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diff --git a/docs/htmldoc/mathcomp.algebra.ssrnum.html b/docs/htmldoc/mathcomp.algebra.ssrnum.html new file mode 100644 index 0000000..aee03a4 --- /dev/null +++ b/docs/htmldoc/mathcomp.algebra.ssrnum.html @@ -0,0 +1,4659 @@ +<!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.ssrnum</title> +</head> + +<body> + +<div id="page"> + +<div id="header"> +</div> + +<div id="main"> + +<h1 class="libtitle">Library mathcomp.algebra.ssrnum</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/> +<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + +<div class="paragraph"> </div> + + This file defines some classes to manipulate number structures, i.e + structures with an order and a norm + +<div class="paragraph"> </div> + +<a name="lab21"></a><h1 class="section">NumDomain (Integral domain with an order and a norm)</h1> + + NumMixin == the mixin that provides an order and a norm over + a ring and their characteristic properties. + numDomainType == interface for a num integral domain. + NumDomainType T m + == packs the num mixin into a numberDomainType. The + carrier T must have a integral domain structure. + [numDomainType of T for S ] + == T-clone of the numDomainType structure S. + [numDomainType of T] + == clone of a canonical numDomainType structure on T. + +<div class="paragraph"> </div> + +<a name="lab22"></a><h1 class="section">NumField (Field with an order and a norm)</h1> + + numFieldType == interface for a num field. + [numFieldType of T] + == clone of a canonical numFieldType structure on T + +<div class="paragraph"> </div> + +<a name="lab23"></a><h1 class="section">NumClosedField (Closed Field with an order and a norm)</h1> + + numClosedFieldType + == interface for a num closed field. + [numClosedFieldType of T] + == clone of a canonical numClosedFieldType structure on T + +<div class="paragraph"> </div> + +<a name="lab24"></a><h1 class="section">RealDomain (Num domain where all elements are positive or negative)</h1> + + realDomainType == interface for a real integral domain. + RealDomainType T r + == packs the real axiom r into a realDomainType. The + carrier T must have a num domain structure. + [realDomainType of T for S ] + == T-clone of the realDomainType structure S. + [realDomainType of T] + == clone of a canonical realDomainType structure on T. + +<div class="paragraph"> </div> + +<a name="lab25"></a><h1 class="section">RealField (Num Field where all elements are positive or negative)</h1> + + realFieldType == interface for a real field. + [realFieldType of T] + == clone of a canonical realFieldType structure on T + +<div class="paragraph"> </div> + +<a name="lab26"></a><h1 class="section">ArchiField (A Real Field with the archimedean axiom)</h1> + + archiFieldType == interface for an archimedean field. + ArchiFieldType T r + == packs the archimeadean axiom r into an archiFieldType. + The carrier T must have a real field type structure. + [archiFieldType of T for S ] + == T-clone of the archiFieldType structure S. + [archiFieldType of T] + == clone of a canonical archiFieldType structure on T + +<div class="paragraph"> </div> + +<a name="lab27"></a><h1 class="section">RealClosedField (Real Field with the real closed axiom)</h1> + + rcfType == interface for a real closed field. + RcfType T r == packs the real closed axiom r into a + rcfType. The carrier T must have a real + field type structure. + [rcfType of T] == clone of a canonical realClosedFieldType structure on + T. + [rcfType of T for S ] + == T-clone of the realClosedFieldType structure S. + +<div class="paragraph"> </div> + +<a name="lab28"></a><h1 class="section">NumClosedField (Partially ordered Closed Field with conjugation)</h1> + + numClosedFieldType == interface for a closed field with conj. + NumClosedFieldType T r == packs the real closed axiom r into a + numClosedFieldType. The carrier T must have a closed + field type structure. + [numClosedFieldType of T] == clone of a canonical numClosedFieldType + structure on T + [numClosedFieldType of T for S ] + == T-clone of the realClosedFieldType structure S. + +<div class="paragraph"> </div> + + Over these structures, we have the following operations + `|x| == norm of x. + x <= y <=> x is less than or equal to y (:= '|y - x| == y - x). + x < y <=> x is less than y (:= (x <= y) && (x != y)). + x <= y ?= iff C <-> x is less than y, or equal iff C is true. + Num.sg x == sign of x: equal to 0 iff x = 0, to 1 iff x > 0, and + to -1 in all other cases (including x < 0). + x \is a Num.pos <=> x is positive (:= x > 0). + x \is a Num.neg <=> x is negative (:= x < 0). + x \is a Num.nneg <=> x is positive or 0 (:= x >= 0). + x \is a Num.real <=> x is real (:= x >= 0 or x < 0). + Num.min x y == minimum of x y + Num.max x y == maximum of x y + Num.bound x == in archimedean fields, and upper bound for x, i.e., + and n such that `|x| < n%:R. + Num.sqrt x == in a real-closed field, a positive square root of x if + x >= 0, or 0 otherwise. + For numeric algebraically closed fields we provide the generic definitions + 'i == the imaginary number (:= sqrtC (-1)). + 'Re z == the real component of z. + 'Im z == the imaginary component of z. + z^* == the complex conjugate of z (:= conjC z). + sqrtC z == a nonnegative square root of z, i.e., 0 <= sqrt x if 0 <= x. + n.-root z == more generally, for n > 0, an nth root of z, chosen with a + minimal non-negative argument for n > 1 (i.e., with a + maximal real part subject to a nonnegative imaginary part). + Note that n.-root (-1) is a primitive 2nth root of unity, + an thus not equal to -1 for n odd > 1 (this will be shown in + file cyclotomic.v). + +<div class="paragraph"> </div> + + There are now three distinct uses of the symbols <, <=, > and >=: + 0-ary, unary (prefix) and binary (infix). + 0. <%R, <=%R, >%R, >=%R stand respectively for lt, le, gt and ge. + 1. (< x), (<= x), (> x), (>= x) stand respectively for + (gt x), (ge x), (lt x), (le x). + So (< x) is a predicate characterizing elements smaller than x. + 2. (x < y), (x <= y), ... mean what they are expected to. + These convention are compatible with haskell's, + where ((< y) x) = (x < y) = ((<) x y), + except that we write <%R instead of (<). + +<div class="paragraph"> </div> + +<ul class="doclist"> +<li> list of prefixes : + p : positive + n : negative + sp : strictly positive + sn : strictly negative + i : interior = in [0, 1] or ]0, 1[ + e : exterior = in [1, +oo[ or ]1; +oo[ + w : non strict (weak) monotony + +</li> +</ul> + +</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>.<br/> + +<br/> +<span class="id" title="keyword">Reserved Notation</span> "<= y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35).<br/> +<span class="id" title="keyword">Reserved Notation</span> ">= y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35).<br/> +<span class="id" title="keyword">Reserved Notation</span> "< y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35).<br/> +<span class="id" title="keyword">Reserved Notation</span> "> y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35).<br/> +<span class="id" title="keyword">Reserved Notation</span> "<= y :> T" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35, <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="keyword">Reserved Notation</span> ">= y :> T" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35, <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="keyword">Reserved Notation</span> "< y :> T" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35, <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="keyword">Reserved Notation</span> "> y :> T" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35, <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/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num"><span class="id" title="module">Num</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Principal mixin; further classes add axioms rather than operations. +</div> +<div class="code"> +<span class="id" title="keyword">Record</span> <a name="Num.mixin_of"><span class="id" title="record">mixin_of</span></a> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>) := <a name="Num.Mixin"><span class="id" title="constructor">Mixin</span></a> {<br/> + <a name="Num.norm_op"><span class="id" title="projection">norm_op</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>;<br/> + <a name="Num.le_op"><span class="id" title="projection">le_op</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>;<br/> + <a name="Num.lt_op"><span class="id" title="projection">lt_op</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>;<br/> + <span class="id" title="var">_</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#le_op"><span class="id" title="method">le_op</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#norm_op"><span class="id" title="method">norm_op</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>)) (<a class="idref" href="mathcomp.algebra.ssrnum.html#norm_op"><span class="id" title="method">norm_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#norm_op"><span class="id" title="method">norm_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>);<br/> + <span class="id" title="var">_</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#lt_op"><span class="id" title="method">lt_op</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#lt_op"><span class="id" title="method">lt_op</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#lt_op"><span class="id" title="method">lt_op</span></a> 0 (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>);<br/> + <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.ssrnum.html#norm_op"><span class="id" title="method">norm_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0;<br/> + <span class="id" title="var">_</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#le_op"><span class="id" title="method">le_op</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#le_op"><span class="id" title="method">le_op</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#le_op"><span class="id" title="method">le_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#le_op"><span class="id" title="method">le_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>;<br/> + <span class="id" title="var">_</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#norm_op"><span class="id" title="method">norm_op</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>;<br/> + <span class="id" title="var">_</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#le_op"><span class="id" title="method">le_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#norm_op"><span class="id" title="method">norm_op</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>;<br/> + <span class="id" title="var">_</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#lt_op"><span class="id" title="method">lt_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#le_op"><span class="id" title="method">le_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a><br/> +}.<br/> + +<br/> + +<br/> +</div> + +<div class="doc"> + Base interface. +</div> +<div class="code"> +<span class="id" title="keyword">Module</span> <a name="Num.NumDomain"><span class="id" title="module">NumDomain</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.NumDomain.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Record</span> <a name="Num.NumDomain.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">T</span> := <a name="Num.NumDomain.Class"><span class="id" title="constructor">Class</span></a> {<br/> + <a name="Num.NumDomain.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.class_of"><span class="id" title="record">GRing.IntegralDomain.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#T"><span class="id" title="variable">T</span></a>;<br/> + <a name="Num.NumDomain.mixin"><span class="id" title="projection">mixin</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.mixin_of"><span class="id" title="record">mixin_of</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ring_for"><span class="id" title="abbreviation">ring_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#base"><span class="id" title="method">base</span></a>)<br/> +}.<br/> +<span class="id" title="keyword">Structure</span> <a name="Num.NumDomain.type"><span class="id" title="record">type</span></a> := <a name="Num.NumDomain.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Num.NumDomain.sort"><span class="id" title="projection">sort</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.NumDomain.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.type"><span class="id" title="record">type</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/> +<span class="id" title="keyword">Let</span> <a name="Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">c</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">b0</span> (<span class="id" title="var">m0</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.mixin_of"><span class="id" title="record">mixin_of</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ring_for"><span class="id" title="abbreviation">ring_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b0"><span class="id" title="variable">b0</span></a>)) :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">bT</span> <span class="id" title="var">b</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.class"><span class="id" title="definition">GRing.IntegralDomain.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#bT"><span class="id" title="variable">bT</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> ⇒<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m0"><span class="id" title="variable">m0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.choiceType"><span class="id" title="definition">choiceType</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.zmodType"><span class="id" title="definition">zmodType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Pack"><span class="id" title="constructor">GRing.Zmodule.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.ringType"><span class="id" title="definition">ringType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Pack"><span class="id" title="constructor">GRing.Ring.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.comRingType"><span class="id" title="definition">comRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Pack"><span class="id" title="constructor">GRing.ComRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.unitRingType"><span class="id" title="definition">unitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Pack"><span class="id" title="constructor">GRing.UnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Pack"><span class="id" title="constructor">GRing.ComUnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumDomain.idomainType"><span class="id" title="definition">idomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Pack"><span class="id" title="constructor">GRing.IntegralDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.NumDomain.Exports"><span class="id" title="module">Exports</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.base"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.base"><span class="id" title="projection">GRing.IntegralDomain.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.mixin"><span class="id" title="projection">mixin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.mixin"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.mixin"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.mixin"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.mixin"><span class="id" title="projection">mixin_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.sort"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.sort"><span class="id" title="projection">Sortclass</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.eqType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.choiceType"><span class="id" title="definition">choiceType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.choiceType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.choiceType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.choiceType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.choiceType"><span class="id" title="definition">Choice.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">choiceType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.zmodType"><span class="id" title="definition">zmodType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.zmodType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.zmodType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.zmodType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.zmodType"><span class="id" title="definition">GRing.Zmodule.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">zmodType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ringType"><span class="id" title="definition">ringType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ringType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ringType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ringType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.ringType"><span class="id" title="definition">GRing.Ring.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ringType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comRingType"><span class="id" title="definition">comRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comRingType"><span class="id" title="definition">GRing.ComRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.unitRingType"><span class="id" title="definition">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.unitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.unitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.unitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.unitRingType"><span class="id" title="definition">GRing.UnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comUnitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comUnitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comUnitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.comUnitRingType"><span class="id" title="definition">GRing.ComUnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comUnitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.idomainType"><span class="id" title="definition">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.idomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.idomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.idomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.idomainType"><span class="id" title="definition">GRing.IntegralDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">idomainType</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.NumDomain.Exports.numDomainType"><span class="id" title="abbreviation">numDomainType</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.type"><span class="id" title="record">type</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.NumDomain.Exports.NumMixin"><span class="id" title="abbreviation">NumMixin</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Mixin"><span class="id" title="constructor">Mixin</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.NumDomain.Exports.NumDomainType"><span class="id" title="abbreviation">NumDomainType</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">m</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>).<br/> +<span class="id" title="keyword">Notation</span> <a name="dfa3a41435329603f4a41bbb73d70957"><span class="id" title="notation">"</span></a>[ 'numDomainType' 'of' T 'for' cT ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">cT</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'numDomainType' 'of' T 'for' cT ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="faa7b03f15fa8c0b383b6f3802b37e9e"><span class="id" title="notation">"</span></a>[ 'numDomainType' 'of' T ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'numDomainType' 'of' T ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Exports"><span class="id" title="module">Exports</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain"><span class="id" title="module">NumDomain</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">NumDomain.Exports</span>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="Num.Def"><span class="id" title="module">Def</span></a>. <span class="id" title="keyword">Section</span> <a name="Num.Def.Def"><span class="id" title="section">Def</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">NumDomain</span>.<br/> +<span class="id" title="keyword">Context</span> {<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.type"><span class="id" title="record">type</span></a>}.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.normr"><span class="id" title="definition">normr</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.norm_op"><span class="id" title="projection">norm_op</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.ler"><span class="id" title="definition">ler</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.le_op"><span class="id" title="projection">le_op</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.ltr"><span class="id" title="definition">ltr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.lt_op"><span class="id" title="projection">lt_op</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.ger"><span class="id" title="definition">ger</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#simpl_rel"><span class="id" title="definition">simpl_rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#82abed2919e25bf64cb56067d0f5feee"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#82abed2919e25bf64cb56067d0f5feee"><span class="id" title="notation">rel</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/8.8.0/stdlib//Coq.ssr.ssrbool.html#82abed2919e25bf64cb56067d0f5feee"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#2038926f673d4ab3e13573d88721ef3c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#82abed2919e25bf64cb56067d0f5feee"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.gtr"><span class="id" title="definition">gtr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#simpl_rel"><span class="id" title="definition">simpl_rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#82abed2919e25bf64cb56067d0f5feee"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#82abed2919e25bf64cb56067d0f5feee"><span class="id" title="notation">rel</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/8.8.0/stdlib//Coq.ssr.ssrbool.html#82abed2919e25bf64cb56067d0f5feee"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e24e94b919f2f836c2c853c0a739656b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#82abed2919e25bf64cb56067d0f5feee"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.lerif"><span class="id" title="definition">lerif</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">C</span> : <span class="id" title="keyword">Prop</span> := (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#2038926f673d4ab3e13573d88721ef3c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>)%<span class="id" title="keyword">type</span>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.sgr"><span class="id" title="definition">sgr</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e24e94b919f2f836c2c853c0a739656b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> -1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> 1.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.minr"><span class="id" title="definition">minr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#2038926f673d4ab3e13573d88721ef3c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.maxr"><span class="id" title="definition">maxr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#2038926f673d4ab3e13573d88721ef3c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.Rpos"><span class="id" title="definition">Rpos</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#qualifier"><span class="id" title="inductive">qualifier</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">qualify</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">|</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#e24e94b919f2f836c2c853c0a739656b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.Rneg"><span class="id" title="definition">Rneg</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#qualifier"><span class="id" title="inductive">qualifier</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">qualify</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e24e94b919f2f836c2c853c0a739656b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.Rnneg"><span class="id" title="definition">Rnneg</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#qualifier"><span class="id" title="inductive">qualifier</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">qualify</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">|</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#2038926f673d4ab3e13573d88721ef3c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Def.Rreal"><span class="id" title="definition">Rreal</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#qualifier"><span class="id" title="inductive">qualifier</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">qualify</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#2038926f673d4ab3e13573d88721ef3c"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#2038926f673d4ab3e13573d88721ef3c"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e542d34820d5d916bfaf26ab326922d1"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.Def"><span class="id" title="section">Def</span></a>. <span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def"><span class="id" title="module">Def</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Shorter qualified names, when Num.Def is not imported. +</div> +<div class="code"> +<span class="id" title="keyword">Notation</span> <a name="Num.norm"><span class="id" title="abbreviation">norm</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.normr"><span class="id" title="definition">normr</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.le"><span class="id" title="abbreviation">le</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ler"><span class="id" title="definition">ler</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.lt"><span class="id" title="abbreviation">lt</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ltr"><span class="id" title="definition">ltr</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.ge"><span class="id" title="abbreviation">ge</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ger"><span class="id" title="definition">ger</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.gt"><span class="id" title="abbreviation">gt</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.gtr"><span class="id" title="definition">gtr</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.sg"><span class="id" title="abbreviation">sg</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sgr"><span class="id" title="definition">sgr</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.max"><span class="id" title="abbreviation">max</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.maxr"><span class="id" title="definition">maxr</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.min"><span class="id" title="abbreviation">min</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.minr"><span class="id" title="definition">minr</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.pos"><span class="id" title="abbreviation">pos</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Rpos"><span class="id" title="definition">Rpos</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.neg"><span class="id" title="abbreviation">neg</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Rneg"><span class="id" title="definition">Rneg</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.nneg"><span class="id" title="abbreviation">nneg</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Rnneg"><span class="id" title="definition">Rnneg</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.real"><span class="id" title="abbreviation">real</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Rreal"><span class="id" title="definition">Rreal</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.Keys"><span class="id" title="module">Keys</span></a>. <span class="id" title="keyword">Section</span> <a name="Num.Keys.Keys"><span class="id" title="section">Keys</span></a>.<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Keys.Keys.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/> +<span class="id" title="keyword">Fact</span> <a name="Num.Keys.Rpos_key"><span class="id" title="lemma">Rpos_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_key"><span class="id" title="inductive">pred_key</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.pos"><span class="id" title="abbreviation">pos</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Keys.R"><span class="id" title="variable">R</span></a>). <br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Keys.Rpos_keyed"><span class="id" title="definition">Rpos_keyed</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#KeyedQualifier"><span class="id" title="definition">KeyedQualifier</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Rpos_key"><span class="id" title="lemma">Rpos_key</span></a>.<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Keys.Rneg_key"><span class="id" title="lemma">Rneg_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_key"><span class="id" title="inductive">pred_key</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Keys.R"><span class="id" title="variable">R</span></a>). <br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Keys.Rneg_keyed"><span class="id" title="definition">Rneg_keyed</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#KeyedQualifier"><span class="id" title="definition">KeyedQualifier</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Rneg_key"><span class="id" title="lemma">Rneg_key</span></a>.<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Keys.Rnneg_key"><span class="id" title="lemma">Rnneg_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_key"><span class="id" title="inductive">pred_key</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Keys.R"><span class="id" title="variable">R</span></a>). <br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Keys.Rnneg_keyed"><span class="id" title="definition">Rnneg_keyed</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#KeyedQualifier"><span class="id" title="definition">KeyedQualifier</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Rnneg_key"><span class="id" title="lemma">Rnneg_key</span></a>.<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Keys.Rreal_key"><span class="id" title="lemma">Rreal_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_key"><span class="id" title="inductive">pred_key</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Keys.R"><span class="id" title="variable">R</span></a>). <br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Keys.Rreal_keyed"><span class="id" title="definition">Rreal_keyed</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#KeyedQualifier"><span class="id" title="definition">KeyedQualifier</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Rreal_key"><span class="id" title="lemma">Rreal_key</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Keys.ler_of_leif"><span class="id" title="definition">ler_of_leif</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">C</span> (<span class="id" title="var">le_xy</span> : @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.lerif"><span class="id" title="definition">lerif</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Keys.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>) := <a class="idref" href="mathcomp.algebra.ssrnum.html#le_xy"><span class="id" title="variable">le_xy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.Keys"><span class="id" title="section">Keys</span></a>. <span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys"><span class="id" title="module">Keys</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + (Exported) symbolic syntax. +</div> +<div class="code"> +<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="Num.Syntax"><span class="id" title="module">Syntax</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">Def</span> <span class="id" title="var">Keys</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">"</span></a>`| x |" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.norm"><span class="id" title="abbreviation">norm</span></a> <span class="id" title="var">x</span>) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="1d9613d915748583958d042a98aee792"><span class="id" title="notation">"</span></a><%R" := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.lt"><span class="id" title="abbreviation">lt</span></a> : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="4f222b0e87b56a0bb524cc002c4daf40"><span class="id" title="notation">"</span></a>>%R" := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.gt"><span class="id" title="abbreviation">gt</span></a> : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="81937a94685a0487cf97a240746fb002"><span class="id" title="notation">"</span></a><=%R" := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.le"><span class="id" title="abbreviation">le</span></a> : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="186aaf7909c11e850d63c2993181ccc7"><span class="id" title="notation">"</span></a>>=%R" := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ge"><span class="id" title="abbreviation">ge</span></a> : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="6fd1556fdb6e0b11ebd82189f1bfb36f"><span class="id" title="notation">"</span></a><?=%R" := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.lerif"><span class="id" title="definition">lerif</span></a> : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="a8f34bc5e467f7971380049ef258ede0"><span class="id" title="notation">"</span></a>< y" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.gt"><span class="id" title="abbreviation">gt</span></a> <span class="id" title="var">y</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="59d3ae83284e7d320c78efee96f330f6"><span class="id" title="notation">"</span></a>< y :> T" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#a8f34bc5e467f7971380049ef258ede0"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a8f34bc5e467f7971380049ef258ede0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#a8f34bc5e467f7971380049ef258ede0"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="a5bfa1108ca863ae15611f2aac402243"><span class="id" title="notation">"</span></a>> y" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.lt"><span class="id" title="abbreviation">lt</span></a> <span class="id" title="var">y</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="cbbb7fa9128771701eefac0819f9b044"><span class="id" title="notation">"</span></a>> y :> T" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#a5bfa1108ca863ae15611f2aac402243"><span class="id" title="notation">></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a5bfa1108ca863ae15611f2aac402243"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#a5bfa1108ca863ae15611f2aac402243"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="e1cffdfd4384e2a91d841324fdf3cf74"><span class="id" title="notation">"</span></a><= y" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ge"><span class="id" title="abbreviation">ge</span></a> <span class="id" title="var">y</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="13d5330a4568f6ceffcf63013bcd1f20"><span class="id" title="notation">"</span></a><= y :> T" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#e1cffdfd4384e2a91d841324fdf3cf74"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e1cffdfd4384e2a91d841324fdf3cf74"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#e1cffdfd4384e2a91d841324fdf3cf74"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="23c398dbba06a0aa4819b2d9e2ed5abb"><span class="id" title="notation">"</span></a>>= y" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.le"><span class="id" title="abbreviation">le</span></a> <span class="id" title="var">y</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="6f89ccb02a8f0a5ad1eea6160e3c21ef"><span class="id" title="notation">"</span></a>>= y :> T" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#23c398dbba06a0aa4819b2d9e2ed5abb"><span class="id" title="notation">≥</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#23c398dbba06a0aa4819b2d9e2ed5abb"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#23c398dbba06a0aa4819b2d9e2ed5abb"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation">"</span></a>x < y" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.lt"><span class="id" title="abbreviation">lt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">"</span></a>x < y :> T" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">"</span></a>x > y" := (<span class="id" title="var">y</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="f54d2e0a46f272d0295ade87cec65608"><span class="id" title="notation">"</span></a>x > y :> T" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="1065783963a393d1eafa2137291f2495"><span class="id" title="notation">"</span></a>x <= y" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.le"><span class="id" title="abbreviation">le</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">"</span></a>x <= y :> T" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">"</span></a>x >= y" := (<span class="id" title="var">y</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="3975356ead321cf4577de4738f745485"><span class="id" title="notation">"</span></a>x >= y :> T" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">≥</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span><a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">"</span></a>x <= y <= z" := (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="cbae941009021cd5693066355a023dd1"><span class="id" title="notation">"</span></a>x < y <= z" := (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="93a29a58d1f90b8a91702885cf86161e"><span class="id" title="notation">"</span></a>x <= y < z" := (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation">"</span></a>x < y < z" := (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">"</span></a>x <= y ?= 'iff' C" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.lerif"><span class="id" title="definition">lerif</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">C</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="6019ab7c51b05c9f67093b53d0c48454"><span class="id" title="notation">"</span></a>x <= y ?