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diff --git a/docs/htmldoc/mathcomp.algebra.polydiv.html b/docs/htmldoc/mathcomp.algebra.polydiv.html deleted file mode 100644 index 7fba25f..0000000 --- a/docs/htmldoc/mathcomp.algebra.polydiv.html +++ /dev/null @@ -1,2053 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" -"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml"> -<head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> -<link href="coqdoc.css" rel="stylesheet" type="text/css" /> -<title>mathcomp.algebra.polydiv</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.algebra.polydiv</h1> - -<div class="code"> -<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> - Distributed under the terms of CeCILL-B. *)</span><br/> - -<br/> -</div> - -<div class="doc"> - This file provides a library for the basic theory of Euclidean and pseudo- - Euclidean division for polynomials over ring structures. - The library defines two versions of the pseudo-euclidean division: one for - coefficients in a (not necessarily commutative) ring structure and one for - coefficients equipped with a structure of integral domain. From the latter - we derive the definition of the usual Euclidean division for coefficients - in a field. Only the definition of the pseudo-division for coefficients in - an integral domain is exported by default and benefits from notations. - Also, the only theory exported by default is the one of division for - polynomials with coefficients in a field. - Other definitions and facts are qualified using name spaces indicating the - hypotheses made on the structure of coefficients and the properties of the - polynomial one divides with. - -<div class="paragraph"> </div> - - Pdiv.Field (exported by the present library): - edivp p q == pseudo-division of p by q with p q : {poly R} where - R is an idomainType. - Computes (k, quo, rem) : nat * {poly r} * {poly R}, - such that size rem < size q and: - + if lead_coef q is not a unit, then: - (lead_coef q ^+ k) *: p = q * quo + rem - + else if lead_coef q is a unit, then: - p = q * quo + rem and k = 0 - p %/ q == quotient (second component) computed by (edivp p q). - p %% q == remainder (third component) computed by (edivp p q). - scalp p q == exponent (first component) computed by (edivp p q). - p %| q == tests the nullity of the remainder of the - pseudo-division of p by q. - rgcdp p q == Pseudo-greater common divisor obtained by performing - the Euclidean algorithm on p and q using redivp as - Euclidean division. - p %= q == p and q are associate polynomials, i.e., p %| q and - q %| p, or equivalently, p = c *: q for some nonzero - constant c. - gcdp p q == Pseudo-greater common divisor obtained by performing - the Euclidean algorithm on p and q using edivp as - Euclidean division. - egcdp p q == The pair of Bezout coefficients: if e := egcdp p q, - then size e.1 <= size q, size e.2 <= size p, and - gcdp p q %= e.1 * p + e.2 * q - coprimep p q == p and q are coprime, i.e., (gcdp p q) is a nonzero - constant. - gdcop q p == greatest divisor of p which is coprime to q. - irreducible_poly p <-> p has only trivial (constant) divisors. - -<div class="paragraph"> </div> - - Pdiv.Idomain: theory available for edivp and the related operation under - the sole assumption that the ring of coefficients is canonically an - integral domain (R : idomainType). - -<div class="paragraph"> </div> - - Pdiv.IdomainMonic: theory available for edivp and the related operations - under the assumption that the ring of coefficients is canonically - and integral domain (R : idomainType) an the divisor is monic. - -<div class="paragraph"> </div> - - Pdiv.IdomainUnit: theory available for edivp and the related operations - under the assumption that the ring of coefficients is canonically an - integral domain (R : idomainType) and the leading coefficient of the - divisor is a unit. - -<div class="paragraph"> </div> - - Pdiv.ClosedField: theory available for edivp and the related operation - under the sole assumption that the ring of coefficients is canonically - an algebraically closed field (R : closedField). - -<div class="paragraph"> </div> - - Pdiv.Ring : - redivp p q == pseudo-division of p by q with p q : {poly R} where R is - a ringType. - Computes (k, quo, rem) : nat * {poly r} * {poly R}, - such that if rem = 0 then quo * q = p * (lead_coef q ^+ k) - -<div class="paragraph"> </div> - - rdivp p q == quotient (second component) computed by (redivp p q). - rmodp p q == remainder (third component) computed by (redivp p q). - rscalp p q == exponent (first component) computed by (redivp p q). - rdvdp p q == tests the nullity of the remainder of the pseudo-division - of p by q. - rgcdp p q == analogue of gcdp for coefficients in a ringType. - rgdcop p q == analogue of gdcop for coefficients in a ringType. -rcoprimep p q == analogue of coprimep p q for coefficients in a ringType. - -<div class="paragraph"> </div> - - Pdiv.RingComRreg : theory of the operations defined in Pdiv.Ring, when the - ring of coefficients is canonically commutative (R : comRingType) and - the leading coefficient of the divisor is both right regular and - commutes as a constant polynomial with the divisor itself - -<div class="paragraph"> </div> - - Pdiv.RingMonic : theory of the operations defined in Pdiv.Ring, under the - assumption that the divisor is monic. - -<div class="paragraph"> </div> - - Pdiv.UnitRing: theory of the operations defined in Pdiv.Ring, when the - ring R of coefficients is canonically with units (R : unitRingType). - -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">GRing.Theory</span>.<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/> - -<br/> -<span class="id" title="keyword">Reserved Notation</span> "p %= q" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv"><span class="id" title="module">Pdiv</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.CommonRing"><span class="id" title="module">CommonRing</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.CommonRing.RingPseudoDivision"><span class="id" title="section">RingPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Pseudo division, defined on an arbitrary ring -</div> -<div class="code"> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.redivp_rec"><span class="id" title="definition">redivp_rec</span></a> (<span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) :=<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">sq</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">cq</span> := <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">fix</span> <span class="id" title="var">loop</span> (<span class="id" title="var">k</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">qq</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>)(<span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#sq"><span class="id" title="variable">sq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">m</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#e809881bcf0cc80f806c17b9ef433187"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#e809881bcf0cc80f806c17b9ef433187"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#e809881bcf0cc80f806c17b9ef433187"><span class="id" title="notation">^(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#sq"><span class="id" title="variable">sq</span></a><a class="idref" href="mathcomp.algebra.poly.html#e809881bcf0cc80f806c17b9ef433187"><span