ac36d488c8d5cdcfec5adcde894e5f"><span class="id" title="notation">-%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#a8ac36d488c8d5cdcfec5adcde894e5f"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">:</span></a> <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.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="lersif_oppE"><span class="id" title="definition">lersif_oppE</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#lersif_0oppr"><span class="id" title="lemma">lersif_0oppr</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#lersif_oppr0"><span class="id" title="lemma">lersif_oppr0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#lersif_opp2"><span class="id" title="lemma">lersif_opp2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_add2l"><span class="id" title="lemma">lersif_add2l</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#a87d5ea2e207e69e5e474db24f56d4cb"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#a87d5ea2e207e69e5e474db24f56d4cb"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#a87d5ea2e207e69e5e474db24f56d4cb"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">:</span></a> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_add2r"><span class="id" title="lemma">lersif_add2r</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#a87d5ea2e207e69e5e474db24f56d4cb"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#a87d5ea2e207e69e5e474db24f56d4cb"><span class="id" title="notation">R</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.algebra.interval.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#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">:</span></a> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="lersif_add2"><span class="id" title="definition">lersif_add2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#lersif_add2l"><span class="id" title="lemma">lersif_add2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#lersif_add2r"><span class="id" title="lemma">lersif_add2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_subl_addr"><span class="id" title="lemma">lersif_subl_addr</span></a> <span class="id" title="var">b</span> <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="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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="lersif_subr_addr"><span class="id" title="lemma">lersif_subr_addr</span></a> <span class="id" title="var">b</span> <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="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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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">Definition</span> <a name="lersif_sub_addr"><span class="id" title="definition">lersif_sub_addr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#lersif_subl_addr"><span class="id" title="lemma">lersif_subl_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#lersif_subr_addr"><span class="id" title="lemma">lersif_subr_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_subl_addl"><span class="id" title="lemma">lersif_subl_addl</span></a> <span class="id" title="var">b</span> <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="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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="lersif_subr_addl"><span class="id" title="lemma">lersif_subr_addl</span></a> <span class="id" title="var">b</span> <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="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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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">Definition</span> <a name="lersif_sub_addl"><span class="id" title="definition">lersif_sub_addl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#lersif_subl_addl"><span class="id" title="lemma">lersif_subl_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#lersif_subr_addl"><span class="id" title="lemma">lersif_subr_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_andb"><span class="id" title="lemma">lersif_andb</span></a> <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.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.interval.html#lersif"><span class="id" title="definition">lersif</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.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.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.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#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.algebra.interval.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_orb"><span class="id" title="lemma">lersif_orb</span></a> <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.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.interval.html#lersif"><span class="id" title="definition">lersif</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.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.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.interval.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#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.algebra.interval.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_imply"><span class="id" title="lemma">lersif_imply</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> <span class="id" title="var">r1</span> <span class="id" title="var">r2</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">==></span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#r1"><span class="id" title="variable">r1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#r2"><span class="id" title="variable">r2</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#r1"><span class="id" title="variable">r1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#r2"><span class="id" title="variable">r2</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersifW"><span class="id" title="lemma">lersifW</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltrW_lersif"><span class="id" title="lemma">ltrW_lersif</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_pmul2l"><span class="id" title="lemma">lersif_pmul2l</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">:</span></a> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_pmul2r"><span class="id" title="lemma">lersif_pmul2r</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">R</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.algebra.interval.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#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">:</span></a> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#02b248fee5f27b186ea3a36733c25088"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_nmul2l"><span class="id" title="lemma">lersif_nmul2l</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> 0 <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.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">:</span></a> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_nmul2r"><span class="id" title="lemma">lersif_nmul2r</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> 0 <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.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">R</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.algebra.interval.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#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">:</span></a> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#04245d0240efdac4719fcf73ee860591"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="real_lersifN"><span class="id" title="lemma">real_lersifN</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.interval.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#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">Num.real</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.algebra.interval.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#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">Num.real</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</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.algebra.interval.html#b"><span class="id" title="variable">b</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="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.