= 'iff' C :> R" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">R</span><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <span class="id" title="var">R</span><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <span class="id" title="var">C</span>)<br/> + (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.ler_of_leif"><span class="id" title="definition">ler_of_leif</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.ler_of_leif"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.ler_of_leif"><span class="id" title="definition">lerif</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.ler_of_leif"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Keys.ler_of_leif"><span class="id" title="definition">is_true</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Rpos_keyed</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Rneg_keyed</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Rnneg_keyed</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Rreal_keyed</span>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Syntax"><span class="id" title="module">Syntax</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.ExtensionAxioms"><span class="id" title="section">ExtensionAxioms</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.ExtensionAxioms.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.numDomainType"><span class="id" title="abbreviation">numDomainType</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.real_axiom"><span class="id" title="definition">real_axiom</span></a> : <span class="id" title="keyword">Prop</span> := <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ExtensionAxioms.R"><span class="id" title="variable">R</span></a>, <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.archimedean_axiom"><span class="id" title="definition">archimedean_axiom</span></a> : <span class="id" title="keyword">Prop</span> := <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ExtensionAxioms.R"><span class="id" title="variable">R</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">ub</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ub"><span class="id" title="variable">ub</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.real_closed_axiom"><span class="id" title="definition">real_closed_axiom</span></a> : <span class="id" title="keyword">Prop</span> :=<br/> + <span class="id" title="keyword">∀</span> (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ExtensionAxioms.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ExtensionAxioms.R"><span class="id" title="variable">R</span></a>),<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ExtensionAxioms"><span class="id" title="section">ExtensionAxioms</span></a>.<br/> + +<br/> + +<br/> +</div> + +<div class="doc"> + The rest of the numbers interface hierarchy. +</div> +<div class="code"> +<span class="id" title="keyword">Module</span> <a name="Num.NumField"><span class="id" title="module">NumField</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.NumField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Record</span> <a name="Num.NumField.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">R</span> :=<br/> + <a name="Num.NumField.Class"><span class="id" title="constructor">Class</span></a> { <a name="Num.NumField.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.class_of"><span class="id" title="record">GRing.Field.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>; <a name="Num.NumField.mixin"><span class="id" title="projection">mixin</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.mixin_of"><span class="id" title="record">mixin_of</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ring_for"><span class="id" title="abbreviation">ring_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#base"><span class="id" title="method">base</span></a>) }.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.base2"><span class="id" title="definition">base2</span></a> <span class="id" title="var">R</span> (<span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>) := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Class"><span class="id" title="constructor">NumDomain.Class</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.mixin"><span class="id" title="projection">mixin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Structure</span> <a name="Num.NumField.type"><span class="id" title="record">type</span></a> := <a name="Num.NumField.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Num.NumField.sort"><span class="id" title="projection">sort</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.NumField.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.type"><span class="id" title="record">type</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/> +<span class="id" title="keyword">Let</span> <a name="Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.pack"><span class="id" title="definition">pack</span></a> :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">bT</span> <span class="id" title="var">b</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.class"><span class="id" title="definition">GRing.Field.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#bT"><span class="id" title="variable">bT</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.class_of"><span class="id" title="record">GRing.Field.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.T"><span class="id" title="variable">T</span></a>) ⇒<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">mT</span> <span class="id" title="var">m</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class"><span class="id" title="definition">NumDomain.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#mT"><span class="id" title="variable">mT</span></a>) (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Class"><span class="id" title="constructor">NumDomain.Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) ⇒<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.choiceType"><span class="id" title="definition">choiceType</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.zmodType"><span class="id" title="definition">zmodType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Pack"><span class="id" title="constructor">GRing.Zmodule.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.ringType"><span class="id" title="definition">ringType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Pack"><span class="id" title="constructor">GRing.Ring.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.comRingType"><span class="id" title="definition">comRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Pack"><span class="id" title="constructor">GRing.ComRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.unitRingType"><span class="id" title="definition">unitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Pack"><span class="id" title="constructor">GRing.UnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Pack"><span class="id" title="constructor">GRing.ComUnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.idomainType"><span class="id" title="definition">idomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Pack"><span class="id" title="constructor">GRing.IntegralDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.numDomainType"><span class="id" title="definition">numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.fieldType"><span class="id" title="definition">fieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Pack"><span class="id" title="constructor">GRing.Field.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.NumField.join_numDomainType"><span class="id" title="definition">join_numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.fieldType"><span class="id" title="definition">fieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.NumField.Exports"><span class="id" title="module">Exports</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base"><span class="id" title="projection">GRing.Field.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base2"><span class="id" title="definition">base2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base2"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base2"><span class="id" title="definition">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base2"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.base2"><span class="id" title="definition">NumDomain.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.sort"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.sort"><span class="id" title="projection">Sortclass</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.eqType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.choiceType"><span class="id" title="definition">choiceType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.choiceType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.choiceType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.choiceType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.choiceType"><span class="id" title="definition">Choice.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">choiceType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.zmodType"><span class="id" title="definition">zmodType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.zmodType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.zmodType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.zmodType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.zmodType"><span class="id" title="definition">GRing.Zmodule.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">zmodType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ringType"><span class="id" title="definition">ringType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ringType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ringType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ringType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.ringType"><span class="id" title="definition">GRing.Ring.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ringType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comRingType"><span class="id" title="definition">comRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comRingType"><span class="id" title="definition">GRing.ComRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.unitRingType"><span class="id" title="definition">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.unitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.unitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.unitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.unitRingType"><span class="id" title="definition">GRing.UnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comUnitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comUnitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comUnitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.comUnitRingType"><span class="id" title="definition">GRing.ComUnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comUnitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.idomainType"><span class="id" title="definition">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.idomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.idomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.idomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.idomainType"><span class="id" title="definition">GRing.IntegralDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">idomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.numDomainType"><span class="id" title="definition">numDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.numDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.numDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.numDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.numDomainType"><span class="id" title="definition">NumDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numDomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.fieldType"><span class="id" title="definition">fieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.fieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.fieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.fieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.fieldType"><span class="id" title="definition">GRing.Field.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fieldType</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.NumField.Exports.numFieldType"><span class="id" title="abbreviation">numFieldType</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.type"><span class="id" title="record">type</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="a7441f0a0e6a98d4d20f782d49891896"><span class="id" title="notation">"</span></a>[ 'numFieldType' 'of' T ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'numFieldType' 'of' T ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Exports"><span class="id" title="module">Exports</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField"><span class="id" title="module">NumField</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">NumField.Exports</span>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.ClosedField"><span class="id" title="module">ClosedField</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.ClosedField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Record</span> <a name="Num.ClosedField.imaginary_mixin_of"><span class="id" title="record">imaginary_mixin_of</span></a> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Exports.numDomainType"><span class="id" title="abbreviation">numDomainType</span></a>) := <a name="Num.ClosedField.ImaginaryMixin"><span class="id" title="constructor">ImaginaryMixin</span></a> {<br/> + <a name="Num.ClosedField.imaginary"><span class="id" title="projection">imaginary</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>;<br/> + <a name="Num.ClosedField.conj_op"><span class="id" title="projection">conj_op</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a>;<br/> + <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#imaginary"><span class="id" title="method">imaginary</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> - 1;<br/> + <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.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#conj_op"><span class="id" title="method">conj_op</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2;<br/> +}.<br/> + +<br/> +<span class="id" title="keyword">Record</span> <a name="Num.ClosedField.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">R</span> := <a name="Num.ClosedField.Class"><span class="id" title="constructor">Class</span></a> {<br/> + <a name="Num.ClosedField.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ClosedField.class_of"><span class="id" title="record">GRing.ClosedField.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>;<br/> + <a name="Num.ClosedField.mixin"><span class="id" title="projection">mixin</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.mixin_of"><span class="id" title="record">mixin_of</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ring_for"><span class="id" title="abbreviation">ring_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#base"><span class="id" title="method">base</span></a>);<br/> + <a name="Num.ClosedField.conj_mixin"><span class="id" title="projection">conj_mixin</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.imaginary_mixin_of"><span class="id" title="record">imaginary_mixin_of</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.num_for"><span class="id" title="abbreviation">num_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Class"><span class="id" title="constructor">NumDomain.Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#mixin"><span class="id" title="method">mixin</span></a>))<br/> +}.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.base2"><span class="id" title="definition">base2</span></a> <span class="id" title="var">R</span> (<span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>) := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Class"><span class="id" title="constructor">NumField.Class</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.mixin"><span class="id" title="projection">mixin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Structure</span> <a name="Num.ClosedField.type"><span class="id" title="record">type</span></a> := <a name="Num.ClosedField.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Num.ClosedField.sort"><span class="id" title="projection">sort</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.ClosedField.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.type"><span class="id" title="record">type</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/> +<span class="id" title="keyword">Let</span> <a name="Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.pack"><span class="id" title="definition">pack</span></a> :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">bT</span> <span class="id" title="var">b</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ClosedField.class"><span class="id" title="definition">GRing.ClosedField.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#bT"><span class="id" title="variable">bT</span></a>)<br/> + (<a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ClosedField.class_of"><span class="id" title="record">GRing.ClosedField.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.T"><span class="id" title="variable">T</span></a>) ⇒<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">mT</span> <span class="id" title="var">m</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class"><span class="id" title="definition">NumField.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#mT"><span class="id" title="variable">mT</span></a>) (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Class"><span class="id" title="constructor">NumField.Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) ⇒<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">mc</span> ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#mc"><span class="id" title="variable">mc</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.clone"><span class="id" title="definition">clone</span></a> := <span class="id" title="keyword">fun</span> <span class="id" title="var">b</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class"><span class="id" title="definition">class</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.T"><span class="id" title="variable">T</span></a>) ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Pack"><span class="id" title="constructor">Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.choiceType"><span class="id" title="definition">choiceType</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.zmodType"><span class="id" title="definition">zmodType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Pack"><span class="id" title="constructor">GRing.Zmodule.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.ringType"><span class="id" title="definition">ringType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Pack"><span class="id" title="constructor">GRing.Ring.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.comRingType"><span class="id" title="definition">comRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Pack"><span class="id" title="constructor">GRing.ComRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.unitRingType"><span class="id" title="definition">unitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Pack"><span class="id" title="constructor">GRing.UnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Pack"><span class="id" title="constructor">GRing.ComUnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.idomainType"><span class="id" title="definition">idomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Pack"><span class="id" title="constructor">GRing.IntegralDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.numDomainType"><span class="id" title="definition">numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.fieldType"><span class="id" title="definition">fieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Pack"><span class="id" title="constructor">GRing.Field.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.numFieldType"><span class="id" title="definition">numFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">NumField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.decFieldType"><span class="id" title="definition">decFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.DecidableField.Pack"><span class="id" title="constructor">GRing.DecidableField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.closedFieldType"><span class="id" title="definition">closedFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ClosedField.Pack"><span class="id" title="constructor">GRing.ClosedField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.join_dec_numDomainType"><span class="id" title="definition">join_dec_numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.decFieldType"><span class="id" title="definition">decFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.join_dec_numFieldType"><span class="id" title="definition">join_dec_numFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">NumField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.decFieldType"><span class="id" title="definition">decFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.join_numDomainType"><span class="id" title="definition">join_numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.closedFieldType"><span class="id" title="definition">closedFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ClosedField.join_numFieldType"><span class="id" title="definition">join_numFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">NumField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.closedFieldType"><span class="id" title="definition">closedFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.ClosedField.Exports"><span class="id" title="module">Exports</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base"><span class="id" title="projection">GRing.ClosedField.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base2"><span class="id" title="definition">base2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base2"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base2"><span class="id" title="definition">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base2"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.base2"><span class="id" title="definition">NumField.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.sort"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.sort"><span class="id" title="projection">Sortclass</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.eqType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.choiceType"><span class="id" title="definition">choiceType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.choiceType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.choiceType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.choiceType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.choiceType"><span class="id" title="definition">Choice.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">choiceType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.zmodType"><span class="id" title="definition">zmodType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.zmodType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.zmodType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.zmodType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.zmodType"><span class="id" title="definition">GRing.Zmodule.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">zmodType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ringType"><span class="id" title="definition">ringType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ringType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ringType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ringType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.ringType"><span class="id" title="definition">GRing.Ring.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ringType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comRingType"><span class="id" title="definition">comRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comRingType"><span class="id" title="definition">GRing.ComRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.unitRingType"><span class="id" title="definition">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.unitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.unitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.unitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.unitRingType"><span class="id" title="definition">GRing.UnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comUnitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comUnitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comUnitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.comUnitRingType"><span class="id" title="definition">GRing.ComUnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comUnitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.idomainType"><span class="id" title="definition">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.idomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.idomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.idomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.idomainType"><span class="id" title="definition">GRing.IntegralDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">idomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numDomainType"><span class="id" title="definition">numDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numDomainType"><span class="id" title="definition">NumDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numDomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.fieldType"><span class="id" title="definition">fieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.fieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.fieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.fieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.fieldType"><span class="id" title="definition">GRing.Field.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fieldType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.decFieldType"><span class="id" title="definition">decFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.decFieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.decFieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.decFieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.decFieldType"><span class="id" title="definition">GRing.DecidableField.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">decFieldType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numFieldType"><span class="id" title="definition">numFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numFieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numFieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numFieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.numFieldType"><span class="id" title="definition">NumField.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numFieldType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.closedFieldType"><span class="id" title="definition">closedFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.closedFieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.closedFieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.closedFieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.closedFieldType"><span class="id" title="definition">GRing.ClosedField.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">closedFieldType</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">join_dec_numDomainType</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">join_dec_numFieldType</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">join_numDomainType</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">join_numFieldType</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.ClosedField.Exports.numClosedFieldType"><span class="id" title="abbreviation">numClosedFieldType</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.type"><span class="id" title="record">type</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.ClosedField.Exports.NumClosedFieldType"><span class="id" title="abbreviation">NumClosedFieldType</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">m</span>).<br/> +<span class="id" title="keyword">Notation</span> <a name="53405ca907938fed833c1fec5ac3d770"><span class="id" title="notation">"</span></a>[ 'numClosedFieldType' 'of' T 'for' cT ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">cT</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'numClosedFieldType' 'of' T 'for' cT ]") :<br/> + <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="f35f0142f80d163945beb05160d401d5"><span class="id" title="notation">"</span></a>[ 'numClosedFieldType' 'of' T ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'numClosedFieldType' 'of' T ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Exports"><span class="id" title="module">Exports</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField"><span class="id" title="module">ClosedField</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">ClosedField.Exports</span>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.RealDomain"><span class="id" title="module">RealDomain</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.RealDomain.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Record</span> <a name="Num.RealDomain.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">R</span> :=<br/> + <a name="Num.RealDomain.Class"><span class="id" title="constructor">Class</span></a> {<a name="Num.RealDomain.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class_of"><span class="id" title="record">NumDomain.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>; <span class="id" title="var">_</span> : @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real_axiom"><span class="id" title="definition">real_axiom</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.num_for"><span class="id" title="abbreviation">num_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#base"><span class="id" title="method">base</span></a>)}.<br/> + +<br/> +<span class="id" title="keyword">Structure</span> <a name="Num.RealDomain.type"><span class="id" title="record">type</span></a> := <a name="Num.RealDomain.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Num.RealDomain.sort"><span class="id" title="projection">sort</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.RealDomain.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.type"><span class="id" title="record">type</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/> +<span class="id" title="keyword">Let</span> <a name="Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">c</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Pack"><span class="id" title="constructor">Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">b0</span> (<span class="id" title="var">m0</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real_axiom"><span class="id" title="definition">real_axiom</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.num_for"><span class="id" title="abbreviation">num_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b0"><span class="id" title="variable">b0</span></a>)) :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">bT</span> <span class="id" title="var">b</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.class"><span class="id" title="definition">NumDomain.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#bT"><span class="id" title="variable">bT</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> ⇒<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m0"><span class="id" title="variable">m0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.choiceType"><span class="id" title="definition">choiceType</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.zmodType"><span class="id" title="definition">zmodType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Pack"><span class="id" title="constructor">GRing.Zmodule.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.ringType"><span class="id" title="definition">ringType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Pack"><span class="id" title="constructor">GRing.Ring.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.comRingType"><span class="id" title="definition">comRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Pack"><span class="id" title="constructor">GRing.ComRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.unitRingType"><span class="id" title="definition">unitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Pack"><span class="id" title="constructor">GRing.UnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Pack"><span class="id" title="constructor">GRing.ComUnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.idomainType"><span class="id" title="definition">idomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Pack"><span class="id" title="constructor">GRing.IntegralDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealDomain.numDomainType"><span class="id" title="definition">numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.RealDomain.Exports"><span class="id" title="module">Exports</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.base"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.base"><span class="id" title="projection">NumDomain.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.sort"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.sort"><span class="id" title="projection">Sortclass</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.eqType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.choiceType"><span class="id" title="definition">choiceType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.choiceType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.choiceType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.choiceType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.choiceType"><span class="id" title="definition">Choice.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">choiceType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.zmodType"><span class="id" title="definition">zmodType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.zmodType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.zmodType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.zmodType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.zmodType"><span class="id" title="definition">GRing.Zmodule.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">zmodType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ringType"><span class="id" title="definition">ringType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ringType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ringType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ringType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.ringType"><span class="id" title="definition">GRing.Ring.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ringType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comRingType"><span class="id" title="definition">comRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comRingType"><span class="id" title="definition">GRing.ComRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.unitRingType"><span class="id" title="definition">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.unitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.unitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.unitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.unitRingType"><span class="id" title="definition">GRing.UnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comUnitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comUnitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comUnitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.comUnitRingType"><span class="id" title="definition">GRing.ComUnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comUnitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.idomainType"><span class="id" title="definition">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.idomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.idomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.idomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.idomainType"><span class="id" title="definition">GRing.IntegralDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">idomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.numDomainType"><span class="id" title="definition">numDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.numDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.numDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.numDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.numDomainType"><span class="id" title="definition">NumDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numDomainType</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.RealDomain.Exports.realDomainType"><span class="id" title="abbreviation">realDomainType</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.type"><span class="id" title="record">type</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.RealDomain.Exports.RealDomainType"><span class="id" title="abbreviation">RealDomainType</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">m</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>).<br/> +<span class="id" title="keyword">Notation</span> <a name="97348e7f29e06cbe2672c3f0899ae4cb"><span class="id" title="notation">"</span></a>[ 'realDomainType' 'of' T 'for' cT ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">cT</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'realDomainType' 'of' T 'for' cT ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="0df319535c5b2c7df349bac58e95e96c"><span class="id" title="notation">"</span></a>[ 'realDomainType' 'of' T ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'realDomainType' 'of' T ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Exports"><span class="id" title="module">Exports</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain"><span class="id" title="module">RealDomain</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">RealDomain.Exports</span>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.RealField"><span class="id" title="module">RealField</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.RealField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Record</span> <a name="Num.RealField.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">R</span> :=<br/> + <a name="Num.RealField.Class"><span class="id" title="constructor">Class</span></a> { <a name="Num.RealField.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class_of"><span class="id" title="record">NumField.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>; <a name="Num.RealField.mixin"><span class="id" title="projection">mixin</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real_axiom"><span class="id" title="definition">real_axiom</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.num_for"><span class="id" title="abbreviation">num_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#base"><span class="id" title="method">base</span></a>) }.