class="id" title="notation">)</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">qq1</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#cq"><span class="id" title="variable">cq</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">r1</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#cq"><span class="id" title="variable">cq</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n1</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.algebra.polydiv.html#loop"><span class="id" title="variable">loop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq1"><span class="id" title="variable">qq1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r1"><span class="id" title="variable">r1</span></a> <span class="id" title="var">n1</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq1"><span class="id" title="variable">qq1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r1"><span class="id" title="variable">r1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.redivp_expanded_def"><span class="id" title="definition">redivp_expanded_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp_rec"><span class="id" title="definition">redivp_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> 0 0 <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> -<span class="id" title="keyword">Fact</span> <a name="Pdiv.CommonRing.redivp_key"><span class="id" title="lemma">redivp_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp_key"><span class="id" title="lemma">redivp_key</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp_expanded_def"><span class="id" title="definition">redivp_expanded_def</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">redivp_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">fun</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">).1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">).2</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">).2</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.rscalp"><span class="id" title="definition">rscalp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">((</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">).1).1</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0.<br/> -</div> - -<div class="doc"> -Definition rmultp := [rel m d | rdvdp d m]. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.redivp_def"><span class="id" title="lemma">redivp_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdiv0p"><span class="id" title="lemma">rdiv0p</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdivp0"><span class="id" title="lemma">rdivp0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdivp_small"><span class="id" title="lemma">rdivp_small</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.leq_rdivp"><span class="id" title="lemma">leq_rdivp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rmod0p"><span class="id" title="lemma">rmod0p</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rmodp0"><span class="id" title="lemma">rmodp0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rscalp_small"><span class="id" title="lemma">rscalp_small</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.ltn_rmodp"><span class="id" title="lemma">ltn_rmodp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.ltn_rmodpN0"><span class="id" title="lemma">ltn_rmodpN0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rmodp1"><span class="id" title="lemma">rmodp1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rmodp_small"><span class="id" title="lemma">rmodp_small</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.leq_rmodp"><span class="id" title="lemma">leq_rmodp</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rmodpC"><span class="id" title="lemma">rmodpC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdvdp0"><span class="id" title="lemma">rdvdp0</span></a> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdvd0p"><span class="id" title="lemma">rdvd0p</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdvd0pP"><span class="id" title="lemma">rdvd0pP</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdvdpN0"><span class="id" title="lemma">rdvdpN0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdvdp1"><span class="id" title="lemma">rdvdp1</span></a> <span class="id" title="var">d</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdvd1p"><span class="id" title="lemma">rdvd1p</span></a> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> 1 <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.Nrdvdp_small"><span class="id" title="lemma">Nrdvdp_small</span></a> (<span class="id" title="var">n</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rmodp_eq0P"><span class="id" title="lemma">rmodp_eq0P</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rmodp_eq0"><span class="id" title="lemma">rmodp_eq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rdvdp_leq"><span class="id" title="lemma">rdvdp_leq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.rgcdp"><span class="id" title="definition">rgcdp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :=<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">p1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">q1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <span class="id" title="var">p1</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">q1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <span class="id" title="var">loop</span> (<span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">pp</span> <span class="id" title="var">qq</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} :=<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">rr</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#pp"><span class="id" title="variable">pp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rr"><span class="id" title="variable">rr</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n1</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.algebra.polydiv.html#loop"><span class="id" title="variable">loop</span></a> <span class="id" title="var">n1</span> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rr"><span class="id" title="variable">rr</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.algebra.polydiv.html#rr"><span class="id" title="variable">rr</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#loop"><span class="id" title="variable">loop</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <span class="id" title="var">p1</span>) <span class="id" title="var">p1</span> <span class="id" title="var">q1</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rgcd0p"><span class="id" title="lemma">rgcd0p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgcdp"><span class="id" title="definition">rgcdp</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rgcdp0"><span class="id" title="lemma">rgcdp0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgcdp"><span class="id" title="definition">rgcdp</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rgcdpE"><span class="id" title="lemma">rgcdpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgcdp"><span class="id" title="definition">rgcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgcdp"><span class="id" title="definition">rgcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgcdp"><span class="id" title="definition">rgcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="Pdiv.CommonRing.comm_redivp_spec"><span