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="real_lersif_norml"><span class="id" title="lemma">real_lersif_norml</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.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#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">Num.real</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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="real_lersif_normr"><span class="id" title="lemma">real_lersif_normr</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.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#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">Num.real</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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="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.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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#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#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_nnormr"><span class="id" title="lemma">lersif_nnormr</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</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.algebra.interval.html#b"><span class="id" title="variable">b</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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#82bf2ea71d9e8bc83fc9c1dc82554e8c"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.interval.html#LersifPo"><span class="id" title="section">LersifPo</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">"</span></a>x <= y ?< 'if' b" := (<a class="idref" href="mathcomp.algebra.interval.html#lersif"><span class="id" title="definition">lersif</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">b</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>,<br/> - <span class="id" title="var">format</span> "x '[hv' <= y '/' ?< 'if' b ']'") : <span class="id" title="var">ring_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="LersifOrdered"><span class="id" title="section">LersifOrdered</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="LersifOrdered.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Exports.realDomainType"><span class="id" title="abbreviation">realDomainType</span></a>) (<a name="LersifOrdered.b"><span class="id" title="variable">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a name="LersifOrdered.x"><span class="id" title="variable">x</span></a> <a name="LersifOrdered.y"><span class="id" title="variable">y</span></a> <a name="LersifOrdered.z"><span class="id" title="variable">z</span></a> <a name="LersifOrdered.e"><span class="id" title="variable">e</span></a> : <a class="idref" href="mathcomp.algebra.interval.html#R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersifN"><span class="id" title="lemma">lersifN</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.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</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.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_norml"><span class="id" title="lemma">lersif_norml</span></a> :<br/> - <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.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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="lersif_normr"><span class="id" title="lemma">lersif_normr</span></a> :<br/> - <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.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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#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#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_distl"><span class="id" title="lemma">lersif_distl</span></a> :<br/> - <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.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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="lersif_minr"><span class="id" title="lemma">lersif_minr</span></a> :<br/> - <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.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">Num.min</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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="lersif_minl"><span class="id" title="lemma">lersif_minl</span></a> :<br/> - <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.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">Num.min</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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#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#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_maxr"><span class="id" title="lemma">lersif_maxr</span></a> :<br/> - <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.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">Num.max</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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#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#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_maxl"><span class="id" title="lemma">lersif_maxl</span></a> :<br/> - <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.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">Num.max</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifOrdered.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered.b"><span class="id" title="variable">b</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">End</span> <a class="idref" href="mathcomp.algebra.interval.html#LersifOrdered"><span class="id" title="section">LersifOrdered</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="LersifField"><span class="id" title="section">LersifField</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="LersifField.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Exports.numFieldType"><span class="id" title="abbreviation">numFieldType</span></a>) (<a name="LersifField.b"><span class="id" title="variable">b</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a name="LersifField.z"><span class="id" title="variable">z</span></a> <a name="LersifField.x"><span class="id" title="variable">x</span></a> <a name="LersifField.y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.interval.html#F"><span class="id" title="variable">F</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_pdivl_mulr"><span class="id" title="lemma">lersif_pdivl_mulr</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#69c431a9c94f6f30a655bd7ddb59037b"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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="lersif_pdivr_mulr"><span class="id" title="lemma">lersif_pdivr_mulr</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#69c431a9c94f6f30a655bd7ddb59037b"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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="lersif_pdivl_mull"><span class="id" title="lemma">lersif_pdivl_mull</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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="lersif_pdivr_mull"><span class="id" title="lemma">lersif_pdivr_mull</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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="lersif_ndivl_mulr"><span class="id" title="lemma">lersif_ndivl_mulr</span></a> : <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> 0 <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.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#69c431a9c94f6f30a655bd7ddb59037b"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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="lersif_ndivr_mulr"><span class="id" title="lemma">lersif_ndivr_mulr</span></a> : <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> 0 <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.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#69c431a9c94f6f30a655bd7ddb59037b"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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="lersif_ndivl_mull"><span class="id" title="lemma">lersif_ndivl_mull</span></a> : <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> 0 <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.