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.base2"><span class="id" title="definition">base2</span></a> <span class="id" title="var">R</span> (<span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>) := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Class"><span class="id" title="constructor">RealDomain.Class</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.mixin"><span class="id" title="projection">mixin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Structure</span> <a name="Num.RealField.type"><span class="id" title="record">type</span></a> := <a name="Num.RealField.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Num.RealField.sort"><span class="id" title="projection">sort</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.RealField.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.type"><span class="id" title="record">type</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/> +<span class="id" title="keyword">Let</span> <a name="Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.pack"><span class="id" title="definition">pack</span></a> :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">bT</span> <span class="id" title="var">b</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class"><span class="id" title="definition">NumField.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#bT"><span class="id" title="variable">bT</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.class_of"><span class="id" title="record">NumField.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.T"><span class="id" title="variable">T</span></a>) ⇒<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">mT</span> <span class="id" title="var">m</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.class"><span class="id" title="definition">RealDomain.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#mT"><span class="id" title="variable">mT</span></a>) (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Class"><span class="id" title="constructor">RealDomain.Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) ⇒<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.choiceType"><span class="id" title="definition">choiceType</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.zmodType"><span class="id" title="definition">zmodType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Pack"><span class="id" title="constructor">GRing.Zmodule.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.ringType"><span class="id" title="definition">ringType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Pack"><span class="id" title="constructor">GRing.Ring.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.comRingType"><span class="id" title="definition">comRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Pack"><span class="id" title="constructor">GRing.ComRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.unitRingType"><span class="id" title="definition">unitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Pack"><span class="id" title="constructor">GRing.UnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Pack"><span class="id" title="constructor">GRing.ComUnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.idomainType"><span class="id" title="definition">idomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Pack"><span class="id" title="constructor">GRing.IntegralDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.numDomainType"><span class="id" title="definition">numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.realDomainType"><span class="id" title="definition">realDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Pack"><span class="id" title="constructor">RealDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.fieldType"><span class="id" title="definition">fieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Pack"><span class="id" title="constructor">GRing.Field.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.numFieldType"><span class="id" title="definition">numFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">NumField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealField.join_realDomainType"><span class="id" title="definition">join_realDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Pack"><span class="id" title="constructor">RealDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numFieldType"><span class="id" title="definition">numFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.RealField.Exports"><span class="id" title="module">Exports</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base"><span class="id" title="projection">NumField.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base2"><span class="id" title="definition">base2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base2"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base2"><span class="id" title="definition">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base2"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.base2"><span class="id" title="definition">RealDomain.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.sort"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.sort"><span class="id" title="projection">Sortclass</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.eqType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.choiceType"><span class="id" title="definition">choiceType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.choiceType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.choiceType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.choiceType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.choiceType"><span class="id" title="definition">Choice.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">choiceType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.zmodType"><span class="id" title="definition">zmodType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.zmodType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.zmodType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.zmodType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.zmodType"><span class="id" title="definition">GRing.Zmodule.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">zmodType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ringType"><span class="id" title="definition">ringType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ringType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ringType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ringType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.ringType"><span class="id" title="definition">GRing.Ring.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ringType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comRingType"><span class="id" title="definition">comRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comRingType"><span class="id" title="definition">GRing.ComRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.unitRingType"><span class="id" title="definition">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.unitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.unitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.unitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.unitRingType"><span class="id" title="definition">GRing.UnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comUnitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comUnitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comUnitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.comUnitRingType"><span class="id" title="definition">GRing.ComUnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comUnitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.idomainType"><span class="id" title="definition">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.idomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.idomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.idomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.idomainType"><span class="id" title="definition">GRing.IntegralDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">idomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numDomainType"><span class="id" title="definition">numDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numDomainType"><span class="id" title="definition">NumDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numDomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.realDomainType"><span class="id" title="definition">realDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.realDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.realDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.realDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.realDomainType"><span class="id" title="definition">RealDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">realDomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.fieldType"><span class="id" title="definition">fieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.fieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.fieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.fieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.fieldType"><span class="id" title="definition">GRing.Field.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fieldType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numFieldType"><span class="id" title="definition">numFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numFieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numFieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numFieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.numFieldType"><span class="id" title="definition">NumField.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numFieldType</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">join_realDomainType</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.RealField.Exports.realFieldType"><span class="id" title="abbreviation">realFieldType</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.type"><span class="id" title="record">type</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="9bd0f21dc8f37cb47d141588c0e6729b"><span class="id" title="notation">"</span></a>[ 'realFieldType' 'of' T ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'realFieldType' 'of' T ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Exports"><span class="id" title="module">Exports</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField"><span class="id" title="module">RealField</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">RealField.Exports</span>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.ArchimedeanField"><span class="id" title="module">ArchimedeanField</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.ArchimedeanField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Record</span> <a name="Num.ArchimedeanField.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">R</span> :=<br/> + <a name="Num.ArchimedeanField.Class"><span class="id" title="constructor">Class</span></a> { <a name="Num.ArchimedeanField.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class_of"><span class="id" title="record">RealField.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.archimedean_axiom"><span class="id" title="definition">archimedean_axiom</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.num_for"><span class="id" title="abbreviation">num_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#base"><span class="id" title="method">base</span></a>) }.<br/> + +<br/> +<span class="id" title="keyword">Structure</span> <a name="Num.ArchimedeanField.type"><span class="id" title="record">type</span></a> := <a name="Num.ArchimedeanField.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Num.ArchimedeanField.sort"><span class="id" title="projection">sort</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.ArchimedeanField.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.type"><span class="id" title="record">type</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/> +<span class="id" title="keyword">Let</span> <a name="Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">c</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.Pack"><span class="id" title="constructor">Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">b0</span> (<span class="id" title="var">m0</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.archimedean_axiom"><span class="id" title="definition">archimedean_axiom</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.num_for"><span class="id" title="abbreviation">num_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b0"><span class="id" title="variable">b0</span></a>)) :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">bT</span> <span class="id" title="var">b</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class"><span class="id" title="definition">RealField.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#bT"><span class="id" title="variable">bT</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> ⇒<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m0"><span class="id" title="variable">m0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.choiceType"><span class="id" title="definition">choiceType</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.zmodType"><span class="id" title="definition">zmodType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Pack"><span class="id" title="constructor">GRing.Zmodule.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.ringType"><span class="id" title="definition">ringType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Pack"><span class="id" title="constructor">GRing.Ring.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.comRingType"><span class="id" title="definition">comRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Pack"><span class="id" title="constructor">GRing.ComRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.unitRingType"><span class="id" title="definition">unitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Pack"><span class="id" title="constructor">GRing.UnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Pack"><span class="id" title="constructor">GRing.ComUnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.idomainType"><span class="id" title="definition">idomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Pack"><span class="id" title="constructor">GRing.IntegralDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.numDomainType"><span class="id" title="definition">numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.realDomainType"><span class="id" title="definition">realDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Pack"><span class="id" title="constructor">RealDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.fieldType"><span class="id" title="definition">fieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Pack"><span class="id" title="constructor">GRing.Field.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.numFieldType"><span class="id" title="definition">numFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">NumField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ArchimedeanField.realFieldType"><span class="id" title="definition">realFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Pack"><span class="id" title="constructor">RealField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.ArchimedeanField.Exports"><span class="id" title="module">Exports</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.base"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.base"><span class="id" title="projection">RealField.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.sort"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.sort"><span class="id" title="projection">Sortclass</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.eqType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.choiceType"><span class="id" title="definition">choiceType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.choiceType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.choiceType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.choiceType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.choiceType"><span class="id" title="definition">Choice.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">choiceType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.zmodType"><span class="id" title="definition">zmodType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.zmodType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.zmodType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.zmodType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.zmodType"><span class="id" title="definition">GRing.Zmodule.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">zmodType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ringType"><span class="id" title="definition">ringType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ringType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ringType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ringType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.ringType"><span class="id" title="definition">GRing.Ring.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ringType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comRingType"><span class="id" title="definition">comRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comRingType"><span class="id" title="definition">GRing.ComRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.unitRingType"><span class="id" title="definition">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.unitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.unitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.unitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.unitRingType"><span class="id" title="definition">GRing.UnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comUnitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comUnitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comUnitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.comUnitRingType"><span class="id" title="definition">GRing.ComUnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comUnitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.idomainType"><span class="id" title="definition">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.idomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.idomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.idomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.idomainType"><span class="id" title="definition">GRing.IntegralDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">idomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numDomainType"><span class="id" title="definition">numDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numDomainType"><span class="id" title="definition">NumDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numDomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realDomainType"><span class="id" title="definition">realDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realDomainType"><span class="id" title="definition">RealDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">realDomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.fieldType"><span class="id" title="definition">fieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.fieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.fieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.fieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.fieldType"><span class="id" title="definition">GRing.Field.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fieldType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numFieldType"><span class="id" title="definition">numFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numFieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numFieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numFieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.numFieldType"><span class="id" title="definition">NumField.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numFieldType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realFieldType"><span class="id" title="definition">realFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realFieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realFieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realFieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.realFieldType"><span class="id" title="definition">RealField.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">realFieldType</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.ArchimedeanField.Exports.archiFieldType"><span class="id" title="abbreviation">archiFieldType</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.type"><span class="id" title="record">type</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.ArchimedeanField.Exports.ArchiFieldType"><span class="id" title="abbreviation">ArchiFieldType</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">m</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>).<br/> +<span class="id" title="keyword">Notation</span> <a name="d70ca3cc0dfddd155fdca7bda79af694"><span class="id" title="notation">"</span></a>[ 'archiFieldType' 'of' T 'for' cT ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">cT</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'archiFieldType' 'of' T 'for' cT ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="a58841a2b4844911030c30bfa80595e1"><span class="id" title="notation">"</span></a>[ 'archiFieldType' 'of' T ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'archiFieldType' 'of' T ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.Exports"><span class="id" title="module">Exports</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField"><span class="id" title="module">ArchimedeanField</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">ArchimedeanField.Exports</span>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.RealClosedField"><span class="id" title="module">RealClosedField</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.RealClosedField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Record</span> <a name="Num.RealClosedField.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">R</span> :=<br/> + <a name="Num.RealClosedField.Class"><span class="id" title="constructor">Class</span></a> { <a name="Num.RealClosedField.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class_of"><span class="id" title="record">RealField.class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real_closed_axiom"><span class="id" title="definition">real_closed_axiom</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.num_for"><span class="id" title="abbreviation">num_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#base"><span class="id" title="method">base</span></a>) }.<br/> + +<br/> +<span class="id" title="keyword">Structure</span> <a name="Num.RealClosedField.type"><span class="id" title="record">type</span></a> := <a name="Num.RealClosedField.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Num.RealClosedField.sort"><span class="id" title="projection">sort</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.RealClosedField.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.type"><span class="id" title="record">type</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/> +<span class="id" title="keyword">Let</span> <a name="Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">c</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.Pack"><span class="id" title="constructor">Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">b0</span> (<span class="id" title="var">m0</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real_closed_axiom"><span class="id" title="definition">real_closed_axiom</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.num_for"><span class="id" title="abbreviation">num_for</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b0"><span class="id" title="variable">b0</span></a>)) :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">bT</span> <span class="id" title="var">b</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.class"><span class="id" title="definition">RealField.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#bT"><span class="id" title="variable">bT</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> ⇒<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m0"><span class="id" title="variable">m0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.choiceType"><span class="id" title="definition">choiceType</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.zmodType"><span class="id" title="definition">zmodType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Pack"><span class="id" title="constructor">GRing.Zmodule.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.ringType"><span class="id" title="definition">ringType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Pack"><span class="id" title="constructor">GRing.Ring.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.comRingType"><span class="id" title="definition">comRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Pack"><span class="id" title="constructor">GRing.ComRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.unitRingType"><span class="id" title="definition">unitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Pack"><span class="id" title="constructor">GRing.UnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Pack"><span class="id" title="constructor">GRing.ComUnitRing.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.idomainType"><span class="id" title="definition">idomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Pack"><span class="id" title="constructor">GRing.IntegralDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.numDomainType"><span class="id" title="definition">numDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Pack"><span class="id" title="constructor">NumDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.realDomainType"><span class="id" title="definition">realDomainType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Pack"><span class="id" title="constructor">RealDomain.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.fieldType"><span class="id" title="definition">fieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Pack"><span class="id" title="constructor">GRing.Field.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.numFieldType"><span class="id" title="definition">numFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumField.Pack"><span class="id" title="constructor">NumField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealClosedField.realFieldType"><span class="id" title="definition">realFieldType</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Pack"><span class="id" title="constructor">RealField.Pack</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.RealClosedField.Exports"><span class="id" title="module">Exports</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.base"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.base"><span class="id" title="projection">RealField.class_of</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.sort"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.sort"><span class="id" title="projection">Sortclass</span></a>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.eqType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.choiceType"><span class="id" title="definition">choiceType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.choiceType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.choiceType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.choiceType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.choiceType"><span class="id" title="definition">Choice.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">choiceType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.zmodType"><span class="id" title="definition">zmodType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.zmodType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.zmodType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.zmodType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.zmodType"><span class="id" title="definition">GRing.Zmodule.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">zmodType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ringType"><span class="id" title="definition">ringType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ringType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ringType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ringType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.ringType"><span class="id" title="definition">GRing.Ring.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ringType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comRingType"><span class="id" title="definition">comRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comRingType"><span class="id" title="definition">GRing.ComRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.unitRingType"><span class="id" title="definition">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.unitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.unitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.unitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.unitRingType"><span class="id" title="definition">GRing.UnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comUnitRingType"><span class="id" title="definition">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comUnitRingType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comUnitRingType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comUnitRingType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.comUnitRingType"><span class="id" title="definition">GRing.ComUnitRing.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comUnitRingType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.idomainType"><span class="id" title="definition">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.idomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.idomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.idomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.idomainType"><span class="id" title="definition">GRing.IntegralDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">idomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numDomainType"><span class="id" title="definition">numDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numDomainType"><span class="id" title="definition">NumDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numDomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realDomainType"><span class="id" title="definition">realDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realDomainType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realDomainType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realDomainType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realDomainType"><span class="id" title="definition">RealDomain.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">realDomainType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.fieldType"><span class="id" title="definition">fieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.fieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.fieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.fieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.fieldType"><span class="id" title="definition">GRing.Field.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fieldType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numFieldType"><span class="id" title="definition">numFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numFieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numFieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numFieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.numFieldType"><span class="id" title="definition">NumField.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">numFieldType</span>.<br/> +<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realFieldType"><span class="id" title="definition">realFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realFieldType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realFieldType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realFieldType"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.realFieldType"><span class="id" title="definition">RealField.type</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">realFieldType</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.RealClosedField.Exports.rcfType"><span class="id" title="abbreviation">rcfType</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.type"><span class="id" title="record">Num.RealClosedField.type</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.RealClosedField.Exports.RcfType"><span class="id" title="abbreviation">RcfType</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">m</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>).<br/> +<span class="id" title="keyword">Notation</span> <a name="d55f7ab3d406a321aadc0a2e4311d80c"><span class="id" title="notation">"</span></a>[ 'rcfType' 'of' T 'for' cT ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">cT</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'rcfType' 'of' T 'for' cT ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="489355e4822051a37f16f0d65e2778f8"><span class="id" title="notation">"</span></a>[ 'rcfType' 'of' T ]" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'rcfType' 'of' T ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.Exports"><span class="id" title="module">Exports</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField"><span class="id" title="module">RealClosedField</span></a>.<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">RealClosedField.Exports</span>.<br/> + +<br/> +</div> + +<div class="doc"> + The elementary theory needed to support the definition of the derived + operations for the extensions described above. +</div> +<div class="code"> +<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="Num.Internals"><span class="id" title="module">Internals</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Internals.Domain"><span class="id" title="section">Domain</span></a>.<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Internals.Domain.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/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Lemmas from the signature +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.normr0_eq0"><span class="id" title="lemma">normr0_eq0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.ler_norm_add"><span class="id" title="lemma">ler_norm_add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.addr_gt0"><span class="id" title="lemma">addr_gt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.ger_leVge"><span class="id" title="lemma">ger_leVge</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.normrM"><span class="id" title="lemma">normrM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.norm"><span class="id" title="abbreviation">norm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.ler_def"><span class="id" title="lemma">ler_def</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.ltr_def"><span class="id" title="lemma">ltr_def</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Basic consequences (just enough to get predicate closure properties). +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.ger0_def"><span class="id" title="lemma">ger0_def</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.subr_ge0"><span class="id" title="lemma">subr_ge0</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.oppr_ge0"><span class="id" title="lemma">oppr_ge0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.ler01"><span class="id" title="lemma">ler01</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.ltr01"><span class="id" title="lemma">ltr01</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.ltrW"><span class="id" title="lemma">ltrW</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.lerr"><span class="id" title="lemma">lerr</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.le0r"><span class="id" title="lemma">le0r</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.addr_ge0"><span class="id" title="lemma">addr_ge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.pmulr_rgt0"><span class="id" title="lemma">pmulr_rgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Closure properties of the real predicates. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.posrE"><span class="id" title="lemma">posrE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.pos"><span class="id" title="abbreviation">pos</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.nnegrE"><span class="id" title="lemma">nnegrE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.realE"><span class="id" title="lemma">realE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Internals.pos_divr_closed"><span class="id" title="lemma">pos_divr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.divr_closed"><span class="id" title="abbreviation">divr_closed</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.pos"><span class="id" title="abbreviation">pos</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pos_mulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.MulrPred"><span class="id" title="definition">MulrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.pos_divr_closed"><span class="id" title="lemma">pos_divr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pos_divrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.DivrPred"><span class="id" title="definition">DivrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.pos_divr_closed"><span class="id" title="lemma">pos_divr_closed</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Internals.nneg_divr_closed"><span class="id" title="lemma">nneg_divr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.divr_closed"><span class="id" title="abbreviation">divr_closed</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nneg_mulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.MulrPred"><span class="id" title="definition">MulrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.nneg_divr_closed"><span class="id" title="lemma">nneg_divr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nneg_divrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.DivrPred"><span class="id" title="definition">DivrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.nneg_divr_closed"><span class="id" title="lemma">nneg_divr_closed</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Internals.nneg_addr_closed"><span class="id" title="lemma">nneg_addr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.addr_closed"><span class="id" title="abbreviation">addr_closed</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>).