class="id" title="inductive">comm_redivp_spec</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <a name="Pdiv.CommonRing.ComEdivnSpec"><span class="id" title="constructor">ComEdivnSpec</span></a> <span class="id" title="var">k</span> (<span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) <span class="id" title="keyword">of</span><br/> - (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) &<br/> - (<a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) : <a class="idref" href="mathcomp.algebra.polydiv.html#comm_redivp_spec"><span class="id" title="inductive">comm_redivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.comm_redivpP"><span class="id" title="lemma">comm_redivpP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.comm_redivp_spec"><span class="id" title="inductive">comm_redivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rmodpp"><span class="id" title="lemma">rmodpp</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.rcoprimep"><span class="id" title="definition">rcoprimep</span></a> (<span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) := <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgcdp"><span class="id" title="definition">rgcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="Pdiv.CommonRing.rgdcop_rec"><span class="id" title="definition">rgdcop_rec</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> <span class="id" title="var">n</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">m</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rcoprimep"><span class="id" title="definition">rcoprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rgdcop_rec"><span class="id" title="definition">rgdcop_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgcdp"><span class="id" title="definition">rgcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>)) <span class="id" title="var">m</span><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonRing.rgdcop"><span class="id" title="definition">rgdcop</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgdcop_rec"><span class="id" title="definition">rgdcop_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonRing.rgdcop0"><span class="id" title="lemma">rgdcop0</span></a> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rgdcop"><span class="id" title="definition">rgdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.RingPseudoDivision"><span class="id" title="section">RingPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing"><span class="id" title="module">CommonRing</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.RingComRreg"><span class="id" title="module">RingComRreg</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">CommonRing</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.RingComRreg.ComRegDivisor"><span class="id" title="section">ComRegDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.RingComRreg.ComRegDivisor.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Pdiv.RingComRreg.ComRegDivisor.Cdl"><span class="id" title="variable">Cdl</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>.<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Pdiv.RingComRreg.ComRegDivisor.Rreg"><span class="id" title="variable">Rreg</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.rreg"><span class="id" title="definition">GRing.rreg</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.redivp_eq"><span class="id" title="lemma">redivp_eq</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">k</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">).1.1</span></a> <span class="id" title="tactic">in</span><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">c</span> := <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - this is a bad name -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.rdivp_eq"><span class="id" title="lemma">rdivp_eq</span></a> <span class="id" title="var">p</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - section variables impose an inconvenient order on parameters -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.eq_rdvdp"><span class="id" title="lemma">eq_rdvdp</span></a> <span class="id" title="var">k</span> <span class="id" title="var">q1</span> <span class="id" title="var">p</span>:<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="Pdiv.RingComRreg.rdvdp_spec"><span class="id" title="inductive">rdvdp_spec</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - | <a name="Pdiv.RingComRreg.Rdvdp"><span class="id" title="constructor">Rdvdp</span></a> <span class="id" title="var">k</span> <span class="id" title="var">q1</span> & <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#rdvdp_spec"><span class="id" title="inductive">rdvdp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> - | <a name="Pdiv.RingComRreg.RdvdpN"><span class="id" title="constructor">RdvdpN</span></a> & <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 : <a class="idref" href="mathcomp.algebra.polydiv.html#rdvdp_spec"><span class="id" title="inductive">rdvdp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Is that version useable ? -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.rdvdp_eqP"><span class="id" title="lemma">rdvdp_eqP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.rdvdp_spec"><span class="id" title="inductive">rdvdp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.rdvdp_mull"><span class="id" title="lemma">rdvdp_mull</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.rmodp_mull"><span class="id" title="lemma">rmodp_mull</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.rmodpp"><span class="id" title="lemma">rmodpp</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.rdivpp"><span class="id" title="lemma">rdivpp</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.rdvdpp"><span class="id" title="lemma">rdvdpp</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingComRreg.rdivpK"><span class="id" title="lemma">rdivpK</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.ComRegDivisor"><span class="id" title="section">ComRegDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg"><span class="id" title="module">RingComRreg</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.RingMonic"><span class="id" title="module">RingMonic</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">CommonRing</span>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">RingComRreg</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.RingMonic.MonicDivisor"><span class="id" title="section">MonicDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.RingMonic.MonicDivisor.