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a><a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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="lersif_ndivr_mull"><span class="id" title="lemma">lersif_ndivr_mull</span></a> : <a class="idref" href="mathcomp.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> 0 <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.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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.algebra.interval.html#LersifField.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#LersifField.b"><span class="id" title="variable">b</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">End</span> <a class="idref" href="mathcomp.algebra.interval.html#LersifField"><span class="id" title="section">LersifField</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="itv_bound"><span class="id" title="inductive">itv_bound</span></a> (<span class="id" title="var">T</span> : <span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> := <a name="BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> & <a class="idref" href="mathcomp.algebra.interval.html#T"><span class="id" title="variable">T</span></a> | <a name="BInfty"><span class="id" title="constructor">BInfty</span></a>.<br/> -<span class="id" title="keyword">Notation</span> <a name="BOpen"><span class="id" title="abbreviation">BOpen</span></a> := (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <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/> -<span class="id" title="keyword">Notation</span> <a name="BClose"><span class="id" title="abbreviation">BClose</span></a> := (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</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/> -<span class="id" title="keyword">Variant</span> <a name="interval"><span class="id" title="inductive">interval</span></a> (<span class="id" title="var">T</span> : <span class="id" title="keyword">Type</span>) := <a name="Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.interval.html#itv_bound"><span class="id" title="inductive">itv_bound</span></a> <a class="idref" href="mathcomp.algebra.interval.html#T"><span class="id" title="variable">T</span></a> & <a class="idref" href="mathcomp.algebra.interval.html#itv_bound"><span class="id" title="inductive">itv_bound</span></a> <a class="idref" href="mathcomp.algebra.interval.html#T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - We provide the 9 following notations to help writing formal intervals -</div> -<div class="code"> -<span class="id" title="keyword">Notation</span> <a name="3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">"</span></a>`[ a , b ]" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <span class="id" title="var">b</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">a</span>, <span class="id" title="var">b</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9 , <span class="id" title="var">format</span> "`[ a , b ]") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">"</span></a>`] a , b ]" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen"><span class="id" title="abbreviation">BOpen</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <span class="id" title="var">b</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">a</span>, <span class="id" title="var">b</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9 , <span class="id" title="var">format</span> "`] a , b ]") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="957313378acc4cb77b1d2f647eca85b3"><span class="id" title="notation">"</span></a>`[ a , b [" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen"><span class="id" title="abbreviation">BOpen</span></a> <span class="id" title="var">b</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">a</span>, <span class="id" title="var">b</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9 , <span class="id" title="var">format</span> "`[ a , b [") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">"</span></a>`] a , b [" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen"><span class="id" title="abbreviation">BOpen</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen"><span class="id" title="abbreviation">BOpen</span></a> <span class="id" title="var">b</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">a</span>, <span class="id" title="var">b</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9 , <span class="id" title="var">format</span> "`] a , b [") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="f0e10d9b9fd61169852de3c2ca19a38d"><span class="id" title="notation">"</span></a>`] '-oo' , b ]" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <span class="id" title="var">b</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">b</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9 , <span class="id" title="var">format</span> "`] '-oo' , b ]") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="f7d0eaaef362bfb530a5cbf036db1453"><span class="id" title="notation">"</span></a>`] '-oo' , b [" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen"><span class="id" title="abbreviation">BOpen</span></a> <span class="id" title="var">b</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">b</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9 , <span class="id" title="var">format</span> "`] '-oo' , b [") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="c5d1424554ae4f6d783beb7e2b68e8c1"><span class="id" title="notation">"</span></a>`[ a , '+oo' [" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> <span class="id" title="var">_</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">a</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9 , <span class="id" title="var">format</span> "`[ a , '+oo' [") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="1133a1612ce44fa1571c2c531aeb13a2"><span class="id" title="notation">"</span></a>`] a , '+oo' [" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen"><span class="id" title="abbreviation">BOpen</span></a> <span class="id" title="var">a</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> <span class="id" title="var">_</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">a</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9 , <span class="id" title="var">format</span> "`] a , '+oo' [") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">"</span></a>`] -oo , '+oo' [" := (<a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> <span class="id" title="var">_</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "`] -oo , '+oo' [") : <span class="id" title="var">ring_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IntervalEq"><span class="id" title="section">IntervalEq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="IntervalEq.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/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="eq_itv_bound"><span class="id" title="definition">eq_itv_bound</span></a> (<span class="id" title="var">b1</span> <span class="id" title="var">b2</span> : <a class="idref" href="mathcomp.algebra.interval.html#itv_bound"><span class="id" title="inductive">itv_bound</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalEq.T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a>, <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b</span> <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><span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <span class="id" title="var">b</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="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><span class="id" title="var">x</span> <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.