<br/> + <span class="id" title="keyword">Canonical</span> <span class="id" title="var">nneg_addrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.AddrPred"><span class="id" title="definition">AddrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.nneg_addr_closed"><span class="id" title="lemma">nneg_addr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nneg_semiringPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SemiringPred"><span class="id" title="definition">SemiringPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.nneg_divr_closed"><span class="id" title="lemma">nneg_divr_closed</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Internals.real_oppr_closed"><span class="id" title="lemma">real_oppr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.oppr_closed"><span class="id" title="abbreviation">oppr_closed</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>).<br/> + <span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_opprPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.OpprPred"><span class="id" title="definition">OpprPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_oppr_closed"><span class="id" title="lemma">real_oppr_closed</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Internals.real_addr_closed"><span class="id" title="lemma">real_addr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.addr_closed"><span class="id" title="abbreviation">addr_closed</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_addrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.AddrPred"><span class="id" title="definition">AddrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_addr_closed"><span class="id" title="lemma">real_addr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_zmodPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.ZmodPred"><span class="id" title="definition">ZmodPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_oppr_closed"><span class="id" title="lemma">real_oppr_closed</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Internals.real_divr_closed"><span class="id" title="lemma">real_divr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.divr_closed"><span class="id" title="abbreviation">divr_closed</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain.R"><span class="id" title="variable">R</span></a>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_mulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.MulrPred"><span class="id" title="definition">MulrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_divr_closed"><span class="id" title="lemma">real_divr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_smulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SmulrPred"><span class="id" title="definition">SmulrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_divr_closed"><span class="id" title="lemma">real_divr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_divrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.DivrPred"><span class="id" title="definition">DivrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_divr_closed"><span class="id" title="lemma">real_divr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_sdivrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SdivrPred"><span class="id" title="definition">SdivrPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_divr_closed"><span class="id" title="lemma">real_divr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_semiringPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SemiringPred"><span class="id" title="definition">SemiringPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_divr_closed"><span class="id" title="lemma">real_divr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_subringPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SubringPred"><span class="id" title="definition">SubringPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_divr_closed"><span class="id" title="lemma">real_divr_closed</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_divringPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.DivringPred"><span class="id" title="definition">DivringPred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.real_divr_closed"><span class="id" title="lemma">real_divr_closed</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.Domain"><span class="id" title="section">Domain</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.num_real"><span class="id" title="lemma">num_real</span></a> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealDomain.Exports.realDomainType"><span class="id" title="abbreviation">realDomainType</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Internals.archi_bound_subproof"><span class="id" title="lemma">archi_bound_subproof</span></a> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.Exports.archiFieldType"><span class="id" title="abbreviation">archiFieldType</span></a>) : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.archimedean_axiom"><span class="id" title="definition">archimedean_axiom</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Internals.RealClosed"><span class="id" title="section">RealClosed</span></a>.<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Internals.RealClosed.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.Exports.rcfType"><span class="id" title="abbreviation">rcfType</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Internals.poly_ivt"><span class="id" title="lemma">poly_ivt</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real_closed_axiom"><span class="id" title="definition">real_closed_axiom</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.RealClosed.R"><span class="id" title="variable">R</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Internals.sqrtr_subproof"><span class="id" title="lemma">sqrtr_subproof</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.RealClosed.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#2511079c59ad38c7f21d89316f028f07"><span class="id" title="notation">if</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#2511079c59ad38c7f21d89316f028f07"><span class="id" title="notation">return</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#2511079c59ad38c7f21d89316f028f07"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#2511079c59ad38c7f21d89316f028f07"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.RealClosed"><span class="id" title="section">RealClosed</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals"><span class="id" title="module">Internals</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.PredInstances"><span class="id" title="module">PredInstances</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pos_mulrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pos_divrPred</span>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nneg_addrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nneg_mulrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nneg_divrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nneg_semiringPred</span>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_addrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_opprPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_zmodPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_mulrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_smulrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_divrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_sdivrPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_semiringPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_subringPred</span>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">real_divringPred</span>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.PredInstances"><span class="id" title="module">PredInstances</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="Num.ExtraDef"><span class="id" title="module">ExtraDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ExtraDef.archi_bound"><span class="id" title="definition">archi_bound</span></a> {<span class="id" title="var">R</span>} <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#sval"><span class="id" title="abbreviation">sval</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#sigW"><span class="id" title="lemma">sigW</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.archi_bound_subproof"><span class="id" title="lemma">archi_bound_subproof</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.ExtraDef.sqrtr"><span class="id" title="definition">sqrtr</span></a> {<span class="id" title="var">R</span>} <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#s2val"><span class="id" title="definition">s2val</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#sig2W"><span class="id" title="lemma">sig2W</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.sqrtr_subproof"><span class="id" title="lemma">sqrtr_subproof</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)).<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ExtraDef"><span class="id" title="module">ExtraDef</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.bound"><span class="id" title="abbreviation">bound</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.archi_bound"><span class="id" title="definition">archi_bound</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrtr"><span class="id" title="definition">sqrtr</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.Theory"><span class="id" title="module">Theory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.NumIntegralDomainTheory"><span class="id" title="section">NumIntegralDomainTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.NumIntegralDomainTheory.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/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Lemmas from the signature (reexported from internals). +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ler_norm_add"><span class="id" title="definition">ler_norm_add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.ler_norm_add"><span class="id" title="lemma">ler_norm_add</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.addr_gt0"><span class="id" title="definition">addr_gt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.addr_gt0"><span class="id" title="lemma">addr_gt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.normr0_eq0"><span class="id" title="definition">normr0_eq0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.normr0_eq0"><span class="id" title="lemma">normr0_eq0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ger_leVge"><span class="id" title="definition">ger_leVge</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> :=<br/> + @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.ger_leVge"><span class="id" title="lemma">ger_leVge</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.normrM"><span class="id" title="definition">normrM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.normr"><span class="id" title="definition">normr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.normrM"><span class="id" title="lemma">normrM</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ler_def"><span class="id" title="definition">ler_def</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.ler_def"><span class="id" title="lemma">ler_def</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ltr_def"><span class="id" title="definition">ltr_def</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> := @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Internals.ltr_def"><span class="id" title="lemma">ltr_def</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Predicate and relation definitions. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gerE"><span class="id" title="lemma">gerE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ge"><span class="id" title="abbreviation">ge</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtrE"><span class="id" title="lemma">gtrE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.gt"><span class="id" title="abbreviation">gt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.posrE"><span class="id" title="lemma">posrE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.pos"><span class="id" title="abbreviation">pos</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.negrE"><span class="id" title="lemma">negrE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.neg"><span class="id" title="abbreviation">neg</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nnegrE"><span class="id" title="lemma">nnegrE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realE"><span class="id" title="lemma">realE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>. <br/> + +<br/> +</div> + +<div class="doc"> + General properties of <= and < +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerr"><span class="id" title="lemma">lerr</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrr"><span class="id" title="lemma">ltrr</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrW"><span class="id" title="lemma">ltrW</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>. <br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">lerr</span> <span class="id" title="var">ltrr</span> <span class="id" title="var">ltrW</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_neqAle"><span class="id" title="lemma">ltr_neqAle</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_eqVlt"><span class="id" title="lemma">ler_eqVlt</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lt0r"><span class="id" title="lemma">lt0r</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.le0r"><span class="id" title="lemma">le0r</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lt0r_neq0"><span class="id" title="lemma">lt0r_neq0</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>) : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr0_neq0"><span class="id" title="lemma">ltr0_neq0</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr_eqF"><span class="id" title="lemma">gtr_eqF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_eqF"><span class="id" title="lemma">ltr_eqF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulr_rgt0"><span class="id" title="lemma">pmulr_rgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulr_rge0"><span class="id" title="lemma">pmulr_rge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Integer comparisons and characteristic 0. +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler01"><span class="id" title="lemma">ler01</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr01"><span class="id" title="lemma">ltr01</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler0n"><span class="id" title="lemma">ler0n</span></a> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">ler01</span> <span class="id" title="var">ltr01</span> <span class="id" title="var">ler0n</span>.<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr0Sn"><span class="id" title="lemma">ltr0Sn</span></a> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr0n"><span class="id" title="lemma">ltr0n</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">ltr0Sn</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pnatr_eq0"><span class="id" title="lemma">pnatr_eq0</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#340b60eb5a3e9913f807040630cb8d43"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#340b60eb5a3e9913f807040630cb8d43"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.char_num"><span class="id" title="lemma">char_num</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred0"><span class="id" title="definition">pred0</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Properties of the norm. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger0_def"><span class="id" title="lemma">ger0_def</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_idP"><span class="id" title="lemma">normr_idP</span></a> {<span class="id" title="var">x</span>} : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + <span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger0_norm"><span class="id" title="lemma">ger0_norm</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr0"><span class="id" title="lemma">normr0</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a>0<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr1"><span class="id" title="lemma">normr1</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a>1<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_nat"><span class="id" title="lemma">normr_nat</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normrMn"><span class="id" title="lemma">normrMn</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_prod"><span class="id" title="lemma">normr_prod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normrX"><span class="id" title="lemma">normrX</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_unit"><span class="id" title="lemma">normr_unit</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#decc5ade2c2879607bbea0935785bb5f"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#decc5ade2c2879607bbea0935785bb5f"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#decc5ade2c2879607bbea0935785bb5f"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.norm"><span class="id" title="abbreviation">norm</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#decc5ade2c2879607bbea0935785bb5f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#decc5ade2c2879607bbea0935785bb5f"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#decc5ade2c2879607bbea0935785bb5f"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#decc5ade2c2879607bbea0935785bb5f"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normrV"><span class="id" title="lemma">normrV</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.normr"><span class="id" title="definition">normr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr0P"><span class="id" title="lemma">normr0P</span></a> {<span class="id" title="var">x</span>} : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.normr_eq0"><span class="id" title="definition">normr_eq0</span></a> <span class="id" title="var">x</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#sameP"><span class="id" title="lemma">sameP</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#ab7a7bc39754581ab8f205db64711d57"><span class="id" title="notation">=</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#ab7a7bc39754581ab8f205db64711d57"><span class="id" title="notation">P</span></a> 0) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr0P"><span class="id" title="lemma">normr0P</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normrN1"><span class="id" title="lemma">normrN1</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a>-1<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normrN"><span class="id" title="lemma">normrN</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.distrC"><span class="id" title="lemma">distrC</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler0_def"><span class="id" title="lemma">ler0_def</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_id"><span class="id" title="lemma">normr_id</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_ge0"><span class="id" title="lemma">normr_ge0</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>. <br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">normr_ge0</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler0_norm"><span class="id" title="lemma">ler0_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.gtr0_norm"><span class="id" title="definition">gtr0_norm</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">hx</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ger0_norm"><span class="id" title="lemma">ger0_norm</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltrW"><span class="id" title="lemma">ltrW</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#hx"><span class="id" title="variable">hx</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ltr0_norm"><span class="id" title="definition">ltr0_norm</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">hx</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0) := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler0_norm"><span class="id" title="lemma">ler0_norm</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltrW"><span class="id" title="lemma">ltrW</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#hx"><span class="id" title="variable">hx</span></a>).<br/> + +<br/> +</div> + +<div class="doc"> + Comparision to 0 of a difference +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.subr_ge0"><span class="id" title="lemma">subr_ge0</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.subr_gt0"><span class="id" title="lemma">subr_gt0</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="Num.Theory.subr_le0"><span class="id" title="lemma">subr_le0</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="Num.Theory.subr_lt0"><span class="id" title="lemma">subr_lt0</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.subr_lte0"><span class="id" title="definition">subr_lte0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.subr_le0"><span class="id" title="lemma">subr_le0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.subr_lt0"><span class="id" title="lemma">subr_lt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.subr_gte0"><span class="id" title="definition">subr_gte0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.subr_ge0"><span class="id" title="lemma">subr_ge0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.subr_gt0"><span class="id" title="lemma">subr_gt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.subr_cp0"><span class="id" title="definition">subr_cp0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.subr_lte0"><span class="id" title="definition">subr_lte0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.subr_gte0"><span class="id" title="definition">subr_gte0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Ordered ring properties. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_asym"><span class="id" title="lemma">ler_asym</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#antisymmetric"><span class="id" title="definition">antisymmetric</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#81937a94685a0487cf97a240746fb002"><span class="id" title="notation"><=%</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#81937a94685a0487cf97a240746fb002"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_le"><span class="id" title="lemma">eqr_le</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_trans"><span class="id" title="lemma">ltr_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.ltr"><span class="id" title="definition">ltr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_lt_trans"><span class="id" title="lemma">ler_lt_trans</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_le_trans"><span class="id" title="lemma">ltr_le_trans</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_trans"><span class="id" title="lemma">ler_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.ler"><span class="id" title="definition">ler</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter01"><span class="id" title="definition">lter01</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler01"><span class="id" title="lemma">ler01</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr01"><span class="id" title="lemma">ltr01</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lterr"><span class="id" title="definition">lterr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.lerr"><span class="id" title="lemma">lerr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltrr"><span class="id" title="lemma">ltrr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addr_ge0"><span class="id" title="lemma">addr_ge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerifP"><span class="id" title="lemma">lerifP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_asym"><span class="id" title="lemma">ltr_asym</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_anti"><span class="id" title="lemma">ler_anti</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#antisymmetric"><span class="id" title="definition">antisymmetric</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.ler"><span class="id" title="definition">ler</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_le_asym"><span class="id" title="lemma">ltr_le_asym</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cbae941009021cd5693066355a023dd1"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cbae941009021cd5693066355a023dd1"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_lt_asym"><span class="id" title="lemma">ler_lt_asym</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#93a29a58d1f90b8a91702885cf86161e"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#93a29a58d1f90b8a91702885cf86161e"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_anti"><span class="id" title="definition">lter_anti</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#f698c37cdb866c2369b5ae93e22451f0"><span class="id" title="notation">=^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.eqr_le"><span class="id" title="lemma">eqr_le</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_asym"><span class="id" title="lemma">ltr_asym</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_le_asym"><span class="id" title="lemma">ltr_le_asym</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_lt_asym"><span class="id" title="lemma">ler_lt_asym</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_geF"><span class="id" title="lemma">ltr_geF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_gtF"><span class="id" title="lemma">ler_gtF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ltr_gtF"><span class="id" title="definition">ltr_gtF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">hxy</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_gtF"><span class="id" title="lemma">ler_gtF</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltrW"><span class="id" title="lemma">ltrW</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#hxy"><span class="id" title="variable">hxy</span></a>).<br/> + +<br/> +</div> + +<div class="doc"> + Norm and order properties. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_le0"><span class="id" title="lemma">normr_le0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_lt0"><span class="id" title="lemma">normr_lt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_gt0"><span class="id" title="lemma">normr_gt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.normrE"><span class="id" title="definition">normrE</span></a> <span class="id" title="var">x</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr_id"><span class="id" title="lemma">normr_id</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr0"><span class="id" title="lemma">normr0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr1"><span class="id" title="lemma">normr1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normrN1"><span class="id" title="lemma">normrN1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr_ge0"><span class="id" title="lemma">normr_ge0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr_eq0"><span class="id" title="definition">normr_eq0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr_lt0"><span class="id" title="lemma">normr_lt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr_le0"><span class="id" title="lemma">normr_le0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normr_gt0"><span class="id" title="lemma">normr_gt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.normrN"><span class="id" title="lemma">normrN</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainTheory"><span class="id" title="section">NumIntegralDomainTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hint Resolve</span> @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler01"><span class="id" title="lemma">ler01</span></a> @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr01"><span class="id" title="lemma">ltr01</span></a> <span class="id" title="var">lerr</span> <span class="id" title="var">ltrr</span> <span class="id" title="var">ltrW</span> <span class="id" title="var">ltr_eqF</span> <span class="id" title="var">ltr0Sn</span> <span class="id" title="var">ler0n</span> <span class="id" title="var">normr_ge0</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.NumIntegralDomainMonotonyTheory"><span class="id" title="section">NumIntegralDomainMonotonyTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="Num.Theory.NumIntegralDomainMonotonyTheory.R"><span class="id" title="variable">R</span></a> <a name="Num.Theory.NumIntegralDomainMonotonyTheory.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/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</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="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.R"><span class="id" title="variable">R</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span> <span class="id" title="var">w</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.R'"><span class="id" title="variable">R'</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes"><span class="id" title="section">AcrossTypes</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a name="Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.R"><span class="id" title="variable">R</span></a>.<br/> +<span class="id" title="keyword">Variable</span> (<a name="Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.R'"><span class="id" title="variable">R'</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrW_homo"><span class="id" title="lemma">ltrW_homo</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrW_nhomo"><span class="id" title="lemma">ltrW_nhomo</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.homo_inj_lt"><span class="id" title="lemma">homo_inj_lt</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nhomo_inj_lt"><span class="id" title="lemma">nhomo_inj_lt</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mono_inj"><span class="id" title="lemma">mono_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmono_inj"><span class="id" title="lemma">nmono_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerW_mono"><span class="id" title="lemma">lerW_mono</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerW_nmono"><span class="id" title="lemma">lerW_nmono</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Monotony in D D' +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrW_homo_in"><span class="id" title="lemma">ltrW_homo_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrW_nhomo_in"><span class="id" title="lemma">ltrW_nhomo_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.homo_inj_in_lt"><span class="id" title="lemma">homo_inj_in_lt</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nhomo_inj_in_lt"><span class="id" title="lemma">nhomo_inj_in_lt</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D'"><span class="id" title="variable">D'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mono_inj_in"><span class="id" title="lemma">mono_inj_in</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmono_inj_in"><span class="id" title="lemma">nmono_inj_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerW_mono_in"><span class="id" title="lemma">lerW_mono_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerW_nmono_in"><span class="id" title="lemma">lerW_nmono_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.AcrossTypes"><span class="id" title="section">AcrossTypes</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.NumIntegralDomainMonotonyTheory.NatToR"><span class="id" title="section">NatToR</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> (<a name="Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltn_ltrW_homo"><span class="id" title="lemma">ltn_ltrW_homo</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltn_ltrW_nhomo"><span class="id" title="lemma">ltn_ltrW_nhomo</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.homo_inj_ltn_lt"><span class="id" title="lemma">homo_inj_ltn_lt</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nhomo_inj_ltn_lt"><span class="id" title="lemma">nhomo_inj_ltn_lt</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.leq_mono_inj"><span class="id" title="lemma">leq_mono_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.leq_nmono_inj"><span class="id" title="lemma">leq_nmono_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.leq_lerW_mono"><span class="id" title="lemma">leq_lerW_mono</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.leq_lerW_nmono"><span class="id" title="lemma">leq_lerW_nmono</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.homo_leq_mono"><span class="id" title="lemma">homo_leq_mono</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nhomo_leq_mono"><span class="id" title="lemma">nhomo_leq_mono</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory.NatToR"><span class="id" title="section">NatToR</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumIntegralDomainMonotonyTheory"><span class="id" title="section">NumIntegralDomainMonotonyTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.NumDomainOperationTheory"><span class="id" title="section">NumDomainOperationTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.NumDomainOperationTheory.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/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Comparision and opposite. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_opp2"><span class="id" title="lemma">ler_opp2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#9fdf1a446ceec36bc97cce801a3ef3f2"><span class="id" title="notation">-%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9fdf1a446ceec36bc97cce801a3ef3f2"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a>.<br/> + <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">ler_opp2</span>.<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_opp2"><span class="id" title="lemma">ltr_opp2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#9fdf1a446ceec36bc97cce801a3ef3f2"><span class="id" title="notation">-%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9fdf1a446ceec36bc97cce801a3ef3f2"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a>.<br/> + <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">ltr_opp2</span>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_opp2"><span class="id" title="definition">lter_opp2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_opp2"><span class="id" title="lemma">ler_opp2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_opp2"><span class="id" title="lemma">ltr_opp2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_oppr"><span class="id" title="lemma">ler_oppr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_oppr"><span class="id" title="lemma">ltr_oppr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_oppr"><span class="id" title="definition">lter_oppr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_oppr"><span class="id" title="lemma">ler_oppr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_oppr"><span class="id" title="lemma">ltr_oppr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_oppl"><span class="id" title="lemma">ler_oppl</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_oppl"><span class="id" title="lemma">ltr_oppl</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_oppl"><span class="id" title="definition">lter_oppl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_oppl"><span class="id" title="lemma">ler_oppl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_oppl"><span class="id" title="lemma">ltr_oppl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.oppr_ge0"><span class="id" title="lemma">oppr_ge0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.oppr_gt0"><span class="id" title="lemma">oppr_gt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.oppr_gte0"><span class="id" title="definition">oppr_gte0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.oppr_ge0"><span class="id" title="lemma">oppr_ge0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.oppr_gt0"><span class="id" title="lemma">oppr_gt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.oppr_le0"><span class="id" title="lemma">oppr_le0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.oppr_lt0"><span class="id" title="lemma">oppr_lt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.oppr_lte0"><span class="id" title="definition">oppr_lte0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.oppr_le0"><span class="id" title="lemma">oppr_le0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.oppr_lt0"><span class="id" title="lemma">oppr_lt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.oppr_cp0"><span class="id" title="definition">oppr_cp0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.oppr_gte0"><span class="id" title="definition">oppr_gte0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.oppr_lte0"><span class="id" title="definition">oppr_lte0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_oppE"><span class="id" title="definition">lter_oppE</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.oppr_cp0"><span class="id" title="definition">oppr_cp0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.lter_opp2"><span class="id" title="definition">lter_opp2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ge0_cp"><span class="id" title="lemma">ge0_cp</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gt0_cp"><span class="id" title="lemma">gt0_cp</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.le0_cp"><span class="id" title="lemma">le0_cp</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lt0_cp"><span class="id" title="lemma">lt0_cp</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Properties of the real subset. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger0_real"><span class="id" title="lemma">ger0_real</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler0_real"><span class="id" title="lemma">ler0_real</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr0_real"><span class="id" title="lemma">gtr0_real</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr0_real"><span class="id" title="lemma">ltr0_real</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real0"><span class="id" title="lemma">real0</span></a> : 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">real0</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real1"><span class="id" title="lemma">real1</span></a> : 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">real1</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realn"><span class="id" title="lemma">realn</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_leVge"><span class="id" title="lemma">ler_leVge</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_leVge"><span class="id" title="lemma">real_leVge</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realB"><span class="id" title="lemma">realB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realN"><span class="id" title="lemma">realN</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.opp"><span class="id" title="definition">GRing.opp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + :TODO: add a rpredBC in ssralg +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realBC"><span class="id" title="lemma">realBC</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realD"><span class="id" title="lemma">realD</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + dichotomy and trichotomy +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.ler_xor_gt"><span class="id" title="inductive">ler_xor_gt</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/> + | <a name="Num.Theory.LerNotGt"><span class="id" title="constructor">LerNotGt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#ler_xor_gt"><span class="id" title="inductive">ler_xor_gt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/> + | <a name="Num.Theory.GtrNotLe"><span class="id" title="constructor">GtrNotLe</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#ler_xor_gt"><span class="id" title="inductive">ler_xor_gt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.ltr_xor_ge"><span class="id" title="inductive">ltr_xor_ge</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/> + | <a name="Num.Theory.LtrNotGe"><span class="id" title="constructor">LtrNotGe</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#ltr_xor_ge"><span class="id" title="inductive">ltr_xor_ge</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> + | <a name="Num.Theory.GerNotLt"><span class="id" title="constructor">GerNotLt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#ltr_xor_ge"><span class="id" title="inductive">ltr_xor_ge</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.comparer"><span class="id" title="inductive">comparer</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/> + | <a name="Num.Theory.ComparerLt"><span class="id" title="constructor">ComparerLt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#comparer"><span class="id" title="inductive">comparer</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/> + | <a name="Num.Theory.ComparerGt"><span class="id" title="constructor">ComparerGt</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#comparer"><span class="id" title="inductive">comparer</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>)<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> + | <a name="Num.Theory.ComparerEq"><span class="id" title="constructor">ComparerEq</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#comparer"><span class="id" title="inductive">comparer</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> 0 0<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_lerP"><span class="id" title="lemma">real_lerP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_xor_gt"><span class="id" title="inductive">ler_xor_gt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ltrP"><span class="id" title="lemma">real_ltrP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_xor_ge"><span class="id" title="inductive">ltr_xor_ge</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ltrNge"><span class="id" title="lemma">real_ltrNge</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_lerNgt"><span class="id" title="lemma">real_lerNgt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ltrgtP"><span class="id" title="lemma">real_ltrgtP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.comparer"><span class="id" title="inductive">comparer</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><br/> + (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.ger0_xor_lt0"><span class="id" title="inductive">ger0_xor_lt0</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/> + | <a name="Num.Theory.Ger0NotLt0"><span class="id" title="constructor">Ger0NotLt0</span></a> <span class="id" title="keyword">of</span> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#ger0_xor_lt0"><span class="id" title="inductive">ger0_xor_lt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> + | <a name="Num.Theory.Ltr0NotGe0"><span class="id" title="constructor">Ltr0NotGe0</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 : <a class="idref" href="mathcomp.algebra.ssrnum.html#ger0_xor_lt0"><span class="id" title="inductive">ger0_xor_lt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.ler0_xor_gt0"><span class="id" title="inductive">ler0_xor_gt0</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/> + | <a name="Num.Theory.Ler0NotLe0"><span class="id" title="constructor">Ler0NotLe0</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 : <a class="idref" href="mathcomp.algebra.ssrnum.html#ler0_xor_gt0"><span class="id" title="inductive">ler0_xor_gt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> + | <a name="Num.Theory.Gtr0NotGt0"><span class="id" title="constructor">Gtr0NotGt0</span></a> <span class="id" title="keyword">of</span> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#ler0_xor_gt0"><span class="id" title="inductive">ler0_xor_gt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.comparer0"><span class="id" title="inductive">comparer0</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/> + | <a name="Num.Theory.ComparerGt0"><span class="id" title="constructor">ComparerGt0</span></a> <span class="id" title="keyword">of</span> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#comparer0"><span class="id" title="inductive">comparer0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> + | <a name="Num.Theory.ComparerLt0"><span class="id" title="constructor">ComparerLt0</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 : <a class="idref" href="mathcomp.algebra.ssrnum.html#comparer0"><span class="id" title="inductive">comparer0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/> + | <a name="Num.Theory.ComparerEq0"><span class="id" title="constructor">ComparerEq0</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 : <a class="idref" href="mathcomp.algebra.ssrnum.html#comparer0"><span class="id" title="inductive">comparer0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ger0P"><span class="id" title="lemma">real_ger0P</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ger0_xor_lt0"><span class="id" title="inductive">ger0_xor_lt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0) (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ler0P"><span class="id" title="lemma">real_ler0P</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler0_xor_gt0"><span class="id" title="inductive">ler0_xor_gt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ltrgt0P"><span class="id" title="lemma">real_ltrgt0P</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.comparer0"><span class="id" title="inductive">comparer0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0) (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_neqr_lt"><span class="id" title="lemma">real_neqr_lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_sub_real"><span class="id" title="lemma">ler_sub_real</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_sub_real"><span class="id" title="lemma">ger_sub_real</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_real"><span class="id" title="lemma">ler_real</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_real"><span class="id" title="lemma">ger_real</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger1_real"><span class="id" title="lemma">ger1_real</span></a> <span class="id" title="var">x</span> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler1_real"><span class="id" title="lemma">ler1_real</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Nreal_leF"><span class="id" title="lemma">Nreal_leF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Nreal_geF"><span class="id" title="lemma">Nreal_geF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Nreal_ltF"><span class="id" title="lemma">Nreal_ltF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Nreal_gtF"><span class="id" title="lemma">Nreal_gtF</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + real wlog +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_wlog_ler"><span class="id" title="lemma">real_wlog_ler</span></a> <span class="id" title="var">P</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>, <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_wlog_ltr"><span class="id" title="lemma">real_wlog_ltr</span></a> <span class="id" title="var">P</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, (<a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>, <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Monotony of addition +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_add2l"><span class="id" title="lemma">ler_add2l</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_add2r"><span class="id" title="lemma">ler_add2r</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_add2r"><span class="id" title="lemma">ltr_add2r</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_add2l"><span class="id" title="lemma">ltr_add2l</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ler_add2"><span class="id" title="definition">ler_add2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_add2l"><span class="id" title="lemma">ler_add2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_add2r"><span class="id" title="lemma">ler_add2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ltr_add2"><span class="id" title="definition">ltr_add2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_add2l"><span class="id" title="lemma">ltr_add2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_add2r"><span class="id" title="lemma">ltr_add2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_add2"><span class="id" title="definition">lter_add2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_add2"><span class="id" title="definition">ler_add2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_add2"><span class="id" title="definition">ltr_add2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Addition, subtraction and transitivity +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_add"><span class="id" title="lemma">ler_add</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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_lt_add"><span class="id" title="lemma">ler_lt_add</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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_le_add"><span class="id" title="lemma">ltr_le_add</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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_add"><span class="id" title="lemma">ltr_add</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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_sub"><span class="id" title="lemma">ler_sub</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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_lt_sub"><span class="id" title="lemma">ler_lt_sub</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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_le_sub"><span class="id" title="lemma">ltr_le_sub</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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_sub"><span class="id" title="lemma">ltr_sub</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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_subl_addr"><span class="id" title="lemma">ler_subl_addr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_subl_addr"><span class="id" title="lemma">ltr_subl_addr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_subr_addr"><span class="id" title="lemma">ler_subr_addr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_subr_addr"><span class="id" title="lemma">ltr_subr_addr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ler_sub_addr"><span class="id" title="definition">ler_sub_addr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_subl_addr"><span class="id" title="lemma">ler_subl_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_subr_addr"><span class="id" title="lemma">ler_subr_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ltr_sub_addr"><span class="id" title="definition">ltr_sub_addr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_subl_addr"><span class="id" title="lemma">ltr_subl_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_subr_addr"><span class="id" title="lemma">ltr_subr_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_sub_addr"><span class="id" title="definition">lter_sub_addr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_sub_addr"><span class="id" title="definition">ler_sub_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_sub_addr"><span class="id" title="definition">ltr_sub_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_subl_addl"><span class="id" title="lemma">ler_subl_addl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_subl_addl"><span class="id" title="lemma">ltr_subl_addl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_subr_addl"><span class="id" title="lemma">ler_subr_addl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_subr_addl"><span class="id" title="lemma">ltr_subr_addl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ler_sub_addl"><span class="id" title="definition">ler_sub_addl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_subl_addl"><span class="id" title="lemma">ler_subl_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_subr_addl"><span class="id" title="lemma">ler_subr_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ltr_sub_addl"><span class="id" title="definition">ltr_sub_addl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_subl_addl"><span class="id" title="lemma">ltr_subl_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_subr_addl"><span class="id" title="lemma">ltr_subr_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_sub_addl"><span class="id" title="definition">lter_sub_addl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_sub_addl"><span class="id" title="definition">ler_sub_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_sub_addl"><span class="id" title="definition">ltr_sub_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_addl"><span class="id" title="lemma">ler_addl</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_addl"><span class="id" title="lemma">ltr_addl</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_addr"><span class="id" title="lemma">ler_addr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_addr"><span class="id" title="lemma">ltr_addr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_addl"><span class="id" title="lemma">ger_addl</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr_addl"><span class="id" title="lemma">gtr_addl</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_addr"><span class="id" title="lemma">ger_addr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr_addr"><span class="id" title="lemma">gtr_addr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.cpr_add"><span class="id" title="definition">cpr_add</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_addl"><span class="id" title="lemma">ler_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_addr"><span class="id" title="lemma">ler_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ger_addl"><span class="id" title="lemma">ger_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ger_addl"><span class="id" title="lemma">ger_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_addl"><span class="id" title="lemma">ltr_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_addr"><span class="id" title="lemma">ltr_addr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.gtr_addl"><span class="id" title="lemma">gtr_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.gtr_addl"><span class="id" title="lemma">gtr_addl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Addition with left member knwon to be positive/negative +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_paddl"><span class="id" title="lemma">ler_paddl</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_paddl"><span class="id" title="lemma">ltr_paddl</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_spaddl"><span class="id" title="lemma">ltr_spaddl</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_spsaddl"><span class="id" title="lemma">ltr_spsaddl</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_naddl"><span class="id" title="lemma">ler_naddl</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_naddl"><span class="id" title="lemma">ltr_naddl</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_snaddl"><span class="id" title="lemma">ltr_snaddl</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_snsaddl"><span class="id" title="lemma">ltr_snsaddl</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Addition with right member we know positive/negative +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_paddr"><span class="id" title="lemma">ler_paddr</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_paddr"><span class="id" title="lemma">ltr_paddr</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_spaddr"><span class="id" title="lemma">ltr_spaddr</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_spsaddr"><span class="id" title="lemma">ltr_spsaddr</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_naddr"><span class="id" title="lemma">ler_naddr</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_naddr"><span class="id" title="lemma">ltr_naddr</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_snaddr"><span class="id" title="lemma">ltr_snaddr</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_snsaddr"><span class="id" title="lemma">ltr_snsaddr</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + x and y have the same sign and their sum is null +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.paddr_eq0"><span class="id" title="lemma">paddr_eq0</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.naddr_eq0"><span class="id" title="lemma">naddr_eq0</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addr_ss_eq0"><span class="id" title="lemma">addr_ss_eq0</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + big sum and ler +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sumr_ge0"><span class="id" title="lemma">sumr_ge0</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_sum"><span class="id" title="lemma">ler_sum</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.psumr_eq0"><span class="id" title="lemma">psumr_eq0</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">==></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + :TODO: Cyril : See which form to keep +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.psumr_eq0P"><span class="id" title="lemma">psumr_eq0P</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + mulr and ler/ltr +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pmul2l"><span class="id" title="lemma">ler_pmul2l</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pmul2l"><span class="id" title="lemma">ltr_pmul2l</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_pmul2l"><span class="id" title="definition">lter_pmul2l</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_pmul2l"><span class="id" title="lemma">ler_pmul2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_pmul2l"><span class="id" title="lemma">ltr_pmul2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pmul2r"><span class="id" title="lemma">ler_pmul2r</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pmul2r"><span class="id" title="lemma">ltr_pmul2r</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_pmul2r"><span class="id" title="definition">lter_pmul2r</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_pmul2r"><span class="id" title="lemma">ler_pmul2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_pmul2r"><span class="id" title="lemma">ltr_pmul2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nmul2l"><span class="id" title="lemma">ler_nmul2l</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_nmul2l"><span class="id" title="lemma">ltr_nmul2l</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_nmul2l"><span class="id" title="definition">lter_nmul2l</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_nmul2l"><span class="id" title="lemma">ler_nmul2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_nmul2l"><span class="id" title="lemma">ltr_nmul2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nmul2r"><span class="id" title="lemma">ler_nmul2r</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_nmul2r"><span class="id" title="lemma">ltr_nmul2r</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_nmul2r"><span class="id" title="definition">lter_nmul2r</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_nmul2r"><span class="id" title="lemma">ler_nmul2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_nmul2r"><span class="id" title="lemma">ltr_nmul2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wpmul2l"><span class="id" title="lemma">ler_wpmul2l</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wpmul2r"><span class="id" title="lemma">ler_wpmul2r</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wnmul2l"><span class="id" title="lemma">ler_wnmul2l</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wnmul2r"><span class="id" title="lemma">ler_wnmul2r</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Binary forms, for backchaining. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pmul"><span class="id" title="lemma">ler_pmul</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">y1</span> <span class="id" title="var">x2</span> <span class="id" title="var">y2</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pmul"><span class="id" title="lemma">ltr_pmul</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">y1</span> <span class="id" title="var">x2</span> <span class="id" title="var">y2</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + complement for x *+ n and <= or < +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pmuln2r"><span class="id" title="lemma">ler_pmuln2r</span></a> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pmuln2r"><span class="id" title="lemma">ltr_pmuln2r</span></a> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrnI"><span class="id" title="lemma">pmulrnI</span></a> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_pmuln2r"><span class="id" title="lemma">eqr_pmuln2r</span></a> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrn_lgt0"><span class="id" title="lemma">pmulrn_lgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrn_llt0"><span class="id" title="lemma">pmulrn_llt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrn_lge0"><span class="id" title="lemma">pmulrn_lge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrn_lle0"><span class="id" title="lemma">pmulrn_lle0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_wmuln2r"><span class="id" title="lemma">ltr_wmuln2r</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_wpmuln2r"><span class="id" title="lemma">ltr_wpmuln2r</span></a> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wmuln2r"><span class="id" title="lemma">ler_wmuln2r</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulrn_wge0"><span class="id" title="lemma">mulrn_wge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulrn_wle0"><span class="id" title="lemma">mulrn_wle0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_muln2r"><span class="id" title="lemma">ler_muln2r</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_muln2r"><span class="id" title="lemma">ltr_muln2r</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>(0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_muln2r"><span class="id" title="lemma">eqr_muln2r</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + More characteristic zero properties. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulrn_eq0"><span class="id" title="lemma">mulrn_eq0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>(<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulrIn"><span class="id" title="lemma">mulrIn</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wpmuln2l"><span class="id" title="lemma">ler_wpmuln2l</span></a> <span class="id" title="var">x</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wnmuln2l"><span class="id" title="lemma">ler_wnmuln2l</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulrn_wgt0"><span class="id" title="lemma">mulrn_wgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulrn_wlt0"><span class="id" title="lemma">mulrn_wlt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pmuln2l"><span class="id" title="lemma">ler_pmuln2l</span></a> <span class="id" title="var">x</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pmuln2l"><span class="id" title="lemma">ltr_pmuln2l</span></a> <span class="id" title="var">x</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nmuln2l"><span class="id" title="lemma">ler_nmuln2l</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_nmuln2l"><span class="id" title="lemma">ltr_nmuln2l</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.natmul"><span class="id" title="definition">GRing.natmul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nat"><span class="id" title="lemma">ler_nat</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_nat"><span class="id" title="lemma">ltr_nat</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_nat"><span class="id" title="lemma">eqr_nat</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#340b60eb5a3e9913f807040630cb8d43"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#340b60eb5a3e9913f807040630cb8d43"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pnatr_eq1"><span class="id" title="lemma">pnatr_eq1</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#340b60eb5a3e9913f807040630cb8d43"><span class="id" title="notation">==</span></a> 1 <a class="idref" href="mathcomp.ssreflect.eqtype.html#340b60eb5a3e9913f807040630cb8d43"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lern0"><span class="id" title="lemma">lern0</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrn0"><span class="id" title="lemma">ltrn0</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler1n"><span class="id" title="lemma">ler1n</span></a> <span class="id" title="var">n</span> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr1n"><span class="id" title="lemma">ltr1n</span></a> <span class="id" title="var">n</span> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lern1"><span class="id" title="lemma">lern1</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> 1)%<span class="id" title="var">N</span>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrn1"><span class="id" title="lemma">ltrn1</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> 1)%<span class="id" title="var">N</span>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrN10"><span class="id" title="lemma">ltrN10</span></a> : -1 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerN10"><span class="id" title="lemma">lerN10</span></a> : -1 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr10"><span class="id" title="lemma">ltr10</span></a> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler10"><span class="id" title="lemma">ler10</span></a> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr0N1"><span class="id" title="lemma">ltr0N1</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> -1 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler0N1"><span class="id" title="lemma">ler0N1</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> -1 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrn_rgt0"><span class="id" title="lemma">pmulrn_rgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrn_rlt0"><span class="id" title="lemma">pmulrn_rlt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrn_rge0"><span class="id" title="lemma">pmulrn_rge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulrn_rle0"><span class="id" title="lemma">pmulrn_rle0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulrn_rgt0"><span class="id" title="lemma">nmulrn_rgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulrn_rge0"><span class="id" title="lemma">nmulrn_rge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulrn_rle0"><span class="id" title="lemma">nmulrn_rle0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0.<br/> + +<br/> +</div> + +<div class="doc"> + (x * y) compared to 0 + Remark : pmulr_rgt0 and pmulr_rge0 are defined above +<div class="paragraph"> </div> + + x positive and y right +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulr_rlt0"><span class="id" title="lemma">pmulr_rlt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulr_rle0"><span class="id" title="lemma">pmulr_rle0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + x positive and y left +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulr_lgt0"><span class="id" title="lemma">pmulr_lgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulr_lge0"><span class="id" title="lemma">pmulr_lge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulr_llt0"><span class="id" title="lemma">pmulr_llt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pmulr_lle0"><span class="id" title="lemma">pmulr_lle0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + x negative and y right +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulr_rgt0"><span class="id" title="lemma">nmulr_rgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulr_rge0"><span class="id" title="lemma">nmulr_rge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulr_rlt0"><span class="id" title="lemma">nmulr_rlt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulr_rle0"><span class="id" title="lemma">nmulr_rle0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + x negative and y left +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulr_lgt0"><span class="id" title="lemma">nmulr_lgt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulr_lge0"><span class="id" title="lemma">nmulr_lge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulr_llt0"><span class="id" title="lemma">nmulr_llt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmulr_lle0"><span class="id" title="lemma">nmulr_lle0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + weak and symmetric lemmas +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_ge0"><span class="id" title="lemma">mulr_ge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_le0"><span class="id" title="lemma">mulr_le0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_ge0_le0"><span class="id" title="lemma">mulr_ge0_le0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_le0_ge0"><span class="id" title="lemma">mulr_le0_ge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0.<br/> + +<br/> +</div> + +<div class="doc"> + mulr_gt0 with only one case +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_gt0"><span class="id" title="lemma">mulr_gt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Iterated products +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.prodr_ge0"><span class="id" title="lemma">prodr_ge0</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.prodr_gt0"><span class="id" title="lemma">prodr_gt0</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_prod"><span class="id" title="lemma">ler_prod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E1</span> <span class="id" title="var">E2</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_prod"><span class="id" title="lemma">ltr_prod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E1</span> <span class="id" title="var">E2</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#93a29a58d1f90b8a91702885cf86161e"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#93a29a58d1f90b8a91702885cf86161e"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_prod_nat"><span class="id" title="lemma">ltr_prod_nat</span></a> (<span class="id" title="var">E1</span> <span class="id" title="var">E2</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">n</span> <span class="id" title="var">m</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) :<br/> + (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#a37ed901e2f16b7d06c569763fc8034f"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#a37ed901e2f16b7d06c569763fc8034f"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#93a29a58d1f90b8a91702885cf86161e"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#93a29a58d1f90b8a91702885cf86161e"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + real of mul +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realMr"><span class="id" title="lemma">realMr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realrM"><span class="id" title="lemma">realrM</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realM"><span class="id" title="lemma">realM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realrMn"><span class="id" title="lemma">realrMn</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + ler/ltr and multiplication between a positive/negative +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_pmull"><span class="id" title="lemma">ger_pmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr_pmull"><span class="id" title="lemma">gtr_pmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_pmulr"><span class="id" title="lemma">ger_pmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr_pmulr"><span class="id" title="lemma">gtr_pmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pmull"><span class="id" title="lemma">ler_pmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pmull"><span class="id" title="lemma">ltr_pmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pmulr"><span class="id" title="lemma">ler_pmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pmulr"><span class="id" title="lemma">ltr_pmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_nmull"><span class="id" title="lemma">ger_nmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr_nmull"><span class="id" title="lemma">gtr_nmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_nmulr"><span class="id" title="lemma">ger_nmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr_nmulr"><span class="id" title="lemma">gtr_nmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nmull"><span class="id" title="lemma">ler_nmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_nmull"><span class="id" title="lemma">ltr_nmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nmulr"><span class="id" title="lemma">ler_nmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_nmulr"><span class="id" title="lemma">ltr_nmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + ler/ltr and multiplication between a positive/negative + and a exterior (1 <= _) or interior (0 <= _ <= 1) +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pemull"><span class="id" title="lemma">ler_pemull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nemull"><span class="id" title="lemma">ler_nemull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pemulr"><span class="id" title="lemma">ler_pemulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nemulr"><span class="id" title="lemma">ler_nemulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pimull"><span class="id" title="lemma">ler_pimull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nimull"><span class="id" title="lemma">ler_nimull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pimulr"><span class="id" title="lemma">ler_pimulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nimulr"><span class="id" title="lemma">ler_nimulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_ile1"><span class="id" title="lemma">mulr_ile1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_ilt1"><span class="id" title="lemma">mulr_ilt1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.mulr_ilte1"><span class="id" title="definition">mulr_ilte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mulr_ile1"><span class="id" title="lemma">mulr_ile1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mulr_ilt1"><span class="id" title="lemma">mulr_ilt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_ege1"><span class="id" title="lemma">mulr_ege1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_egt1"><span class="id" title="lemma">mulr_egt1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.mulr_egte1"><span class="id" title="definition">mulr_egte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mulr_ege1"><span class="id" title="lemma">mulr_ege1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mulr_egt1"><span class="id" title="lemma">mulr_egt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.mulr_cp1"><span class="id" title="definition">mulr_cp1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mulr_ilte1"><span class="id" title="definition">mulr_ilte1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mulr_egte1"><span class="id" title="definition">mulr_egte1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + ler and ^-1 +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_gt0"><span class="id" title="lemma">invr_gt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_ge0"><span class="id" title="lemma">invr_ge0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_lt0"><span class="id" title="lemma">invr_lt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_le0"><span class="id" title="lemma">invr_le0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.invr_gte0"><span class="id" title="definition">invr_gte0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_ge0"><span class="id" title="lemma">invr_ge0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_gt0"><span class="id" title="lemma">invr_gt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.invr_lte0"><span class="id" title="definition">invr_lte0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_le0"><span class="id" title="lemma">invr_le0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_lt0"><span class="id" title="lemma">invr_lt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.divr_ge0"><span class="id" title="lemma">divr_ge0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.divr_gt0"><span class="id" title="lemma">divr_gt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realV"><span class="id" title="lemma">realV</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#69ee97879e4a4ae19a99125173c5741e"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + ler and exprn +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_ge0"><span class="id" title="lemma">exprn_ge0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realX"><span class="id" title="lemma">realX</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_gt0"><span class="id" title="lemma">exprn_gt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.exprn_gte0"><span class="id" title="definition">exprn_gte0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.exprn_ge0"><span class="id" title="lemma">exprn_ge0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.exprn_gt0"><span class="id" title="lemma">exprn_gt0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_ile1"><span