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Pdiv.RingMonic.MonicDivisor.mond"><span class="id" title="variable">mond</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.redivp_eq"><span class="id" title="lemma">redivp_eq</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">k</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">).1.1</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.redivp"><span class="id" title="definition">redivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdivp_eq"><span class="id" title="lemma">rdivp_eq</span></a> <span class="id" title="var">p</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdivpp"><span class="id" title="lemma">rdivpp</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdivp_addl_mul_small"><span class="id" title="lemma">rdivp_addl_mul_small</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdivp_addl_mul"><span class="id" title="lemma">rdivp_addl_mul</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdivp_addl"><span class="id" title="lemma">rdivp_addl</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdivp_addr"><span class="id" title="lemma">rdivp_addr</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdivp_mull"><span class="id" title="lemma">rdivp_mull</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rmodp_mull"><span class="id" title="lemma">rmodp_mull</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rmodpp"><span class="id" title="lemma">rmodpp</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rmodp_addl_mul_small"><span class="id" title="lemma">rmodp_addl_mul_small</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rmodp_add"><span class="id" title="lemma">rmodp_add</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rmodp_mulmr"><span class="id" title="lemma">rmodp_mulmr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rmodp"><span class="id" title="definition">rmodp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdvdpp"><span class="id" title="lemma">rdvdpp</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - section variables impose an inconvenient order on parameters -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.eq_rdvdp"><span class="id" title="lemma">eq_rdvdp</span></a> <span class="id" title="var">q1</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdvdp_mull"><span class="id" title="lemma">rdvdp_mull</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdvdpP"><span class="id" title="lemma">rdvdpP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">qq</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.RingMonic.rdivpK"><span class="id" title="lemma">rdivpK</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic.MonicDivisor"><span class="id" title="section">MonicDivisor</span></a>.<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingMonic"><span class="id" title="module">RingMonic</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.Ring"><span class="id" title="module">Ring</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Include</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonRing"><span class="id" title="module">CommonRing</span></a>.<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">RingMonic</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.Ring.ExtraMonicDivisor"><span class="id" title="section">ExtraMonicDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.Ring.ExtraMonicDivisor.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.ExtraMonicDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Ring.rdivp1"><span class="id" title="lemma">rdivp1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Ring.rdvdp_XsubCl"><span class="id" title="lemma">rdvdp_XsubCl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdvdp"><span class="id" title="definition">rdvdp</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Ring.polyXsubCP"><span class="id" title="lemma">polyXsubCP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdvdp"><span class="id" title="definition">rdvdp</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Ring.root_factor_theorem"><span class="id" title="lemma">root_factor_theorem</span></a> <span class="id" title="var">p</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.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdvdp"><span class="id" title="definition">rdvdp</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.ExtraMonicDivisor"><span class="id" title="section">ExtraMonicDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring"><span class="id" title="module">Ring</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.ComRing"><span class="id" title="module">ComRing</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Ring</span>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">RingComRreg</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.ComRing.CommutativeRingPseudoDivision"><span class="id" title="section">CommutativeRingPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.ComRing.CommutativeRingPseudoDivision.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Exports.comRingType"><span class="id" title="abbreviation">comRingType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ComRing.CommutativeRingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="Pdiv.ComRing.redivp_spec"><span class="id" title="inductive">redivp_spec</span></a> (<span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ComRing.CommutativeRingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ComRing.CommutativeRingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ComRing.CommutativeRingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <a name="Pdiv.ComRing.EdivnSpec"><span class="id" title="constructor">EdivnSpec</span></a> <span class="id" title="var">k</span> (<span class="id" title="var">q</span> <span class="id" title="var">r</span>: <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ComRing.CommutativeRingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) <span class="id" title="keyword">of</span><br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> &<br/> - (<a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) : <a class="idref" href="mathcomp.algebra.polydiv.html#redivp_spec"><span class="id" title="inductive">redivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.ComRing.redivpP"><span class="id" title="lemma">redivpP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ComRing.redivp_spec"><span class="id" title="inductive">redivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.ComRing.rdivp_eq"><span class="id" title="lemma">rdivp_eq</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> :<br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.ComRing.rdvdp_eqP"><span class="id" title="lemma">rdvdp_eqP</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.RingComRreg.rdvdp_spec"><span class="id" title="inductive">rdvdp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.ComRing.rdvdp_eq"><span class="id" title="lemma">rdvdp_eq</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ComRing.CommutativeRingPseudoDivision"><span class="id" title="section">CommutativeRingPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ComRing"><span class="id" title="module">ComRing</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.UnitRing"><span class="id" title="module">UnitRing</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Ring</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.UnitRing.UnitRingPseudoDivision"><span class="id" title="section">UnitRingPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.UnitRing.UnitRingPseudoDivision.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Exports.unitRingType"><span class="id" title="abbreviation">unitRingType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.UnitRing.UnitRingPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.UnitRing.uniq_roots_rdvdp"><span class="id" title="lemma">uniq_roots_rdvdp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">rs</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</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.