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a>, <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> ⇒ <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/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <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/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_itv_boundP"><span class="id" title="lemma">eq_itv_boundP</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.axiom"><span class="id" title="definition">Equality.axiom</span></a> <a class="idref" href="mathcomp.algebra.interval.html#eq_itv_bound"><span class="id" title="definition">eq_itv_bound</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">itv_bound_eqMixin</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqMixin"><span class="id" title="abbreviation">EqMixin</span></a> <a class="idref" href="mathcomp.algebra.interval.html#eq_itv_boundP"><span class="id" title="lemma">eq_itv_boundP</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">itv_bound_eqType</span> :=<br/> - <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#itv_bound"><span class="id" title="inductive">itv_bound</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalEq.T"><span class="id" title="variable">T</span></a>) <a class="idref" href="mathcomp.algebra.interval.html#itv_bound_eqMixin"><span class="id" title="definition">itv_bound_eqMixin</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="eqitv"><span class="id" title="definition">eqitv</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalEq.T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">x</span> <span class="id" title="var">x'</span> := <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">y</span> <span class="id" title="var">y'</span> := <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <span class="id" title="tactic">in</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.algebra.interval.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.algebra.interval.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#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><span class="id" title="var">x'</span> <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.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqitvP"><span class="id" title="lemma">eqitvP</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.axiom"><span class="id" title="definition">Equality.axiom</span></a> <a class="idref" href="mathcomp.algebra.interval.html#eqitv"><span class="id" title="definition">eqitv</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">interval_eqMixin</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqMixin"><span class="id" title="abbreviation">EqMixin</span></a> <a class="idref" href="mathcomp.algebra.interval.html#eqitvP"><span class="id" title="lemma">eqitvP</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">interval_eqType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalEq.T"><span class="id" title="variable">T</span></a>) <a class="idref" href="mathcomp.algebra.interval.html#interval_eqMixin"><span class="id" title="definition">interval_eqMixin</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.interval.html#IntervalEq"><span class="id" title="section">IntervalEq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IntervalPo"><span class="id" title="section">IntervalPo</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="IntervalPo.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Exports.numDomainType"><span class="id" title="abbreviation">numDomainType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="pred_of_itv"><span class="id" title="definition">pred_of_itv</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.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#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a> :=<br/> - <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">x</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> <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">l</span> <span class="id" title="var">u</span> := <a class="idref" href="mathcomp.algebra.interval.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b</span> <span class="id" title="var">lb</span> ⇒ <span class="id" title="var">lb</span> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <span class="id" title="var">b</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> ⇒ <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/> - <span class="id" title="keyword">end</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><br/> - <span class="id" title="keyword">match</span> <span class="id" title="var">u</span> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b</span> <span class="id" title="var">ub</span> ⇒ <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <span class="id" title="var">ub</span> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <span class="id" title="var">b</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> ⇒ <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/> - <span class="id" title="keyword">end</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>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="keyword">Structure</span> <span class="id" title="var">itvPredType</span> := <a class="idref" href="mathcomp.ssreflect.ssrbool.html#PredType"><span class="id" title="definition">PredType</span></a> <a class="idref" href="mathcomp.algebra.interval.html#pred_of_itv"><span class="id" title="definition">pred_of_itv</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - we compute a set of rewrite rules associated to an interval -</div> -<div class="code"> -<span class="id" title="keyword">Definition</span> <a name="itv_rewrite"><span class="id" title="definition">itv_rewrite</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">x</span> : <span class="id" title="keyword">Type</span> :=<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">l</span> <span class="id" title="var">u</span> := <a class="idref" href="mathcomp.algebra.interval.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="tactic">in</span><br/> - (<span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/> - | <span class="id" title="var">BClose</span> <span class="id" title="var">a</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.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#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <span class="id" title="var">a</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a><br/> - | <span class="id" title="var">BOpen</span> <span class="id" title="var">a</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.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#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <span class="id" title="var">a</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <span class="id" title="var">a</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> ⇒ <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>, <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#x"><span class="id" title="variable">x</span></a><br/> - <span class="id" title="keyword">end</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><br/> - <span class="id" title="keyword">match</span> <span class="id" title="var">u</span> <span class="id" title="keyword">with</span><br/> - | <span class="id" title="var">BClose</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a><br/> - | <span class="id" title="var">BOpen</span> <span class="id" title="var">b</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.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.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="var">b</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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" 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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.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">`[</span></a><span class="id" title="var">a</span><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" 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title="var">b</span> <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.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">`[</span></a><span class="id" title="var">a</span><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" 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title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</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#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.