class="id" title="lemma">exprn_ile1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_ilt1"><span class="id" title="lemma">exprn_ilt1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.exprn_ilte1"><span class="id" title="definition">exprn_ilte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.exprn_ile1"><span class="id" title="lemma">exprn_ile1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.exprn_ilt1"><span class="id" title="lemma">exprn_ilt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_ege1"><span class="id" title="lemma">exprn_ege1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_egt1"><span class="id" title="lemma">exprn_egt1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.exprn_egte1"><span class="id" title="definition">exprn_egte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.exprn_ege1"><span class="id" title="lemma">exprn_ege1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.exprn_egt1"><span class="id" title="lemma">exprn_egt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.exprn_cp1"><span class="id" title="definition">exprn_cp1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.exprn_ilte1"><span class="id" title="definition">exprn_ilte1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.exprn_egte1"><span class="id" title="definition">exprn_egte1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_iexpr"><span class="id" title="lemma">ler_iexpr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_iexpr"><span class="id" title="lemma">ltr_iexpr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_iexpr"><span class="id" title="definition">lter_iexpr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_iexpr"><span class="id" title="lemma">ler_iexpr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_iexpr"><span class="id" title="lemma">ltr_iexpr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_eexpr"><span class="id" title="lemma">ler_eexpr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_eexpr"><span class="id" title="lemma">ltr_eexpr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_eexpr"><span class="id" title="definition">lter_eexpr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_eexpr"><span class="id" title="lemma">ler_eexpr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_eexpr"><span class="id" title="lemma">ltr_eexpr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_expr"><span class="id" title="definition">lter_expr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.lter_iexpr"><span class="id" title="definition">lter_iexpr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.lter_eexpr"><span class="id" title="definition">lter_eexpr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wiexpn2l"><span class="id" title="lemma">ler_wiexpn2l</span></a> <span class="id" title="var">x</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_weexpn2l"><span class="id" title="lemma">ler_weexpn2l</span></a> <span class="id" title="var">x</span> :<br/> + 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f95d8eadb7771a72d2066af1aebc7110"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ieexprn_weq1"><span class="id" title="lemma">ieexprn_weq1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ieexprIn"><span class="id" title="lemma">ieexprIn</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_iexpn2l"><span class="id" title="lemma">ler_iexpn2l</span></a> <span class="id" title="var">x</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_iexpn2l"><span class="id" title="lemma">ltr_iexpn2l</span></a> <span class="id" title="var">x</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_iexpn2l"><span class="id" title="definition">lter_iexpn2l</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_iexpn2l"><span class="id" title="lemma">ler_iexpn2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_iexpn2l"><span class="id" title="lemma">ltr_iexpn2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_eexpn2l"><span class="id" title="lemma">ler_eexpn2l</span></a> <span class="id" title="var">x</span> :<br/> + 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_eexpn2l"><span class="id" title="lemma">ltr_eexpn2l</span></a> <span class="id" title="var">x</span> :<br/> + 1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">:</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#aed5e67b67900c70888bda2ff72f1de4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_eexpn2l"><span class="id" title="definition">lter_eexpn2l</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_eexpn2l"><span class="id" title="lemma">ler_eexpn2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_eexpn2l"><span class="id" title="lemma">ltr_eexpn2l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_expn2r"><span class="id" title="lemma">ltr_expn2r</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_expn2r"><span class="id" title="lemma">ler_expn2r</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_expn2r"><span class="id" title="definition">lter_expn2r</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_expn2r"><span class="id" title="lemma">ler_expn2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_expn2r"><span class="id" title="lemma">ltr_expn2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_wpexpn2r"><span class="id" title="lemma">ltr_wpexpn2r</span></a> <span class="id" title="var">n</span> :<br/> + (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pexpn2r"><span class="id" title="lemma">ler_pexpn2r</span></a> <span class="id" title="var">n</span> :<br/> + (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pexpn2r"><span class="id" title="lemma">ltr_pexpn2r</span></a> <span class="id" title="var">n</span> :<br/> + (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_pexpn2r"><span class="id" title="definition">lter_pexpn2r</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_pexpn2r"><span class="id" title="lemma">ler_pexpn2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_pexpn2r"><span class="id" title="lemma">ltr_pexpn2r</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pexpIrn"><span class="id" title="lemma">pexpIrn</span></a> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.exp"><span class="id" title="definition">GRing.exp</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + expr and ler/ltr +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.expr_le1"><span class="id" title="lemma">expr_le1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.expr_lt1"><span class="id" title="lemma">expr_lt1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.expr_lte1"><span class="id" title="definition">expr_lte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.expr_le1"><span class="id" title="lemma">expr_le1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.expr_lt1"><span class="id" title="lemma">expr_lt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.expr_ge1"><span class="id" title="lemma">expr_ge1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.expr_gt1"><span class="id" title="lemma">expr_gt1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.expr_gte1"><span class="id" title="definition">expr_gte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.expr_ge1"><span class="id" title="lemma">expr_ge1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.expr_gt1"><span class="id" title="lemma">expr_gt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pexpr_eq1"><span class="id" title="lemma">pexpr_eq1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.pexprn_eq1"><span class="id" title="lemma">pexprn_eq1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_expn2"><span class="id" title="lemma">eqr_expn2</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrp_eq1"><span class="id" title="lemma">sqrp_eq1</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrn_eq1"><span class="id" title="lemma">sqrn_eq1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> -1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_sqr"><span class="id" title="lemma">ler_sqr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">(</span></a><span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_sqr"><span class="id" title="lemma">ltr_sqr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">(</span></a><span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pinv"><span class="id" title="lemma">ler_pinv</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">pred</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">|</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_ninv"><span class="id" title="lemma">ler_ninv</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">pred</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pinv"><span class="id" title="lemma">ltr_pinv</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">pred</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">|</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_ninv"><span class="id" title="lemma">ltr_ninv</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">pred</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_gt1"><span class="id" title="lemma">invr_gt1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_ge1"><span class="id" title="lemma">invr_ge1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.invr_gte1"><span class="id" title="definition">invr_gte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_ge1"><span class="id" title="lemma">invr_ge1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_gt1"><span class="id" title="lemma">invr_gt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_le1"><span class="id" title="lemma">invr_le1</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">ux</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a>) (<span class="id" title="var">hx</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_lt1"><span class="id" title="lemma">invr_lt1</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">ux</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a>) (<span class="id" title="var">hx</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.invr_lte1"><span class="id" title="definition">invr_lte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_le1"><span class="id" title="lemma">invr_le1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_lt1"><span class="id" title="lemma">invr_lt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.invr_cp1"><span class="id" title="definition">invr_cp1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_gte1"><span class="id" title="definition">invr_gte1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invr_lte1"><span class="id" title="definition">invr_lte1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + norm +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ler_norm"><span class="id" title="lemma">real_ler_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + norm + add +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_real"><span class="id" title="lemma">normr_real</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>. <br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">normr_real</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_norm_sum"><span class="id" title="lemma">ler_norm_sum</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>):<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_norm_sub"><span class="id" title="lemma">ler_norm_sub</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_dist_add"><span class="id" title="lemma">ler_dist_add</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_sub_norm_add"><span class="id" title="lemma">ler_sub_norm_add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_sub_dist"><span class="id" title="lemma">ler_sub_dist</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_dist_dist"><span class="id" title="lemma">ler_dist_dist</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_dist_norm_add"><span class="id" title="lemma">ler_dist_norm_add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ler_norml"><span class="id" title="lemma">real_ler_norml</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ler_normlP"><span class="id" title="lemma">real_ler_normlP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_eqr_norml"><span class="id" title="lemma">real_eqr_norml</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_eqr_norm2"><span class="id" title="lemma">real_eqr_norm2</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ltr_norml"><span class="id" title="lemma">real_ltr_norml</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.real_lter_norml"><span class="id" title="definition">real_lter_norml</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.real_ler_norml"><span class="id" title="lemma">real_ler_norml</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.real_ltr_norml"><span class="id" title="lemma">real_ltr_norml</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ltr_normlP"><span class="id" title="lemma">real_ltr_normlP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ler_normr"><span class="id" title="lemma">real_ler_normr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ltr_normr"><span class="id" title="lemma">real_ltr_normr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.real_lter_normr"><span class="id" title="definition">real_lter_normr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.real_ler_normr"><span class="id" title="lemma">real_ler_normr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.real_ltr_normr"><span class="id" title="lemma">real_ltr_normr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_nnorml"><span class="id" title="lemma">ler_nnorml</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_nnorml"><span class="id" title="lemma">ltr_nnorml</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_nnormr"><span class="id" title="definition">lter_nnormr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_nnorml"><span class="id" title="lemma">ler_nnorml</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_nnorml"><span class="id" title="lemma">ltr_nnorml</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ler_distl"><span class="id" title="lemma">real_ler_distl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">e</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_ltr_distl"><span class="id" title="lemma">real_ltr_distl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">e</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.real_lter_distl"><span class="id" title="definition">real_lter_distl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.real_ler_distl"><span class="id" title="lemma">real_ler_distl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.real_ltr_distl"><span class="id" title="lemma">real_ltr_distl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="comment">(* GG: pointless duplication }-( *)</span><br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_norm_id"><span class="id" title="lemma">eqr_norm_id</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_normN"><span class="id" title="lemma">eqr_normN</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.eqr_norm_idVN"><span class="id" title="definition">eqr_norm_idVN</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#f698c37cdb866c2369b5ae93e22451f0"><span class="id" title="notation">=^~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ger0_def"><span class="id" title="lemma">ger0_def</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler0_def"><span class="id" title="lemma">ler0_def</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_exprn_even_ge0"><span class="id" title="lemma">real_exprn_even_ge0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_exprn_even_gt0"><span class="id" title="lemma">real_exprn_even_gt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_exprn_even_le0"><span class="id" title="lemma">real_exprn_even_le0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_exprn_even_lt0"><span class="id" title="lemma">real_exprn_even_lt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_exprn_odd_ge0"><span class="id" title="lemma">real_exprn_odd_ge0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_exprn_odd_gt0"><span class="id" title="lemma">real_exprn_odd_gt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_exprn_odd_le0"><span class="id" title="lemma">real_exprn_odd_le0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_exprn_odd_lt0"><span class="id" title="lemma">real_exprn_odd_lt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + GG: Could this be a better definition of "real" ? +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realEsqr"><span class="id" title="lemma">realEsqr</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_normK"><span class="id" title="lemma">real_normK</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2.<br/> + +<br/> +</div> + +<div class="doc"> + Binary sign ((-1) ^+ s). +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_sign"><span class="id" title="lemma">normr_sign</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normrMsign"><span class="id" title="lemma">normrMsign</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.signr_gt0"><span class="id" title="lemma">signr_gt0</span></a> (<span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.signr_lt0"><span class="id" title="lemma">signr_lt0</span></a> (<span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#3c0e77197870a42e4951057d43bba909"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.signr_ge0"><span class="id" title="lemma">signr_ge0</span></a> (<span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.signr_le0"><span class="id" title="lemma">signr_le0</span></a> (<span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#d6cd0798ccf4decf11598a746ae90abf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + This actually holds for char R != 2. +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.signr_inj"><span class="id" title="lemma">signr_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> ⇒ <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +</div> + +<div class="doc"> + Ternary sign (sg). +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_def"><span class="id" title="lemma">sgr_def</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.neqr0_sign"><span class="id" title="lemma">neqr0_sign</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.sgr"><span class="id" title="definition">sgr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.gtr0_sg"><span class="id" title="lemma">gtr0_sg</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr0_sg"><span class="id" title="lemma">ltr0_sg</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> -1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr0"><span class="id" title="lemma">sgr0</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr1"><span class="id" title="lemma">sgr1</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgrN1"><span class="id" title="lemma">sgrN1</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> (-1) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> -1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.sgrE"><span class="id" title="definition">sgrE</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sgr0"><span class="id" title="lemma">sgr0</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sgr1"><span class="id" title="lemma">sgr1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sgrN1"><span class="id" title="lemma">sgrN1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqr_sg"><span class="id" title="lemma">sqr_sg</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_sg_eq1"><span class="id" title="lemma">mulr_sg_eq1</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_sg_eqN1"><span class="id" title="lemma">mulr_sg_eqN1</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> -1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_eq0"><span class="id" title="lemma">sgr_eq0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_odd"><span class="id" title="lemma">sgr_odd</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgrMn"><span class="id" title="lemma">sgrMn</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_nat"><span class="id" title="lemma">sgr_nat</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_id"><span class="id" title="lemma">sgr_id</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_lt0"><span class="id" title="lemma">sgr_lt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_le0"><span class="id" title="lemma">sgr_le0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.sgr"><span class="id" title="definition">sgr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + sign and norm +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realEsign"><span class="id" title="lemma">realEsign</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realNEsign"><span class="id" title="lemma">realNEsign</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_normrEsign"><span class="id" title="lemma">real_normrEsign</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">xR</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>) : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + GG: pointless duplication... +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_mulr_sign_norm"><span class="id" title="lemma">real_mulr_sign_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_mulr_Nsign_norm"><span class="id" title="lemma">real_mulr_Nsign_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.realEsg"><span class="id" title="lemma">realEsg</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.sgr"><span class="id" title="definition">sgr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normr_sg"><span class="id" title="lemma">normr_sg</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_norm"><span class="id" title="lemma">sgr_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + lerif +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_refl"><span class="id" title="lemma">lerif_refl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_trans"><span class="id" title="lemma">lerif_trans</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">x3</span> <span class="id" title="var">C12</span> <span class="id" title="var">C23</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C12"><span class="id" title="variable">C12</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x3"><span class="id" title="variable">x3</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C23"><span class="id" title="variable">C23</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x3"><span class="id" title="variable">x3</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C12"><span class="id" title="variable">C12</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C23"><span class="id" title="variable">C23</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_le"><span class="id" title="lemma">lerif_le</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">≥</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_eq"><span class="id" title="lemma">lerif_eq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger_lerif"><span class="id" title="lemma">ger_lerif</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_lerif"><span class="id" title="lemma">ltr_lerif</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_nat"><span class="id" title="lemma">lerif_nat</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6019ab7c51b05c9f67093b53d0c48454"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6019ab7c51b05c9f67093b53d0c48454"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6019ab7c51b05c9f67093b53d0c48454"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6019ab7c51b05c9f67093b53d0c48454"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f0472472bc7977d4afb419d87444a23b"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>)%<span class="id" title="var">N</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mono_in_lerif"><span class="id" title="lemma">mono_in_lerif</span></a> (<span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">C</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mono_lerif"><span class="id" title="lemma">mono_lerif</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">C</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmono_in_lerif"><span class="id" title="lemma">nmono_in_lerif</span></a> (<span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">C</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nmono_lerif"><span class="id" title="lemma">nmono_lerif</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">C</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_subLR"><span class="id" title="lemma">lerif_subLR</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">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_subRL"><span class="id" title="lemma">lerif_subRL</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">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_add"><span class="id" title="lemma">lerif_add</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">y1</span> <span class="id" title="var">C1</span> <span class="id" title="var">x2</span> <span class="id" title="var">y2</span> <span class="id" title="var">C2</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C2"><span class="id" title="variable">C2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C2"><span class="id" title="variable">C2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_sum"><span class="id" title="lemma">lerif_sum</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> <span class="id" title="var">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E1</span> <span class="id" title="var">E2</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">∀</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_0_sum"><span class="id" title="lemma">lerif_0_sum</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> <span class="id" title="var">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">∀</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_lerif_norm"><span class="id" title="lemma">real_lerif_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_pmul"><span class="id" title="lemma">lerif_pmul</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span> <span class="id" title="var">C1</span> <span class="id" title="var">C2</span> :<br/> + 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C2"><span class="id" title="variable">C2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C2"><span class="id" title="variable">C2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_nmul"><span class="id" title="lemma">lerif_nmul</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span> <span class="id" title="var">C1</span> <span class="id" title="var">C2</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C2"><span class="id" title="variable">C2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C1"><span class="id" title="variable">C1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C2"><span class="id" title="variable">C2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_pprod"><span class="id" title="lemma">lerif_pprod</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> <span class="id" title="var">C</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E1</span> <span class="id" title="var">E2</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">let</span> <span class="id" title="var">pi</span> <span class="id" title="var">E</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="tactic">in</span><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E1"><span class="id" title="variable">E1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E2"><span class="id" title="variable">E2</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">∀</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#71aa9b4d33ee64c2b31b6cd545727657"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Mean inequalities. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_lerif_mean_square_scaled"><span class="id" title="lemma">real_lerif_mean_square_scaled</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_lerif_AGM2_scaled"><span class="id" title="lemma">real_lerif_AGM2_scaled</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> 4 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_AGM_scaled"><span class="id" title="lemma">lerif_AGM_scaled</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">n</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">∀</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">∀</span></a> <span class="id" title="var">j</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Polynomial bound. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.poly_disk_bound"><span class="id" title="lemma">poly_disk_bound</span></a> <span class="id" title="var">p</span> <span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">ub</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ub"><span class="id" title="variable">ub</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainOperationTheory"><span class="id" title="section">NumDomainOperationTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">ler_opp2</span> <span class="id" title="var">ltr_opp2</span> <span class="id" title="var">real0</span> <span class="id" title="var">real1</span> <span class="id" title="var">normr_real</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.NumDomainMonotonyTheoryForReals"><span class="id" title="section">NumDomainMonotonyTheoryForReals</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.Theory.NumDomainMonotonyTheoryForReals.R"><span class="id" title="variable">R</span></a> <a name="Num.Theory.NumDomainMonotonyTheoryForReals.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>) (<a name="Num.Theory.NumDomainMonotonyTheoryForReals.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>) (<a name="Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R'"><span class="id" title="variable">R'</span></a>).<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">u</span> <span class="id" title="var">v</span> <span class="id" title="var">w</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.R'"><span class="id" title="variable">R'</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_mono"><span class="id" title="lemma">real_mono</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_nmono"><span class="id" title="lemma">real_nmono</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + GG: Domain should precede condition. +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_mono_in"><span class="id" title="lemma">real_mono_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">pred</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_nmono_in"><span class="id" title="lemma">real_nmono_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">pred</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#de871826ce6370a1556511e2ea704fe8"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumDomainMonotonyTheoryForReals"><span class="id" title="section">NumDomainMonotonyTheoryForReals</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.FinGroup"><span class="id" title="section">FinGroup</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.Theory.FinGroup.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>) (<a name="Num.Theory.FinGroup.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.FinGroup.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.natrG_gt0"><span class="id" title="lemma">natrG_gt0</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f54d2e0a46f272d0295ade87cec65608"><span class="id" title="notation">></span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#f54d2e0a46f272d0295ade87cec65608"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.FinGroup.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.natrG_neq0"><span class="id" title="lemma">natrG_neq0</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.FinGroup.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.natr_indexg_gt0"><span class="id" title="lemma">natr_indexg_gt0</span></a> <span class="id" title="var">G</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f54d2e0a46f272d0295ade87cec65608"><span class="id" title="notation">></span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#f54d2e0a46f272d0295ade87cec65608"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.FinGroup.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.natr_indexg_neq0"><span class="id" title="lemma">natr_indexg_neq0</span></a> <span class="id" title="var">G</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.FinGroup.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.FinGroup"><span class="id" title="section">FinGroup</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.NumFieldTheory"><span class="id" title="section">NumFieldTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.NumFieldTheory.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>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.unitf_gt0"><span class="id" title="lemma">unitf_gt0</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.unitf_lt0"><span class="id" title="lemma">unitf_lt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lef_pinv"><span class="id" title="lemma">lef_pinv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.pos"><span class="id" title="abbreviation">pos</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lef_ninv"><span class="id" title="lemma">lef_ninv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.neg"><span class="id" title="abbreviation">neg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltf_pinv"><span class="id" title="lemma">ltf_pinv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.pos"><span class="id" title="abbreviation">pos</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltf_ninv"><span class="id" title="lemma">ltf_ninv</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.neg"><span class="id" title="abbreviation">neg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.inv"><span class="id" title="definition">GRing.inv</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ltef_pinv"><span class="id" title="definition">ltef_pinv</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.lef_pinv"><span class="id" title="lemma">lef_pinv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltf_pinv"><span class="id" title="lemma">ltf_pinv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.ltef_ninv"><span class="id" title="definition">ltef_ninv</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.lef_ninv"><span class="id" title="lemma">lef_ninv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltf_ninv"><span class="id" title="lemma">ltf_ninv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invf_gt1"><span class="id" title="lemma">invf_gt1</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invf_ge1"><span class="id" title="lemma">invf_ge1</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.invf_gte1"><span class="id" title="definition">invf_gte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invf_ge1"><span class="id" title="lemma">invf_ge1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invf_gt1"><span class="id" title="lemma">invf_gt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invf_le1"><span class="id" title="lemma">invf_le1</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invf_lt1"><span class="id" title="lemma">invf_lt1</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.invf_lte1"><span class="id" title="definition">invf_lte1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invf_le1"><span class="id" title="lemma">invf_le1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invf_lt1"><span class="id" title="lemma">invf_lt1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.invf_cp1"><span class="id" title="definition">invf_cp1</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invf_gte1"><span class="id" title="definition">invf_gte1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.invf_lte1"><span class="id" title="definition">invf_lte1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + These lemma are all combinations of mono(LR|RL) with ler</i> [pn]mul2[rl]. +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pdivl_mulr"><span class="id" title="lemma">ler_pdivl_mulr</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pdivl_mulr"><span class="id" title="lemma">ltr_pdivl_mulr</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_pdivl_mulr"><span class="id" title="definition">lter_pdivl_mulr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_pdivl_mulr"><span class="id" title="lemma">ler_pdivl_mulr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_pdivl_mulr"><span class="id" title="lemma">ltr_pdivl_mulr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pdivr_mulr"><span class="id" title="lemma">ler_pdivr_mulr</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pdivr_mulr"><span class="id" title="lemma">ltr_pdivr_mulr</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_pdivr_mulr"><span class="id" title="definition">lter_pdivr_mulr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_pdivr_mulr"><span class="id" title="lemma">ler_pdivr_mulr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_pdivr_mulr"><span class="id" title="lemma">ltr_pdivr_mulr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pdivl_mull"><span class="id" title="lemma">ler_pdivl_mull</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pdivl_mull"><span class="id" title="lemma">ltr_pdivl_mull</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_pdivl_mull"><span class="id" title="definition">lter_pdivl_mull</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_pdivl_mull"><span class="id" title="lemma">ler_pdivl_mull</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_pdivl_mull"><span class="id" title="lemma">ltr_pdivl_mull</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_pdivr_mull"><span class="id" title="lemma">ler_pdivr_mull</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_pdivr_mull"><span class="id" title="lemma">ltr_pdivr_mull</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_pdivr_mull"><span class="id" title="definition">lter_pdivr_mull</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_pdivr_mull"><span class="id" title="lemma">ler_pdivr_mull</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_pdivr_mull"><span class="id" title="lemma">ltr_pdivr_mull</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_ndivl_mulr"><span class="id" title="lemma">ler_ndivl_mulr</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_ndivl_mulr"><span class="id" title="lemma">ltr_ndivl_mulr</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_ndivl_mulr"><span class="id" title="definition">lter_ndivl_mulr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_ndivl_mulr"><span class="id" title="lemma">ler_ndivl_mulr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_ndivl_mulr"><span class="id" title="lemma">ltr_ndivl_mulr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_ndivr_mulr"><span class="id" title="lemma">ler_ndivr_mulr</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_ndivr_mulr"><span class="id" title="lemma">ltr_ndivr_mulr</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_ndivr_mulr"><span class="id" title="definition">lter_ndivr_mulr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_ndivr_mulr"><span class="id" title="lemma">ler_ndivr_mulr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_ndivr_mulr"><span class="id" title="lemma">ltr_ndivr_mulr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_ndivl_mull"><span class="id" title="lemma">ler_ndivl_mull</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_ndivl_mull"><span class="id" title="lemma">ltr_ndivl_mull</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_ndivl_mull"><span class="id" title="definition">lter_ndivl_mull</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_ndivl_mull"><span class="id" title="lemma">ler_ndivl_mull</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_ndivl_mull"><span class="id" title="lemma">ltr_ndivl_mull</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_ndivr_mull"><span class="id" title="lemma">ler_ndivr_mull</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_ndivr_mull"><span class="id" title="lemma">ltr_ndivr_mull</span></a> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_ndivr_mull"><span class="id" title="definition">lter_ndivr_mull</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_ndivr_mull"><span class="id" title="lemma">ler_ndivr_mull</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_ndivr_mull"><span class="id" title="lemma">ltr_ndivr_mull</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.natf_div"><span class="id" title="lemma">natf_div</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#aa34fd1c61c5cf0a3356b624a5d2afed"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#df17451da28eb630dbb51b12706ba39e"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normfV"><span class="id" title="lemma">normfV</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.norm"><span class="id" title="abbreviation">norm</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normf_div"><span class="id" title="lemma">normf_div</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.norm"><span class="id" title="abbreviation">norm</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invr_sg"><span class="id" title="lemma">invr_sg</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">)^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.sgr"><span class="id" title="definition">sgr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgrV"><span class="id" title="lemma">sgrV</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.sgr"><span class="id" title="definition">sgr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.sgr"><span class="id" title="definition">sgr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Interval midpoint. +</div> +<div class="code"> + +<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.midf_le"><span class="id" title="lemma">midf_le</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mid"><span class="id" title="abbreviation">mid</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mid"><span class="id" title="abbreviation">mid</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.midf_lt"><span class="id" title="lemma">midf_lt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mid"><span class="id" title="abbreviation">mid</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mid"><span class="id" title="abbreviation">mid</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.midf_lte"><span class="id" title="definition">midf_lte</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.midf_le"><span class="id" title="lemma">midf_le</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.midf_lt"><span class="id" title="lemma">midf_lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + The AGM, unscaled but without the nth root. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_lerif_mean_square"><span class="id" title="lemma">real_lerif_mean_square</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mid"><span class="id" title="abbreviation">mid</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2) <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.real_lerif_AGM2"><span class="id" title="lemma">real_lerif_AGM2</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.mid"><span class="id" title="abbreviation">mid</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_AGM"><span class="id" title="lemma">lerif_AGM</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a>) :<br/> + <span class="id" title="keyword">let</span> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">mu</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <span class="id" title="tactic">in</span><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#mu"><span class="id" title="variable">mu</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">∀</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">∀</span></a> <span class="id" title="var">j</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Cauchy_root_bound"><span class="id" title="lemma">Cauchy_root_bound</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.natf_indexg"><span class="id" title="lemma">natf_indexg</span></a> (<span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory.F"><span class="id" title="variable">F</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.NumFieldTheory"><span class="id" title="section">NumFieldTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.RealDomainTheory"><span class="id" title="section">RealDomainTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">lerr</span>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.RealDomainTheory.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/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.num_real"><span class="id" title="lemma">num_real</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>. <br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">num_real</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_total"><span class="id" title="lemma">ler_total</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#total"><span class="id" title="definition">total</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.le"><span class="id" title="abbreviation">le</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainTheory.R"><span class="id" title="variable">R</span></a>). <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_total"><span class="id" title="lemma">ltr_total</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.wlog_ler"><span class="id" title="lemma">wlog_ler</span></a> <span class="id" title="var">P</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainTheory.R"><span class="id" title="variable">R</span></a>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.wlog_ltr"><span class="id" title="lemma">wlog_ltr</span></a> <span class="id" title="var">P</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, (<a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainTheory.R"><span class="id" title="variable">R</span></a>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrNge"><span class="id" title="lemma">ltrNge</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">)</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerNgt"><span class="id" title="lemma">lerNgt</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">)</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerP"><span class="id" title="lemma">lerP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_xor_gt"><span class="id" title="inductive">ler_xor_gt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrP"><span class="id" title="lemma">ltrP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_xor_ge"><span class="id" title="inductive">ltr_xor_ge</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrgtP"><span class="id" title="lemma">ltrgtP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.comparer"><span class="id" title="inductive">comparer</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>)<br/> + (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) .<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ger0P"><span class="id" title="lemma">ger0P</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ger0_xor_lt0"><span class="id" title="inductive">ger0_xor_lt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0) (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler0P"><span class="id" title="lemma">ler0P</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler0_xor_gt0"><span class="id" title="inductive">ler0_xor_gt0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltrgt0P"><span class="id" title="lemma">ltrgt0P</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.comparer0"><span class="id" title="inductive">comparer0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0) (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.neqr_lt"><span class="id" title="lemma">neqr_lt</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_leLR"><span class="id" title="lemma">eqr_leLR</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">t</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_leRL"><span class="id" title="lemma">eqr_leRL</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">t</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_ltLR"><span class="id" title="lemma">eqr_ltLR</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">t</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_ltRL"><span class="id" title="lemma">eqr_ltRL</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">t</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + sign +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_lt0"><span class="id" title="lemma">mulr_lt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">(+)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.neq0_mulr_lt0"><span class="id" title="lemma">neq0_mulr_lt0</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">(+)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_sign_lt0"><span class="id" title="lemma">mulr_sign_lt0</span></a> (<span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <span class="id" title="var">x</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ef177bde7d01ae97c98f9cba81f6c95b"><span class="id" title="notation">(+)</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + sign & norm +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_sign_norm"><span class="id" title="lemma">mulr_sign_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_Nsign_norm"><span class="id" title="lemma">mulr_Nsign_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.numEsign"><span class="id" title="lemma">numEsign</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.numNEsign"><span class="id" title="lemma">numNEsign</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normrEsign"><span class="id" title="lemma">normrEsign</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0)%<span class="id" title="var">R</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainTheory"><span class="id" title="section">RealDomainTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">num_real</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.RealDomainMonotony"><span class="id" title="section">RealDomainMonotony</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.Theory.RealDomainMonotony.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="Num.Theory.RealDomainMonotony.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>) (<a name="Num.Theory.RealDomainMonotony.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>) (<a name="Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R'"><span class="id" title="variable">R'</span></a>).<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">u</span> <span class="id" title="var">v</span> <span class="id" title="var">w</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.R'"><span class="id" title="variable">R'</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Hint Resolve</span> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.num_real"><span class="id" title="lemma">num_real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.homo_mono"><span class="id" title="lemma">homo_mono</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nhomo_mono"><span class="id" title="lemma">nhomo_mono</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.homo_mono_in"><span class="id" title="lemma">homo_mono_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nhomo_mono_in"><span class="id" title="lemma">nhomo_mono_in</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#9caa0748cced1c5dad561bd5f28bd7db"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">/~</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#744665b4ecc83382846c028dbdceae50"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainMonotony"><span class="id" title="section">RealDomainMonotony</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.RealDomainOperations"><span class="id" title="section">RealDomainOperations</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + sgr section +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.RealDomainOperations.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/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <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">t</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a>.<br/> +<span class="id" title="keyword">Hint Resolve</span> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.num_real"><span class="id" title="lemma">num_real</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_cp0"><span class="id" title="lemma">sgr_cp0</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> -1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.sgr_val"><span class="id" title="inductive">sgr_val</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span> :=<br/> + | <a name="Num.Theory.SgrNull"><span class="id" title="constructor">SgrNull</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 : <a class="idref" href="mathcomp.algebra.ssrnum.html#sgr_val"><span class="id" title="inductive">sgr_val</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> 0<br/> + | <a name="Num.Theory.SgrPos"><span class="id" title="constructor">SgrPos</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0 : <a class="idref" href="mathcomp.algebra.ssrnum.html#sgr_val"><span class="id" title="inductive">sgr_val</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> 1<br/> + | <a name="Num.Theory.SgrNeg"><span class="id" title="constructor">SgrNeg</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 : <a class="idref" href="mathcomp.algebra.ssrnum.html#sgr_val"><span class="id" title="inductive">sgr_val</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> (-1).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgrP"><span class="id" title="lemma">sgrP</span></a> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sgr_val"><span class="id" title="inductive">sgr_val</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> (0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0) (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0) (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)<br/> + (0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (-1 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (1 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>)<br/> + (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> -1) (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1) (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normrEsg"><span class="id" title="lemma">normrEsg</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.numEsg"><span class="id" title="lemma">numEsg</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + GG: duplicate! +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulr_sg_norm"><span class="id" title="lemma">mulr_sg_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgrM"><span class="id" title="lemma">sgrM</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgrN"><span class="id" title="lemma">sgrN</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgrX"><span class="id" title="lemma">sgrX</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_smul"><span class="id" title="lemma">sgr_smul</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_gt0"><span class="id" title="lemma">sgr_gt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sg"><span class="id" title="abbreviation">sg</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sgr_ge0"><span class="id" title="lemma">sgr_ge0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.sgr"><span class="id" title="definition">sgr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">≥</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">≥</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + norm section +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_norm"><span class="id" title="lemma">ler_norm</span></a> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_norml"><span class="id" title="lemma">ler_norml</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_normlP"><span class="id" title="lemma">ler_normlP</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/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_norml"><span class="id" title="lemma">eqr_norml</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_norm2"><span class="id" title="lemma">eqr_norm2</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_norml"><span class="id" title="lemma">ltr_norml</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_norml"><span class="id" title="definition">lter_norml</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_norml"><span class="id" title="lemma">ler_norml</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_norml"><span class="id" title="lemma">ltr_norml</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_normlP"><span class="id" title="lemma">ltr_normlP</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/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_normr"><span class="id" title="lemma">ler_normr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_normr"><span class="id" title="lemma">ltr_normr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_normr"><span class="id" title="definition">lter_normr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_normr"><span class="id" title="lemma">ler_normr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_normr"><span class="id" title="lemma">ltr_normr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_distl"><span class="id" title="lemma">ler_distl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">e</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_distl"><span class="id" title="lemma">ltr_distl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">e</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#7cd47ea66f219bc403cc6631c817f8b3"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_distl"><span class="id" title="definition">lter_distl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_distl"><span class="id" title="lemma">ler_distl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_distl"><span class="id" title="lemma">ltr_distl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_even_ge0"><span class="id" title="lemma">exprn_even_ge0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_even_gt0"><span class="id" title="lemma">exprn_even_gt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_even_le0"><span class="id" title="lemma">exprn_even_le0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_even_lt0"><span class="id" title="lemma">exprn_even_lt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_odd_ge0"><span class="id" title="lemma">exprn_odd_ge0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_odd_gt0"><span class="id" title="lemma">exprn_odd_gt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_odd_le0"><span class="id" title="lemma">exprn_odd_le0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprn_odd_lt0"><span class="id" title="lemma">exprn_odd_lt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Special lemmas for squares. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqr_ge0"><span class="id" title="lemma">sqr_ge0</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqr_norm_eq1"><span class="id" title="lemma">sqr_norm_eq1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_mean_square_scaled"><span class="id" title="lemma">lerif_mean_square_scaled</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_AGM2_scaled"><span class="id" title="lemma">lerif_AGM2_scaled</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> 4 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.RealDomainOperations.MinMax"><span class="id" title="section">MinMax</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + GG: Many of the first lemmas hold unconditionally, and others hold for + the real subset of a general domain. +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minrC"><span class="id" title="lemma">minrC</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minrr"><span class="id" title="lemma">minrr</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#idempotent"><span class="id" title="definition">idempotent</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_l"><span class="id" title="lemma">minr_l</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_r"><span class="id" title="lemma">minr_r</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxrC"><span class="id" title="lemma">maxrC</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxrr"><span class="id" title="lemma">maxrr</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#idempotent"><span class="id" title="definition">idempotent</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_l"><span class="id" title="lemma">maxr_l</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_r"><span class="id" title="lemma">maxr_r</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addr_min_max"><span class="id" title="lemma">addr_min_max</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addr_max_min"><span class="id" title="lemma">addr_max_min</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_to_max"><span class="id" title="lemma">minr_to_max</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_to_min"><span class="id" title="lemma">maxr_to_min</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minrA"><span class="id" title="lemma">minrA</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minrCA"><span class="id" title="lemma">minrCA</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_commutative"><span class="id" title="definition">left_commutative</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minrAC"><span class="id" title="lemma">minrAC</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_commutative"><span class="id" title="definition">right_commutative</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.minr_spec"><span class="id" title="inductive">minr_spec</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/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> +| <a name="Num.Theory.Minr_r"><span class="id" title="constructor">Minr_r</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#minr_spec"><span class="id" title="inductive">minr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><br/> +| <a name="Num.Theory.Minr_l"><span class="id" title="constructor">Minr_l</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#minr_spec"><span class="id" title="inductive">minr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minrP"><span class="id" title="lemma">minrP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.minr_spec"><span class="id" title="inductive">minr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.oppr_max"><span class="id" title="lemma">oppr_max</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.oppr_min"><span class="id" title="lemma">oppr_min</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxrA"><span class="id" title="lemma">maxrA</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxrCA"><span class="id" title="lemma">maxrCA</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_commutative"><span class="id" title="definition">left_commutative</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxrAC"><span class="id" title="lemma">maxrAC</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_commutative"><span class="id" title="definition">right_commutative</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.maxr_spec"><span class="id" title="inductive">maxr_spec</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/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> +| <a name="Num.Theory.Maxr_r"><span class="id" title="constructor">Maxr_r</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#maxr_spec"><span class="id" title="inductive">maxr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><br/> +| <a name="Num.Theory.Maxr_l"><span class="id" title="constructor">Maxr_l</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#maxr_spec"><span class="id" title="inductive">maxr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxrP"><span class="id" title="lemma">maxrP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.maxr_spec"><span class="id" title="inductive">maxr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Def.maxr"><span class="id" title="definition">maxr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_minl"><span class="id" title="lemma">eqr_minl</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_minr"><span class="id" title="lemma">eqr_minr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_maxl"><span class="id" title="lemma">eqr_maxl</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_maxr"><span class="id" title="lemma">eqr_maxr</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_minr"><span class="id" title="lemma">ler_minr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_minl"><span class="id" title="lemma">ler_minl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_maxr"><span class="id" title="lemma">ler_maxr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_maxl"><span class="id" title="lemma">ler_maxl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_minr"><span class="id" title="lemma">ltr_minr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_minl"><span class="id" title="lemma">ltr_minl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_maxr"><span class="id" title="lemma">ltr_maxr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_maxl"><span class="id" title="lemma">ltr_maxl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_minr"><span class="id" title="definition">lter_minr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_minr"><span class="id" title="lemma">ler_minr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_minr"><span class="id" title="lemma">ltr_minr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_minl"><span class="id" title="definition">lter_minl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_minl"><span class="id" title="lemma">ler_minl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_minl"><span class="id" title="lemma">ltr_minl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_maxr"><span class="id" title="definition">lter_maxr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_maxr"><span class="id" title="lemma">ler_maxr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_maxr"><span class="id" title="lemma">ltr_maxr</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.lter_maxl"><span class="id" title="definition">lter_maxl</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ler_maxl"><span class="id" title="lemma">ler_maxl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ltr_maxl"><span class="id" title="lemma">ltr_maxl</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addr_minl"><span class="id" title="lemma">addr_minl</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_distributive"><span class="id" title="definition">left_distributive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addr_minr"><span class="id" title="lemma">addr_minr</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addr_maxl"><span class="id" title="lemma">addr_maxl</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_distributive"><span class="id" title="definition">left_distributive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addr_maxr"><span class="id" title="lemma">addr_maxr</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minrK"><span class="id" title="lemma">minrK</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minKr"><span class="id" title="lemma">minKr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_minl"><span class="id" title="lemma">maxr_minl</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_distributive"><span class="id" title="definition">left_distributive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_minr"><span class="id" title="lemma">maxr_minr</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_maxl"><span class="id" title="lemma">minr_maxl</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_distributive"><span class="id" title="definition">left_distributive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_maxr"><span class="id" title="lemma">minr_maxr</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_pmulr"><span class="id" title="lemma">minr_pmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_nmulr"><span class="id" title="lemma">minr_nmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_pmulr"><span class="id" title="lemma">maxr_pmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_nmulr"><span class="id" title="lemma">maxr_nmulr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_pmull"><span class="id" title="lemma">minr_pmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minr_nmull"><span class="id" title="lemma">minr_nmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_pmull"><span class="id" title="lemma">maxr_pmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxr_nmull"><span class="id" title="lemma">maxr_nmull</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxrN"><span class="id" title="lemma">maxrN</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.maxNr"><span class="id" title="lemma">maxNr</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.max"><span class="id" title="abbreviation">max</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minrN"><span class="id" title="lemma">minrN</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.minNr"><span class="id" title="lemma">minNr</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.min"><span class="id" title="abbreviation">min</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.MinMax"><span class="id" title="section">MinMax</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.RealDomainOperations.PolyBounds"><span class="id" title="section">PolyBounds</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.RealDomainOperations.PolyBounds.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.poly_itv_bound"><span class="id" title="lemma">poly_itv_bound</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">ub</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ff736a21b738c796d1200c3222013b46"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.PolyBounds.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#ub"><span class="id" title="variable">ub</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.monic_Cauchy_bound"><span class="id" title="lemma">monic_Cauchy_bound</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.PolyBounds.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">≥</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.PolyBounds.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations.PolyBounds"><span class="id" title="section">PolyBounds</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealDomainOperations"><span class="id" title="section">RealDomainOperations</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.RealField"><span class="id" title="section">RealField</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.Theory.RealField.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealField.Exports.realFieldType"><span class="id" title="abbreviation">realFieldType</span></a>) (<a name="Num.Theory.RealField.x"><span class="id" title="variable">x</span></a> <a name="Num.Theory.RealField.y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_mean_square"><span class="id" title="lemma">lerif_mean_square</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_AGM2"><span class="id" title="lemma">lerif_AGM2</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField.y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealField"><span class="id" title="section">RealField</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.ArchimedeanFieldTheory"><span class="id" title="section">ArchimedeanFieldTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.Theory.ArchimedeanFieldTheory.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ArchimedeanField.Exports.archiFieldType"><span class="id" title="abbreviation">archiFieldType</span></a>) (<a name="Num.Theory.ArchimedeanFieldTheory.x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.archi_boundP"><span class="id" title="lemma">archi_boundP</span></a> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ArchimedeanFieldTheory.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ArchimedeanFieldTheory.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.bound"><span class="id" title="abbreviation">bound</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ArchimedeanFieldTheory.x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.upper_nthrootP"><span class="id" title="lemma">upper_nthrootP</span></a> <span class="id" title="var">i</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.bound"><span class="id" title="abbreviation">bound</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ArchimedeanFieldTheory.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ArchimedeanFieldTheory.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ArchimedeanFieldTheory"><span class="id" title="section">ArchimedeanFieldTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.RealClosedFieldTheory"><span class="id" title="section">RealClosedFieldTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealClosedField.Exports.rcfType"><span class="id" title="abbreviation">rcfType</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.poly_ivt"><span class="id" title="lemma">poly_ivt</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real_closed_axiom"><span class="id" title="definition">real_closed_axiom</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a>. <br/> + +<br/> +</div> + +<div class="doc"> + Square Root theory +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtr_ge0"><span class="id" title="lemma">sqrtr_ge0</span></a> <span class="id" title="var">a</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a>.<br/> + <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">sqrtr_ge0</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqr_sqrtr"><span class="id" title="lemma">sqr_sqrtr</span></a> <span class="id" title="var">a</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler0_sqrtr"><span class="id" title="lemma">ler0_sqrtr</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr0_sqrtr"><span class="id" title="lemma">ltr0_sqrtr</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.sqrtr_spec"><span class="id" title="inductive">sqrtr_spec</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> +| <a name="Num.Theory.IsNoSqrtr"><span class="id" title="constructor">IsNoSqrtr</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0 : <a class="idref" href="mathcomp.algebra.ssrnum.html#sqrtr_spec"><span class="id" title="inductive">sqrtr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> 0<br/> +| <a name="Num.Theory.IsSqrtr"><span class="id" title="constructor">IsSqrtr</span></a> <span class="id" title="var">b</span> <span class="id" title="keyword">of</span> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#sqrtr_spec"><span class="id" title="inductive">sqrtr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtrP"><span class="id" title="lemma">sqrtrP</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtr_spec"><span class="id" title="inductive">sqrtr_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtr_sqr"><span class="id" title="lemma">sqrtr_sqr</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtrM"><span class="id" title="lemma">sqrtrM</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtr0"><span class="id" title="lemma">sqrtr0</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtr1"><span class="id" title="lemma">sqrtr1</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtr_eq0"><span class="id" title="lemma">sqrtr_eq0</span></a> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtr_gt0"><span class="id" title="lemma">sqrtr_gt0</span></a> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_sqrt"><span class="id" title="lemma">eqr_sqrt</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_wsqrtr"><span class="id" title="lemma">ler_wsqrtr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">homo</span></a> @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#bf2e58b992e0391262bcf1e3bb7b0fc0"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_psqrt"><span class="id" title="lemma">ler_psqrt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> @<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.pos"><span class="id" title="abbreviation">pos</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_sqrt"><span class="id" title="lemma">ler_sqrt</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_sqrt"><span class="id" title="lemma">ltr_sqrt</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.sqrt"><span class="id" title="abbreviation">sqrt</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RealClosedFieldTheory"><span class="id" title="section">RealClosedFieldTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.conjC"><span class="id" title="definition">conjC</span></a> {<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Exports.numClosedFieldType"><span class="id" title="abbreviation">numClosedFieldType</span></a>} : <a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a> :=<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.conj_op"><span class="id" title="projection">ClosedField.conj_op</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.conj_mixin"><span class="id" title="projection">ClosedField.conj_mixin</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class"><span class="id" title="definition">ClosedField.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>)).