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#uniq_roots"><span class="id" title="definition">uniq_roots</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdvdp"><span class="id" title="definition">rdvdp</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rs"><span class="id" title="variable">rs</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.UnitRing.UnitRingPseudoDivision"><span class="id" title="section">UnitRingPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.UnitRing"><span class="id" title="module">UnitRing</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.IdomainDefs"><span class="id" title="module">IdomainDefs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Ring</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.IdomainDefs.IDomainPseudoDivisionDefs"><span class="id" title="section">IDomainPseudoDivisionDefs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.IdomainDefs.IDomainPseudoDivisionDefs.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>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.IDomainPseudoDivisionDefs.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.IdomainDefs.edivp_expanded_def"><span class="id" title="definition">edivp_expanded_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :=<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">k</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">d</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">edvpq</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)^-</span></a><span class="id" title="var">k</span> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <span class="id" title="var">d</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)^-</span></a><span class="id" title="var">k</span> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <span class="id" title="var">r</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <span class="id" title="var">edvpq</span>.<br/> -<span class="id" title="keyword">Fact</span> <a name="Pdiv.IdomainDefs.edivp_key"><span class="id" title="lemma">edivp_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp_key"><span class="id" title="lemma">edivp_key</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp_expanded_def"><span class="id" title="definition">edivp_expanded_def</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">edivp_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">fun</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#84464b412faf5a30a7c5c6423d9b3956"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.IdomainDefs.divp"><span class="id" title="definition">divp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">).1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">).2</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.IdomainDefs.modp"><span class="id" title="definition">modp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">).2</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">((</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">).1).1</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.IdomainDefs.dvdp"><span class="id" title="definition">dvdp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.modp"><span class="id" title="definition">modp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0.<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.IdomainDefs.eqp"><span class="id" title="definition">eqp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.dvdp"><span class="id" title="definition">dvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.dvdp"><span class="id" title="definition">dvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.IDomainPseudoDivisionDefs"><span class="id" title="section">IDomainPseudoDivisionDefs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">"</span></a>m %/ d" := (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.divp"><span class="id" title="definition">divp</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">"</span></a>m %% d" := (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.modp"><span class="id" title="definition">modp</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">"</span></a>p %| q" := (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.dvdp"><span class="id" title="definition">dvdp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">"</span></a>p %= q" := (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.eqp"><span class="id" title="definition">eqp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs"><span class="id" title="module">IdomainDefs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.WeakIdomain"><span class="id" title="module">WeakIdomain</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Ring</span> <span class="id" title="var">ComRing</span> <span class="id" title="var">UnitRing</span> <span class="id" title="var">IdomainDefs</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision"><span class="id" title="section">WeakTheoryForIDomainPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision.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>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.edivp_def"><span class="id" title="lemma">edivp_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.divp"><span class="id" title="definition">divp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.modp"><span class="id" title="definition">modp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.edivp_redivp"><span class="id" title="lemma">edivp_redivp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.divpE"><span class="id" title="lemma">divpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)^-(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.modpE"><span class="id" title="lemma">modpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)^-(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.scalpE"><span class="id" title="lemma">scalpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.dvdpE"><span class="id" title="lemma">dvdpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.lc_expn_scalp_neq0"><span class="id" title="lemma">lc_expn_scalp_neq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">lc_expn_scalp_neq0</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="Pdiv.WeakIdomain.edivp_spec"><span class="id" title="inductive">edivp_spec</span></a> (<span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> -|<a name="Pdiv.WeakIdomain.Redivp_spec"><span class="id" title="constructor">Redivp_spec</span></a> <span class="id" title="var">k</span> (<span class="id" title="var">q</span> <span class="id" title="var">r</span>: <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) <span class="id" title="keyword">of</span><br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> & <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> &<br/> - (<a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) : <a class="idref" href="mathcomp.algebra.polydiv.html#edivp_spec"><span class="id" title="inductive">edivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/> -|<a name="Pdiv.WeakIdomain.Fedivp_spec"><span class="id" title="constructor">Fedivp_spec</span></a> (<span class="id" title="var">q</span> <span class="id" title="var">r</span>: <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> & (<a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a>) &<br/> - (<a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) : <a class="idref" href="mathcomp.algebra.polydiv.html#edivp_spec"><span class="id" title="inductive">edivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - There are several ways to state this fact. The most appropriate statement - might be polished in light of usage. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.edivpP"><span class="id" title="lemma">edivpP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain.edivp_spec"><span class="id" title="inductive">edivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) (<a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.edivp_eq"><span class="id" title="lemma">edivp_eq</span></a> <span class="id" title="var">d</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.divp_eq"><span class="id" title="lemma">divp_eq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.dvdp_eq"><span class="id" title="lemma">dvdp_eq</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.divpK"><span class="id" title="lemma">divpK</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.divpKC"><span class="id" title="lemma">divpKC</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.dvdpP"><span class="id" title="lemma">dvdpP</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">cqq</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#cqq"><span class="id" title="variable">cqq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#cqq"><span class="id" title="variable">cqq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#cqq"><span class="id" title="variable">cqq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.mulpK"><span class="id" title="lemma">mulpK</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.mulKp"><span class="id" title="lemma">mulKp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.WeakIdomain.divpp"><span class="id" title="lemma">divpp</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain.WeakTheoryForIDomainPseudoDivision"><span class="id" title="section">WeakTheoryForIDomainPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">lc_expn_scalp_neq0</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain"><span class="id" title="module">WeakIdomain</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.CommonIdomain"><span class="id" title="module">CommonIdomain</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Ring</span> <span class="id" title="var">ComRing</span> <span class="id" title="var">UnitRing</span> <span class="id" title="var">IdomainDefs</span> <span class="id" title="var">WeakIdomain</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.CommonIdomain.IDomainPseudoDivision"><span class="id" title="section">IDomainPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.CommonIdomain.IDomainPseudoDivision.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>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.scalp0"><span class="id" title="lemma">scalp0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.divp_small"><span class="id" title="lemma">divp_small</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.leq_divp"><span class="id" title="lemma">leq_divp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.div0p"><span class="id" title="lemma">div0p</span></a> <span class="id" title="var">p</span> : 0 <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.divp0"><span class="id" title="lemma">divp0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.divp1"><span class="id" title="lemma">divp1</span></a> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp0"><span class="id" title="lemma">modp0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.mod0p"><span class="id" title="lemma">mod0p</span></a> <span class="id" title="var">p</span> : 0 <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp1"><span class="id" title="lemma">modp1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">divp0</span> <span class="id" title="var">divp1</span> <span class="id" title="var">mod0p</span> <span class="id" title="var">modp0</span> <span class="id" title="var">modp1</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp_small"><span class="id" title="lemma">modp_small</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modpC"><span class="id" title="lemma">modpC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp_mull"><span class="id" title="lemma">modp_mull</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp_mulr"><span class="id" title="lemma">modp_mulr</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modpp"><span class="id" title="lemma">modpp</span></a> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.ltn_modp"><span class="id" title="lemma">ltn_modp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.ltn_divpl"><span class="id" title="lemma">ltn_divpl</span></a> <span class="id" title="var">d</span> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.leq_divpr"><span class="id" title="lemma">leq_divpr</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.divpN0"><span class="id" title="lemma">divpN0</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.size_divp"><span class="id" title="lemma">size_divp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">).-1</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.ltn_modpN0"><span class="id" title="lemma">ltn_modpN0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp_mod"><span class="id" title="lemma">modp_mod</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.leq_modp"><span class="id" title="lemma">leq_modp</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp0"><span class="id" title="lemma">dvdp0</span></a> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">dvdp0</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvd0p"><span class="id" title="lemma">dvd0p</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvd0pP"><span class="id" title="lemma">dvd0pP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (0 <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdpN0"><span class="id" title="lemma">dvdpN0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp1"><span class="id" title="lemma">dvdp1</span></a> <span class="id" title="var">d</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvd1p"><span class="id" title="lemma">dvd1p</span></a> <span class="id" title="var">m</span> : 1 <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gtNdvdp"><span class="id" title="lemma">gtNdvdp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp_eq0P"><span class="id" title="lemma">modp_eq0P</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp_eq0"><span class="id" title="lemma">modp_eq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.leq_divpl"><span class="id" title="lemma">leq_divpl</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_leq"><span class="id" title="lemma">dvdp_leq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eq_dvdp"><span class="id" title="lemma">eq_dvdp</span></a> <span class="id" title="var">c</span> <span class="id" title="var">quo</span> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#quo"><span class="id" title="variable">quo</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdpp"><span class="id" title="lemma">dvdpp</span></a> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">dvdpp</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.divp_dvd"><span class="id" title="lemma">divp_dvd</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mull"><span