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">`]</span></a><span class="id" title="var">a</span><a class="idref" href="mathcomp.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="mathcomp.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a><br/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>, <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#x"><span class="id" title="variable">x</span></a><br/> - <span class="id" title="keyword">end</span>)%<span class="id" title="keyword">type</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="itv_decompose"><span class="id" title="definition">itv_decompose</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">x</span> : <span class="id" title="keyword">Prop</span> :=<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">l</span> <span class="id" title="var">u</span> := <a class="idref" href="mathcomp.algebra.interval.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="tactic">in</span><br/> - (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b</span> <span class="id" title="var">lb</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">(</span></a><span class="id" title="var">lb</span> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">Prop</span></a><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/> - <span class="id" title="keyword">end</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">Prop</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> <span class="id" title="var">u</span> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b</span> <span class="id" title="var">ub</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <span class="id" title="var">ub</span> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">Prop</span></a><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/> - <span class="id" title="keyword">end</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">Prop</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>)%<span class="id" title="keyword">type</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_dec"><span class="id" title="lemma">itv_dec</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>),<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="mathcomp.algebra.interval.html#itv_decompose"><span class="id" title="definition">itv_decompose</span></a> <a class="idref" href="mathcomp.algebra.interval.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="le_boundl"><span class="id" title="definition">le_boundl</span></a> (<span class="id" title="var">b1</span> <span class="id" title="var">b2</span> : <a class="idref" href="mathcomp.algebra.interval.html#itv_bound"><span class="id" title="inductive">itv_bound</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a>, <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">x1</span>, <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b2</span> <span class="id" title="var">x2</span> ⇒ <span class="id" title="var">x1</span> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</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.algebra.interval.html#b2"><span class="id" title="variable">b2</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.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>, <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</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/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</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/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="le_boundr"><span class="id" title="definition">le_boundr</span></a> (<span class="id" title="var">b1</span> <span class="id" title="var">b2</span> : <a class="idref" href="mathcomp.algebra.interval.html#itv_bound"><span class="id" title="inductive">itv_bound</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a>, <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">x1</span>, <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">b2</span> <span class="id" title="var">x2</span> ⇒ <span class="id" title="var">x1</span> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</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.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a>, <a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ <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/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</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/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_boundlr"><span class="id" title="lemma">itv_boundlr</span></a> <span class="id" title="var">bl</span> <span class="id" title="var">br</span> <span class="id" title="var">x</span> :<br/> - <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.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bl"><span class="id" title="variable">bl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#br"><span class="id" title="variable">br</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><br/> - <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.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bl"><span class="id" title="variable">bl</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.algebra.interval.html#br"><span class="id" title="variable">br</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="le_boundl_refl"><span class="id" title="lemma">le_boundl_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.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">le_boundl_refl</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="le_boundr_refl"><span class="id" title="lemma">le_boundr_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.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">le_boundr_refl</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="le_boundl_trans"><span class="id" title="lemma">le_boundl_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.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="le_boundr_trans"><span class="id" title="lemma">le_boundr_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.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="le_boundl_bb"><span class="id" title="lemma">le_boundl_bb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="mathcomp.algebra.interval.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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">==></span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</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="le_boundr_bb"><span class="id" title="lemma">le_boundr_bb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="mathcomp.algebra.interval.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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">==></span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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="le_boundl_anti"><span class="id" title="lemma">le_boundl_anti</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</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.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</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.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</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="le_boundr_anti"><span class="id" title="lemma">le_boundr_anti</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</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.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</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.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</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.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</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="itv_xx"><span class="id" title="lemma">itv_xx</span></a> <span class="id" title="var">x</span> <span class="id" title="var">bl</span> <span class="id" title="var">br</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bl"><span class="id" title="variable">bl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#br"><span class="id" title="variable">br</span></a> <a class="idref" href="mathcomp.algebra.interval.