<br/> +<span class="id" title="keyword">Notation</span> <a name="b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">"</span></a>z ^*" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a> <span class="id" title="var">_</span> <span class="id" title="var">z</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "z ^*") : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.imaginaryC"><span class="id" title="definition">imaginaryC</span></a> {<span class="id" title="var">C</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Exports.numClosedFieldType"><span class="id" title="abbreviation">numClosedFieldType</span></a>} : <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a> :=<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.imaginary"><span class="id" title="projection">ClosedField.imaginary</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.conj_mixin"><span class="id" title="projection">ClosedField.conj_mixin</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.class"><span class="id" title="definition">ClosedField.class</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#C"><span class="id" title="variable">C</span></a>)).<br/> +<span class="id" title="keyword">Notation</span> <a name="f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">"</span></a>'i" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.imaginaryC"><span class="id" title="definition">imaginaryC</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.Theory.ClosedFieldTheory"><span class="id" title="section">ClosedFieldTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.ClosedField.Exports.numClosedFieldType"><span class="id" title="abbreviation">numClosedFieldType</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">a</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="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.normCK"><span class="id" title="definition">normCK</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrCi"><span class="id" title="lemma">sqrCi</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> -1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conjCK"><span class="id" title="lemma">conjCK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#involutive"><span class="id" title="definition">involutive</span></a> (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Let</span> <a name="Num.Theory.ClosedFieldTheory.Re2"><span class="id" title="variable">Re2</span></a> <span class="id" title="var">z</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.nnegIm"><span class="id" title="definition">nnegIm</span></a> <span class="id" title="var">z</span> := (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.imaginaryC"><span class="id" title="definition">imaginaryC</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.argCle"><span class="id" title="definition">argCle</span></a> <span class="id" title="var">y</span> <span class="id" title="var">z</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.nnegIm"><span class="id" title="definition">nnegIm</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">==></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.nnegIm"><span class="id" title="definition">nnegIm</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.Re2"><span class="id" title="variable">Re2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.Re2"><span class="id" title="variable">Re2</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="Num.Theory.rootC_spec"><span class="id" title="inductive">rootC_spec</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>) : <span class="id" title="keyword">Type</span> :=<br/> + <a name="Num.Theory.RootCspec"><span class="id" title="constructor">RootCspec</span></a> (<span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>) <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0<br/> + & <span class="id" title="keyword">∀</span> <span class="id" title="var">z</span>, (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.argCle"><span class="id" title="definition">argCle</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Theory.rootC_subproof"><span class="id" title="lemma">rootC_subproof</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.rootC_spec"><span class="id" title="inductive">rootC_spec</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.nthroot"><span class="id" title="definition">nthroot</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.RootCspec"><span class="id" title="constructor">RootCspec</span></a> <span class="id" title="var">y</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.rootC_subproof"><span class="id" title="lemma">rootC_subproof</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">y</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="faffc7ebdc59e33c8506558c91f1ae94"><span class="id" title="notation">"</span></a>n .-root" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.nthroot"><span class="id" title="definition">nthroot</span></a> <span class="id" title="var">n</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "n .-root") : <span class="id" title="var">ring_core_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">"</span></a>n .-root" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.nthroot"><span class="id" title="definition">nthroot</span></a> <span class="id" title="var">n</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> := 2<a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.Re"><span class="id" title="definition">Re</span></a> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.Theory.Im"><span class="id" title="definition">Im</span></a> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">"</span></a>'Re z" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Re"><span class="id" title="definition">Re</span></a> <span class="id" title="var">z</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">z</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">"</span></a>'Im z" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Im"><span class="id" title="definition">Im</span></a> <span class="id" title="var">z</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">z</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Let</span> <a name="Num.Theory.ClosedFieldTheory.nz2"><span class="id" title="variable">nz2</span></a> : 2<a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normCKC"><span class="id" title="lemma">normCKC</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mul_conjC_ge0"><span class="id" title="lemma">mul_conjC_ge0</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mul_conjC_gt0"><span class="id" title="lemma">mul_conjC_gt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mul_conjC_eq0"><span class="id" title="lemma">mul_conjC_eq0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conjC_ge0"><span class="id" title="lemma">conjC_ge0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conjC_nat"><span class="id" title="lemma">conjC_nat</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">)^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conjC0"><span class="id" title="lemma">conjC0</span></a> : 0<a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conjC1"><span class="id" title="lemma">conjC1</span></a> : 1<a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conjC_eq0"><span class="id" title="lemma">conjC_eq0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invC_norm"><span class="id" title="lemma">invC_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c3d3108f22e21916f6afd1f17c0f8125"><span class="id" title="notation">^-</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Real number subset. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.CrealE"><span class="id" title="lemma">CrealE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.CrealP"><span class="id" title="lemma">CrealP</span></a> {<span class="id" title="var">x</span>} : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conj_Creal"><span class="id" title="lemma">conj_Creal</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conj_normC"><span class="id" title="lemma">conj_normC</span></a> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.geC0_conj"><span class="id" title="lemma">geC0_conj</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.geC0_unit_exp"><span class="id" title="lemma">geC0_unit_exp</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Elementary properties of roots. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Ltac</span> <span class="id" title="var">case_rootC</span> := <span class="id" title="tactic">rewrite</span> /<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.nthroot"><span class="id" title="definition">nthroot</span></a>; <span class="id" title="tactic">case</span>: (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.rootC_subproof"><span class="id" title="lemma">rootC_subproof</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.root0C"><span class="id" title="lemma">root0C</span></a> <span class="id" title="var">x</span> : 0<a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootCK"><span class="id" title="lemma">rootCK</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.root1C"><span class="id" title="lemma">root1C</span></a> <span class="id" title="var">x</span> : 1<a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC0"><span class="id" title="lemma">rootC0</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_inj"><span class="id" title="lemma">rootC_inj</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_rootC"><span class="id" title="lemma">eqr_rootC</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_eq0"><span class="id" title="lemma">rootC_eq0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Rectangular coordinates. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.nonRealCi"><span class="id" title="lemma">nonRealCi</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c21f83c2830c19977e3cc98c7b8d8632"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c21f83c2830c19977e3cc98c7b8d8632"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c21f83c2830c19977e3cc98c7b8d8632"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c21f83c2830c19977e3cc98c7b8d8632"><span class="id" title="notation">isn't</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.neq0Ci"><span class="id" title="lemma">neq0Ci</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normCi"><span class="id" title="lemma">normCi</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invCi"><span class="id" title="lemma">invCi</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conjCi"><span class="id" title="lemma">conjCi</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Crect"><span class="id" title="lemma">Crect</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Creal_Re"><span class="id" title="lemma">Creal_Re</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Creal_Im"><span class="id" title="lemma">Creal_Im</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>.<br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">Creal_Re</span> <span class="id" title="var">Creal_Im</span>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Theory.Re_is_additive"><span class="id" title="lemma">Re_is_additive</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.additive"><span class="id" title="abbreviation">additive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Re"><span class="id" title="definition">Re</span></a>.<br/> + <span class="id" title="keyword">Canonical</span> <span class="id" title="var">Re_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Re_is_additive"><span class="id" title="lemma">Re_is_additive</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.Theory.Im_is_additive"><span class="id" title="lemma">Im_is_additive</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.additive"><span class="id" title="abbreviation">additive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Im"><span class="id" title="definition">Im</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Im_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Im_is_additive"><span class="id" title="lemma">Im_is_additive</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Creal_ImP"><span class="id" title="lemma">Creal_ImP</span></a> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Creal_ReP"><span class="id" title="lemma">Creal_ReP</span></a> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ReMl"><span class="id" title="lemma">ReMl</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Re"><span class="id" title="definition">Re</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ReMr"><span class="id" title="lemma">ReMr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Re"><span class="id" title="definition">Re</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ImMl"><span class="id" title="lemma">ImMl</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Im"><span class="id" title="definition">Im</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ImMr"><span class="id" title="lemma">ImMr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Im"><span class="id" title="definition">Im</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Re_i"><span class="id" title="lemma">Re_i</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Im_i"><span class="id" title="lemma">Im_i</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Re_conj"><span class="id" title="lemma">Re_conj</span></a> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Im_conj"><span class="id" title="lemma">Im_conj</span></a> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Re_rect"><span class="id" title="lemma">Re_rect</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Im_rect"><span class="id" title="lemma">Im_rect</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.conjC_rect"><span class="id" title="lemma">conjC_rect</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">)^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.addC_rect"><span class="id" title="lemma">addC_rect</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">y1</span> <span class="id" title="var">x2</span> <span class="id" title="var">y2</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.oppC_rect"><span class="id" title="lemma">oppC_rect</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.subC_rect"><span class="id" title="lemma">subC_rect</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">y1</span> <span class="id" title="var">x2</span> <span class="id" title="var">y2</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.mulC_rect"><span class="id" title="lemma">mulC_rect</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">y1</span> <span class="id" title="var">x2</span> <span class="id" title="var">y2</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y1"><span class="id" title="variable">y1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC2_rect"><span class="id" title="lemma">normC2_rect</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC2_Re_Im"><span class="id" title="lemma">normC2_Re_Im</span></a> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.invC_rect"><span class="id" title="lemma">invC_rect</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">)^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2<a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_normC_Re_Creal"><span class="id" title="lemma">lerif_normC_Re_Creal</span></a> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_Re_Creal"><span class="id" title="lemma">lerif_Re_Creal</span></a> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Equality from polar coordinates, for the upper plane. +</div> +<div class="code"> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqC_semipolar"><span class="id" title="lemma">eqC_semipolar</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Nth roots. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Let</span> <a name="Num.Theory.ClosedFieldTheory.argCleP"><span class="id" title="variable">argCleP</span></a> <span class="id" title="var">y</span> <span class="id" title="var">z</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.argCle"><span class="id" title="definition">argCle</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>).<br/> +</div> + +<div class="doc"> + case Du: sqrCi => [u u2N1] /=. + have/eqP := u2N1; rewrite -sqrCi eqf_sqr => /pred2P[ ] //. + have:= conjCi; rewrite /'i; case_rootC => /= v v2n1 min_v conj_v Duv. + have{min_v} /idPn[ ] := min_v u isT u2N1; rewrite negb_imply /nnegIm Du /= Duv. + rewrite rmorphN conj_v opprK -opprD mulrNN mulNr -mulr2n mulrnAr -expr2 v2n1. + by rewrite mulNrn opprK ler0n oppr_ge0 (ler_nat _ 2 0). +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_Re_max"><span class="id" title="lemma">rootC_Re_max</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Let</span> <a name="Num.Theory.ClosedFieldTheory.neg_unity_root"><span class="id" title="variable">neg_unity_root</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 1)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">w</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#w"><span class="id" title="variable">w</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#w"><span class="id" title="variable">w</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.Im_rootC_ge0"><span class="id" title="lemma">Im_rootC_ge0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 1)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_lt0"><span class="id" title="lemma">rootC_lt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_ge0"><span class="id" title="lemma">rootC_ge0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_gt0"><span class="id" title="lemma">rootC_gt0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_le0"><span class="id" title="lemma">rootC_le0</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_rootCl"><span class="id" title="lemma">ler_rootCl</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">Num.nneg</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_rootC"><span class="id" title="lemma">ler_rootC</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">Num.nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_rootCl"><span class="id" title="lemma">ltr_rootCl</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">Num.nneg</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_rootC"><span class="id" title="lemma">ltr_rootC</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">Num.nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.exprCK"><span class="id" title="lemma">exprCK</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.norm_rootC"><span class="id" title="lemma">norm_rootC</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootCX"><span class="id" title="lemma">rootCX</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">k</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#k"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#k"><span class="id" title="variable">k</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC1"><span class="id" title="lemma">rootC1</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootCpX"><span class="id" title="lemma">rootCpX</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">k</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#k"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#k"><span class="id" title="variable">k</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootCV"><span class="id" title="lemma">rootCV</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">)^-1</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_eq1"><span class="id" title="lemma">rootC_eq1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_ge1"><span class="id" title="lemma">rootC_ge1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">≥</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#4a55c8439dfd5912be472b2910ab4015"><span class="id" title="notation">≥</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_gt1"><span class="id" title="lemma">rootC_gt1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_le1"><span class="id" title="lemma">rootC_le1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootC_lt1"><span class="id" title="lemma">rootC_lt1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootCMl"><span class="id" title="lemma">rootCMl</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.rootCMr"><span class="id" title="lemma">rootCMr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.imaginaryCE"><span class="id" title="lemma">imaginaryCE</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> (-1).<br/> + +<br/> +</div> + +<div class="doc"> + More properties of n.-root will be established in cyclotomic.v. +<div class="paragraph"> </div> + + The proper form of the Arithmetic - Geometric Mean inequality. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.lerif_rootC_AGM"><span class="id" title="lemma">lerif_rootC_AGM</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">n</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>) <span class="id" title="var">E</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#27eefe7225317752adf12d2ec6054502"><span class="id" title="notation">root</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#50f6ed3c9dd83e0dda7460830646e9b1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c791dbdc1655ae690f0a6c159a384c0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#79bf01a504ffcda65f8e3d816a5515cd"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">∀</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">∀</span></a> <span class="id" title="var">j</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#569613cf8a3bdd9ea86bbbe48a5b61c3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Square root. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtC0"><span class="id" title="lemma">sqrtC0</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtC1"><span class="id" title="lemma">sqrtC1</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtCK"><span class="id" title="lemma">sqrtCK</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrCK"><span class="id" title="lemma">sqrCK</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtC_ge0"><span class="id" title="lemma">sqrtC_ge0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtC_eq0"><span class="id" title="lemma">sqrtC_eq0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtC_gt0"><span class="id" title="lemma">sqrtC_gt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#07bcd9d86ae6b6828fbc17b15193853f"><span class="id" title="notation">></span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtC_lt0"><span class="id" title="lemma">sqrtC_lt0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtC_le0"><span class="id" title="lemma">sqrtC_le0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ler_sqrtC"><span class="id" title="lemma">ler_sqrtC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">Num.nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="Num.Theory.ltr_sqrtC"><span class="id" title="lemma">ltr_sqrtC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">Num.nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="Num.Theory.eqr_sqrtC"><span class="id" title="lemma">eqr_sqrtC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cedeb4d71ab31ea655ad3d06a9c296b3"><span class="id" title="notation">}</span></a>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtC_inj"><span class="id" title="lemma">sqrtC_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrtCM"><span class="id" title="lemma">sqrtCM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.nneg"><span class="id" title="abbreviation">Num.nneg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.sqrCK_P"><span class="id" title="lemma">sqrCK_P</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#388c172bf8d34ef0bf11898cd56f8d7b"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">)</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC_def"><span class="id" title="lemma">normC_def</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.norm_conjC"><span class="id" title="lemma">norm_conjC</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#b07d6e6599ef6e468ce182ffe6029532"><span class="id" title="notation">^*</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC_rect"><span class="id" title="lemma">normC_rect</span></a> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real"><span class="id" title="abbreviation">real</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC_Re_Im"><span class="id" title="lemma">normC_Re_Im</span></a> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> (<a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#dcb39ac1261bac8232156a37fb17d31c"><span class="id" title="notation">Re</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#03c300d501fea655f6f62a3c297714e9"><span class="id" title="notation">Im</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> 2).<br/> + +<br/> +</div> + +<div class="doc"> + Norm sum (in)equalities. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC_add_eq"><span class="id" title="lemma">normC_add_eq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC_sum_eq"><span class="id" title="lemma">normC_sum_eq</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>) :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC_sum_eq1"><span class="id" title="lemma">normC_sum_eq1</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>) :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC_sum_upper"><span class="id" title="lemma">normC_sum_upper</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory.C"><span class="id" title="variable">C</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#1065783963a393d1eafa2137291f2495"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#622398b62523a74328f94700e42198d0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.Theory.normC_sub_eq"><span class="id" title="lemma">normC_sub_eq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#c536f9a86d3c053391521360ac3f5a61"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.ClosedFieldTheory"><span class="id" title="section">ClosedFieldTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="04479a6ff05539b339540fbd2eba4ebf"><span class="id" title="notation">"</span></a>n .-root" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.nthroot"><span class="id" title="definition">nthroot</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "n .-root") : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> := 2<a class="idref" href="mathcomp.algebra.ssrnum.html#04479a6ff05539b339540fbd2eba4ebf"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#04479a6ff05539b339540fbd2eba4ebf"><span class="id" title="notation">root</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="f2048b82a608311baf565ccc9caefe54"><span class="id" title="notation">"</span></a>'i" := (@<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.imaginaryC"><span class="id" title="definition">imaginaryC</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="9153b7cf9b84603850c08e5131d1133a"><span class="id" title="notation">"</span></a>'Re z" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Re"><span class="id" title="definition">Re</span></a> <span class="id" title="var">z</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">z</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">ring_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="581a18ced675f8f9c5eb585ba4ae312b"><span class="id" title="notation">"</span></a>'Im z" := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.Im"><span class="id" title="definition">Im</span></a> <span class="id" title="var">z</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">z</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory"><span class="id" title="module">Theory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Module</span> <a name="Num.RealMixin"><span class="id" title="module">RealMixin</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.RealMixin.RealMixins"><span class="id" title="section">RealMixins</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="Num.RealMixin.RealMixins.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Exports.idomainType"><span class="id" title="abbreviation">idomainType</span></a>) (<a name="Num.RealMixin.RealMixins.le"><span class="id" title="variable">le</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>) (<a name="Num.RealMixin.RealMixins.lt"><span class="id" title="variable">lt</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>) (<a name="Num.RealMixin.RealMixins.norm"><span class="id" title="variable">norm</span></a> : <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.RealMixin.RealMixins.LeMixin"><span class="id" title="section">LeMixin</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LeMixin.le0_add"><span class="id" title="variable">le0_add</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LeMixin.le0_mul"><span class="id" title="variable">le0_mul</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LeMixin.le0_anti"><span class="id" title="variable">le0_anti</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LeMixin.sub_ge0"><span class="id" title="variable">sub_ge0</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LeMixin.le0_total"><span class="id" title="variable">le0_total</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LeMixin.normN"><span class="id" title="variable">normN</span></a>: <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LeMixin.ge0_norm"><span class="id" title="variable">ge0_norm</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LeMixin.lt_def"><span class="id" title="variable">lt_def</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Let</span> <a name="Num.RealMixin.RealMixins.LeMixin.le0N"><span class="id" title="variable">le0N</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>. <br/> +<span class="id" title="keyword">Let</span> <a name="Num.RealMixin.RealMixins.LeMixin.leN_total"><span class="id" title="variable">leN_total</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Let</span> <a name="Num.RealMixin.RealMixins.LeMixin.le00"><span class="id" title="variable">le00</span></a> : (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> 0). <br/> +<span class="id" title="keyword">Let</span> <a name="Num.RealMixin.RealMixins.LeMixin.le01"><span class="id" title="variable">le01</span></a> : (0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> 1).<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.lt0_add"><span class="id" title="lemma">lt0_add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.eq0_norm"><span class="id" title="lemma">eq0_norm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.le_def"><span class="id" title="lemma">le_def</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.normM"><span class="id" title="lemma">normM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.RealMixins.norm"><span class="id" title="variable">norm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><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/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.le_normD"><span class="id" title="lemma">le_normD</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.RealMixin.le_total"><span class="id" title="lemma">le_total</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/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealMixin.Le"><span class="id" title="definition">Le</span></a> :=<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Mixin"><span class="id" title="constructor">Mixin</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.le_normD"><span class="id" title="lemma">le_normD</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.lt0_add"><span class="id" title="lemma">lt0_add</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.eq0_norm"><span class="id" title="lemma">eq0_norm</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#in2W"><span class="id" title="lemma">in2W</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.le_total"><span class="id" title="lemma">le_total</span></a>) <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.normM"><span class="id" title="lemma">normM</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.le_def"><span class="id" title="lemma">le_def</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.RealMixins.LeMixin.lt_def"><span class="id" title="variable">lt_def</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Num.RealMixin.Real"><span class="id" title="lemma">Real</span></a> (<span class="id" title="var">R'</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Exports.numDomainType"><span class="id" title="abbreviation">numDomainType</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#phant"><span class="id" title="inductive">phant</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R'"><span class="id" title="variable">R'</span></a> :<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#R'"><span class="id" title="variable">R'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.NumDomain.Exports.NumDomainType"><span class="id" title="abbreviation">NumDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.RealMixins.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.real_axiom"><span class="id" title="definition">real_axiom</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#R'"><span class="id" title="variable">R'</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.RealMixins.LeMixin"><span class="id" title="section">LeMixin</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Num.RealMixin.RealMixins.LtMixin"><span class="id" title="section">LtMixin</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LtMixin.lt0_add"><span class="id" title="variable">lt0_add</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LtMixin.lt0_mul"><span class="id" title="variable">lt0_mul</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LtMixin.lt0_ngt0"><span class="id" title="variable">lt0_ngt0</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LtMixin.sub_gt0"><span class="id" title="variable">sub_gt0</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LtMixin.lt0_total"><span class="id" title="variable">lt0_total</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LtMixin.normN"><span class="id" title="variable">normN</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LtMixin.ge0_norm"><span class="id" title="variable">ge0_norm</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#de7043ea17a224ebc5072aadb91ca5c8"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="Num.RealMixin.RealMixins.LtMixin.le_def"><span class="id" title="variable">le_def</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.le0_add"><span class="id" title="lemma">le0_add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.le0_mul"><span class="id" title="lemma">le0_mul</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.le0_anti"><span class="id" title="lemma">le0_anti</span></a> <span class="id" title="var">x</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.sub_ge0"><span class="id" title="lemma">sub_ge0</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.lt_def"><span class="id" title="lemma">lt_def</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/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#6742b89c3f5ac60b756597eec71a6ec6"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Fact</span> <a name="Num.RealMixin.le0_total"><span class="id" title="lemma">le0_total</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#982e7680c015af99a74da7cd6b581a82"><span class="id" title="notation">≤</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Num.RealMixin.Lt"><span class="id" title="definition">Lt</span></a> :=<br/> + <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.Le"><span class="id" title="definition">Le</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.le0_add"><span class="id" title="lemma">le0_add</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.le0_mul"><span class="id" title="lemma">le0_mul</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.le0_anti"><span class="id" title="lemma">le0_anti</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.sub_ge0"><span class="id" title="lemma">sub_ge0</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.le0_total"><span class="id" title="lemma">le0_total</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.RealMixins.LtMixin.normN"><span class="id" title="variable">normN</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.RealMixins.LtMixin.ge0_norm"><span class="id" title="variable">ge0_norm</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.lt_def"><span class="id" title="lemma">lt_def</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.RealMixins.LtMixin"><span class="id" title="section">LtMixin</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin.RealMixins"><span class="id" title="section">RealMixins</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.RealMixin"><span class="id" title="module">RealMixin</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num"><span class="id" title="module">Num</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Export</span> <span class="id" title="var">Num.NumDomain.Exports</span> <span class="id" title="var">Num.NumField.Exports</span> <span class="id" title="var">Num.ClosedField.Exports</span>.<br/> +<span class="id" title="keyword">Export</span> <span class="id" title="var">Num.RealDomain.Exports</span> <span class="id" title="var">Num.RealField.Exports</span>.<br/> +<span class="id" title="keyword">Export</span> <span class="id" title="var">Num.ArchimedeanField.Exports</span> <span class="id" title="var">Num.RealClosedField.Exports</span>.<br/> +<span class="id" title="keyword">Export</span> <span class="id" title="var">Num.Syntax</span> <span class="id" title="var">Num.PredInstances</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="RealLeMixin"><span class="id" title="abbreviation">RealLeMixin</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Le"><span class="id" title="definition">Num.RealMixin.Le</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="RealLtMixin"><span class="id" title="abbreviation">RealLtMixin</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#Lt"><span class="id" title="definition">Num.RealMixin.Lt</span></a>.<br/> +<span class="id" title="keyword">Notation</span> <a name="RealLeAxiom"><span class="id" title="abbreviation">RealLeAxiom</span></a> <span class="id" title="var">R</span> := (<a class="idref" href="mathcomp.algebra.ssrnum.html#Real"><span class="id" title="lemma">Num.RealMixin.Real</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <span class="id" title="var">R</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#erefl"><span class="id" title="abbreviation">erefl</span></a> <span class="id" title="var">_</span>)).<br/> +<span class="id" title="keyword">Notation</span> <a name="ImaginaryMixin"><span class="id" title="abbreviation">ImaginaryMixin</span></a> := <a class="idref" href="mathcomp.algebra.ssrnum.html#ImaginaryMixin"><span class="id" title="constructor">Num.ClosedField.ImaginaryMixin</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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