class="id" title="lemma">dvdp_mull</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mulr"><span class="id" title="lemma">dvdp_mulr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">dvdp_mull</span> <span class="id" title="var">dvdp_mulr</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mul"><span class="id" title="lemma">dvdp_mul</span></a> <span class="id" title="var">d1</span> <span class="id" title="var">d2</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d1"><span class="id" title="variable">d1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d2"><span class="id" title="variable">d2</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d1"><span class="id" title="variable">d1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d2"><span class="id" title="variable">d2</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m2"><span class="id" title="variable">m2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_addr"><span class="id" title="lemma">dvdp_addr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_addl"><span class="id" title="lemma">dvdp_addl</span></a> <span class="id" title="var">n</span> <span class="id" title="var">d</span> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_add"><span class="id" title="lemma">dvdp_add</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_add_eq"><span class="id" title="lemma">dvdp_add_eq</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_subr"><span class="id" title="lemma">dvdp_subr</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_subl"><span class="id" title="lemma">dvdp_subl</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_sub"><span class="id" title="lemma">dvdp_sub</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mod"><span class="id" title="lemma">dvdp_mod</span></a> <span class="id" title="var">d</span> <span class="id" title="var">n</span> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_trans"><span class="id" title="lemma">dvdp_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> (@<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.dvdp"><span class="id" title="definition">dvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mulIl"><span class="id" title="lemma">dvdp_mulIl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mulIr"><span class="id" title="lemma">dvdp_mulIr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mul2r"><span class="id" title="lemma">dvdp_mul2r</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mul2l"><span class="id" title="lemma">dvdp_mul2l</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.ltn_divpr"><span class="id" title="lemma">ltn_divpr</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_exp"><span class="id" title="lemma">dvdp_exp</span></a> <span class="id" title="var">d</span> <span class="id" title="var">k</span> <span class="id" title="var">p</span> : 0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_exp2l"><span class="id" title="lemma">dvdp_exp2l</span></a> <span class="id" title="var">d</span> <span class="id" title="var">k</span> <span class="id" title="var">l</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#l"><span class="id" title="variable">l</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_Pexp2l"><span class="id" title="lemma">dvdp_Pexp2l</span></a> <span class="id" title="var">d</span> <span class="id" title="var">k</span> <span class="id" title="var">l</span> : 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_exp2r"><span class="id" title="lemma">dvdp_exp2r</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">k</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_exp_sub"><span class="id" title="lemma">dvdp_exp_sub</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">k</span> <span class="id" title="var">l</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#l"><span class="id" title="variable">l</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_XsubCl"><span class="id" title="lemma">dvdp_XsubCl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.polyXsubCP"><span class="id" title="lemma">polyXsubCP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_div_XsubC"><span class="id" title="lemma">eqp_div_XsubC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.root_factor_theorem"><span class="id" title="lemma">root_factor_theorem</span></a> <span class="id" title="var">p</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.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.uniq_roots_dvdp"><span class="id" title="lemma">uniq_roots_dvdp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">rs</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</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.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#uniq_roots"><span class="id" title="definition">uniq_roots</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rs"><span class="id" title="variable">rs</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.root_bigmul"><span class="id" title="lemma">root_bigmul</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> (<span class="id" title="var">ps</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>),<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">(</span></a><span class="id" title="var">p</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#ps"><span class="id" title="variable">ps</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">p</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#ps"><span class="id" title="variable">ps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqpP"><span class="id" title="lemma">eqpP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">c12</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c12"><span class="id" title="variable">c12</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c12"><span class="id" title="variable">c12</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c12"><span class="id" title="variable">c12</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c12"><span class="id" title="variable">c12</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>)<br/> - (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_eq"><span class="id" title="lemma">eqp_eq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqpxx"><span class="id" title="lemma">eqpxx</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflexive"><span class="id" title="definition">reflexive</span></a> (@<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.eqp"><span class="id" title="definition">eqp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">eqpxx</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_sym"><span class="id" title="lemma">eqp_sym</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#symmetric"><span class="id" title="definition">symmetric</span></a> (@<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.eqp"><span class="id" title="definition">eqp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_trans"><span class="id" title="lemma">eqp_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> (@<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.eqp"><span class="id" title="definition">eqp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_ltrans"><span class="id" title="lemma">eqp_ltrans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#left_transitive"><span class="id" title="definition">left_transitive</span></a> (@<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.eqp"><span