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#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> <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="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="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.algebra.interval.html#bl"><span class="id" title="variable">bl</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.algebra.interval.html#br"><span class="id" title="variable">br</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="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.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.algebra.interval.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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred0"><span class="id" title="definition">pred0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_gte"><span class="id" title="lemma">itv_gte</span></a> <span class="id" title="var">ba</span> <span class="id" title="var">xa</span> <span class="id" title="var">bb</span> <span class="id" title="var">xb</span> : <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</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="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.algebra.interval.html#ba"><span class="id" title="variable">ba</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.algebra.interval.html#bb"><span class="id" title="variable">bb</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> <br/> - <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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred0"><span class="id" title="definition">pred0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="boundl_in_itv"><span class="id" title="lemma">boundl_in_itv</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">ba</span> <span class="id" title="var">xa</span> <span class="id" title="var">b</span>,<br/> - <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a>) <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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/> - <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.algebra.interval.html#ba"><span class="id" title="variable">ba</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</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.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a>) <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="boundr_in_itv"><span class="id" title="lemma">boundr_in_itv</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">bb</span> <span class="id" title="var">xb</span> <span class="id" title="var">a</span>,<br/> - <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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/> - <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.algebra.interval.html#bb"><span class="id" title="variable">bb</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</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.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BClose"><span class="id" title="abbreviation">BClose</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="bound_in_itv"><span class="id" title="definition">bound_in_itv</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#boundl_in_itv"><span class="id" title="lemma">boundl_in_itv</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#boundr_in_itv"><span class="id" title="lemma">boundr_in_itv</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itvP"><span class="id" title="lemma">itvP</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>), <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.interval.html#itv_rewrite"><span class="id" title="definition">itv_rewrite</span></a> <a class="idref" href="mathcomp.algebra.interval.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="itv_intersection"><span class="id" title="definition">itv_intersection</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a> :=<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">x</span> <span class="id" title="var">x'</span> := <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">y</span> <span class="id" title="var">y'</span> := <a class="idref" href="mathcomp.algebra.interval.html#y"><span class="id" title="variable">y</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</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">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.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.algebra.interval.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">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> <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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">x'</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>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="itv_intersection1i"><span class="id" title="definition">itv_intersection1i</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">`]-</span></a><a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">oo</span></a><a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">+</span></a><a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">oo</span></a><a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">[</span></a> <a class="idref" href="mathcomp.algebra.interval.html#itv_intersection"><span class="id" title="definition">itv_intersection</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="itv_intersectioni1"><span class="id" title="definition">itv_intersectioni1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">`]-</span></a><a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">oo</span></a><a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">+</span></a><a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">oo</span></a><a class="idref" href="mathcomp.algebra.interval.html#2d0f4f3a8486d97a7c68fd4d2f96ccac"><span class="id" title="notation">[</span></a> <a class="idref" href="mathcomp.algebra.interval.html#itv_intersection"><span class="id" title="definition">itv_intersection</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_intersectionii"><span class="id" title="lemma">itv_intersectionii</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#idempotent"><span class="id" title="definition">idempotent</span></a> <a class="idref" href="mathcomp.algebra.interval.html#itv_intersection"><span class="id" title="definition">itv_intersection</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="subitv"><span class="id" title="definition">subitv</span></a> (<span class="id" title="var">i1</span> <span class="id" title="var">i2</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.algebra.interval.html#i1"><span class="id" title="variable">i1</span></a>, <a class="idref" href="mathcomp.algebra.interval.html#i2"><span class="id" title="variable">i2</span></a> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">b1</span>, <a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <span class="id" title="var">a2</span> <span class="id" title="var">b2</span> ⇒ <a class="idref" href="mathcomp.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> <span class="id" title="var">a2</span> <span class="id" title="var">a1</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.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span><br/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subitvP"><span class="id" title="lemma">subitvP</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">i2</span> <span class="id" title="var">i1</span> : <a class="idref" href="mathcomp.algebra.interval.html#interval"><span class="id" title="inductive">interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>), <a class="idref" href="mathcomp.algebra.interval.html#subitv"><span class="id" title="definition">subitv</span></a> <a class="idref" href="mathcomp.algebra.interval.html#i1"><span class="id" title="variable">i1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#i2"><span class="id" title="variable">i2</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#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.algebra.interval.html#i1"><span class="id" title="variable">i1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#i2"><span class="id" title="variable">i2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subitvPr"><span class="id" title="lemma">subitvPr</span></a> (<span class="id" title="var">a</span> <span class="id" title="var">b1</span> <span class="id" title="var">b2</span> : <a class="idref" href="mathcomp.algebra.interval.html#itv_bound"><span class="id" title="inductive">itv_bound</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</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#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b1"><span class="id" title="variable">b1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b2"><span class="id" title="variable">b2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">)}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subitvPl"><span class="id" title="lemma">subitvPl</span></a> (<span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.interval.html#itv_bound"><span class="id" title="inductive">itv_bound</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a1"><span class="id" title="variable">a1</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#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">)}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lersif_in_itv"><span class="id" title="lemma">lersif_in_itv</span></a> <span class="id" title="var">ba</span> <span class="id" title="var">bb</span> <span class="id" title="var">xa</span> <span class="id" title="var">xb</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</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.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltr_in_itv"><span class="id" title="lemma">ltr_in_itv</span></a> <span class="id" title="var">ba</span> <span class="id" title="var">bb</span> <span class="id" title="var">xa</span> <span class="id" title="var">xb</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</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.algebra.interval.html#bb"><span class="id" title="variable">bb</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.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ler_in_itv"><span class="id" title="lemma">ler_in_itv</span></a> <span class="id" title="var">ba</span> <span class="id" title="var">bb</span> <span class="id" title="var">xa</span> <span class="id" title="var">xb</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem0_itvcc_xNx"><span class="id" title="lemma">mem0_itvcc_xNx</span></a> <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>0 <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.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">`[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem0_itvoo_xNx"><span class="id" title="lemma">mem0_itvoo_xNx</span></a> <span class="id" title="var">x</span> : 0 <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.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">`](</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">),</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><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>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_splitI"><span class="id" title="lemma">itv_splitI</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span>,<br/> - <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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/> - <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.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> <span class="id" title="var">_</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="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.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BInfty"><span class="id" title="constructor">BInfty</span></a> <span class="id" title="var">_</span>) <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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="oppr_itv"><span class="id" title="lemma">oppr_itv</span></a> <span class="id" title="var">ba</span> <span class="id" title="var">bb</span> (<span class="id" title="var">xa</span> <span class="id" title="var">xb</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</span></a>) :<br/> - <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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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> <br/> - <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.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a>)) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</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="oppr_itvoo"><span class="id" title="lemma">oppr_itvoo</span></a> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">`]</span></a><a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><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="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.algebra.interval.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.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">`](</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">),</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="oppr_itvco"><span class="id" title="lemma">oppr_itvco</span></a> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#957313378acc4cb77b1d2f647eca85b3"><span class="id" title="notation">`[</span></a><a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.interval.html#957313378acc4cb77b1d2f647eca85b3"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.interval.html#957313378acc4cb77b1d2f647eca85b3"><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="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.algebra.interval.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.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">`](</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">),</span></a> <a class="idref" href="mathcomp.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="oppr_itvoc"><span class="id" title="lemma">oppr_itvoc</span></a> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">`]</span></a><a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.interval.html#9ad860ccdfe75680d287151e25f339b9"><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="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.algebra.interval.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.algebra.interval.html#957313378acc4cb77b1d2f647eca85b3"><span class="id" title="notation">`[(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.interval.html#957313378acc4cb77b1d2f647eca85b3"><span class="id" title="notation">),</span></a> <a class="idref" href="mathcomp.algebra.interval.html#957313378acc4cb77b1d2f647eca85b3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.interval.html#957313378acc4cb77b1d2f647eca85b3"><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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="oppr_itvcc"><span class="id" title="lemma">oppr_itvcc</span></a> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo.R"><span class="id" title="variable">R</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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">`[</span></a><a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><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="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.algebra.interval.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.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">`[(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">),</span></a> <a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><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>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.interval.html#IntervalPo"><span class="id" title="section">IntervalPo</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IntervalOrdered"><span class="id" title="section">IntervalOrdered</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="IntervalOrdered.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Exports.realDomainType"><span class="id" title="abbreviation">realDomainType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="le_boundl_total"><span class="id" title="lemma">le_boundl_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.algebra.interval.html#le_boundl"><span class="id" title="definition">le_boundl</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="le_boundr_total"><span class="id" title="lemma">le_boundr_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.algebra.interval.html#le_boundr"><span class="id" title="definition">le_boundr</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_splitU"><span class="id" title="lemma">itv_splitU</span></a> (<span class="id" title="var">xc</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">bc</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.interval.html#xc"><span class="id" title="variable">xc</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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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">y</span>, <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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/> - <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.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</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.algebra.interval.html#bc"><span class="id" title="variable">bc</span></a>) <a class="idref" href="mathcomp.algebra.interval.html#xc"><span class="id" title="variable">xc</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><br/> - <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#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bc"><span class="id" title="variable">bc</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xc"><span class="id" title="variable">xc</span></a>) <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_splitU2"><span class="id" title="lemma">itv_splitU2</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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">y</span>, <a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</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/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> <a class="idref" href="mathcomp.algebra.interval.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen"><span class="id" title="abbreviation">BOpen</span></a> <a class="idref" href="mathcomp.algebra.interval.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#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.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#6e3f5b731a46299b833a2834f381d536"><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#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.interval.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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen"><span class="id" title="abbreviation">BOpen</span></a> <a class="idref" href="mathcomp.algebra.interval.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.algebra.interval.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">)]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_intersectionC"><span class="id" title="lemma">itv_intersectionC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> (@<a class="idref" href="mathcomp.algebra.interval.html#itv_intersection"><span class="id" title="definition">itv_intersection</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="itv_intersectionA"><span class="id" title="lemma">itv_intersectionA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> (@<a class="idref" href="mathcomp.algebra.interval.html#itv_intersection"><span class="id" title="definition">itv_intersection</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">itv_intersection_monoid</span> :=<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.Law"><span class="id" title="constructor">Monoid.Law</span></a> <a class="idref" href="mathcomp.algebra.interval.html#itv_intersectionA"><span class="id" title="lemma">itv_intersectionA</span></a> (@<a class="idref" href="mathcomp.algebra.interval.html#itv_intersection1i"><span class="id" title="definition">itv_intersection1i</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered.R"><span class="id" title="variable">R</span></a>) (@<a class="idref" href="mathcomp.algebra.interval.html#itv_intersectioni1"><span class="id" title="definition">itv_intersectioni1</span></a> <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">itv_intersection_comoid</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.ComLaw"><span class="id" title="constructor">Monoid.ComLaw</span></a> <a class="idref" href="mathcomp.algebra.interval.html#itv_intersectionC"><span class="id" title="lemma">itv_intersectionC</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.interval.html#IntervalOrdered"><span class="id" title="section">IntervalOrdered</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IntervalField"><span class="id" title="section">IntervalField</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="IntervalField.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Exports.realFieldType"><span class="id" title="abbreviation">realFieldType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mid_in_itv"><span class="id" title="lemma">mid_in_itv</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">ba</span> <span class="id" title="var">bb</span> (<span class="id" title="var">xa</span> <span class="id" title="var">xb</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalField.R"><span class="id" title="variable">R</span></a>), <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">?<</span></a> <a class="idref" href="mathcomp.algebra.interval.html#2bafcba5e020e03f24077f5d83dfd2f0"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</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.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a><br/> - <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.algebra.interval.html#mid"><span class="id" title="abbreviation">mid</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#Interval"><span class="id" title="constructor">Interval</span></a> (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#ba"><span class="id" title="variable">ba</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a>) (<a class="idref" href="mathcomp.algebra.interval.html#BOpen_if"><span class="id" title="constructor">BOpen_if</span></a> <a class="idref" href="mathcomp.algebra.interval.html#bb"><span class="id" title="variable">bb</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mid_in_itvoo"><span class="id" title="lemma">mid_in_itvoo</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">xa</span> <span class="id" title="var">xb</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalField.R"><span class="id" title="variable">R</span></a>), <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#mid"><span class="id" title="abbreviation">mid</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">`]</span></a><a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a><a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a><a class="idref" href="mathcomp.algebra.interval.html#ebff9f8c4fe58806999ff1c12ccee445"><span class="id" title="notation">[</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mid_in_itvcc"><span class="id" title="lemma">mid_in_itvcc</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">xa</span> <span class="id" title="var">xb</span> : <a class="idref" href="mathcomp.algebra.interval.html#IntervalField.R"><span class="id" title="variable">R</span></a>), <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#mid"><span class="id" title="abbreviation">mid</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</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.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">`[</span></a><a class="idref" href="mathcomp.algebra.interval.html#xa"><span class="id" title="variable">xa</span></a><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.interval.html#xb"><span class="id" title="variable">xb</span></a><a class="idref" href="mathcomp.algebra.interval.html#3faa329f2bb2903beba94067ccb07f1b"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.interval.html#IntervalField"><span class="id" title="section">IntervalField</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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