class="id" title="definition">eqp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_rtrans"><span class="id" title="lemma">eqp_rtrans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#right_transitive"><span class="id" title="definition">right_transitive</span></a> (@<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.eqp"><span class="id" title="definition">eqp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp0"><span class="id" title="lemma">eqp0</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp01"><span class="id" title="lemma">eqp01</span></a> : 0 <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_scale"><span class="id" title="lemma">eqp_scale</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_size"><span class="id" title="lemma">eqp_size</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.size_poly_eq1"><span class="id" title="lemma">size_poly_eq1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.polyXsubC_eqp1"><span class="id" title="lemma">polyXsubC_eqp1</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_eqp1"><span class="id" title="lemma">dvdp_eqp1</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_dvdr"><span class="id" title="lemma">eqp_dvdr</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> <span class="id" title="var">d</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_dvdl"><span class="id" title="lemma">eqp_dvdl</span></a> <span class="id" title="var">d2</span> <span class="id" title="var">d1</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d1"><span class="id" title="variable">d1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d2"><span class="id" title="variable">d2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d1"><span class="id" title="variable">d1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d2"><span class="id" title="variable">d2</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_scaler"><span class="id" title="lemma">dvdp_scaler</span></a> <span class="id" title="var">c</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_scalel"><span class="id" title="lemma">dvdp_scalel</span></a> <span class="id" title="var">c</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_opp"><span class="id" title="lemma">dvdp_opp</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_mul2r"><span class="id" title="lemma">eqp_mul2r</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_mul2l"><span class="id" title="lemma">eqp_mul2l</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_mull"><span class="id" title="lemma">eqp_mull</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_mulr"><span class="id" title="lemma">eqp_mulr</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> <span class="id" title="var">r</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_exp"><span class="id" title="lemma">eqp_exp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">k</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.polyC_eqp1"><span class="id" title="lemma">polyC_eqp1</span></a> (<span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdUp"><span class="id" title="lemma">dvdUp</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_size_eqp"><span class="id" title="lemma">dvdp_size_eqp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_root"><span class="id" title="lemma">eqp_root</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_rmod_mod"><span class="id" title="lemma">eqp_rmod_mod</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.modp"><span class="id" title="definition">modp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_rdiv_div"><span class="id" title="lemma">eqp_rdiv_div</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.divp"><span class="id" title="definition">divp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvd_eqp_divl"><span class="id" title="lemma">dvd_eqp_divl</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> (<span class="id" title="var">dvd_dp</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) (<span class="id" title="var">eq_pq</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonIdomain.gcdp_rec"><span class="id" title="definition">gcdp_rec</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :=<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">p1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">q1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <span class="id" title="var">p1</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">q1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <span class="id" title="var">loop</span> (<span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">pp</span> <span class="id" title="var">qq</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} :=<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">rr</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.modp"><span class="id" title="definition">modp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#pp"><span class="id" title="variable">pp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rr"><span class="id" title="variable">rr</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n1</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.algebra.polydiv.html#loop"><span class="id" title="variable">loop</span></a> <span class="id" title="var">n1</span> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#rr"><span class="id" title="variable">rr</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.algebra.polydiv.html#rr"><span class="id" title="variable">rr</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#loop"><span class="id" title="variable">loop</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <span class="id" title="var">p1</span>) <span class="id" title="var">p1</span> <span class="id" title="var">q1</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#nosimpl"><span class="id" title="abbreviation">nosimpl</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp_rec"><span class="id" title="definition">gcdp_rec</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcd0p"><span class="id" title="lemma">gcd0p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp0"><span class="id" title="lemma">gcdp0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdpE"><span class="id" title="lemma">gcdpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.modp"><span class="id" title="definition">modp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.modp"><span class="id" title="definition">modp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.size_gcd1p"><span class="id" title="lemma">size_gcd1p</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> 1 <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.size_gcdp1"><span class="id" title="lemma">size_gcdp1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 1) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdpp"><span class="id" title="lemma">gcdpp</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#idempotent"><span class="id" title="definition">idempotent</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_gcdlr"><span class="id" title="lemma">dvdp_gcdlr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_gcdl"><span class="id" title="lemma">dvdp_gcdl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_gcdr"><span class="id" title="lemma">dvdp_gcdr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.leq_gcdpl"><span class="id" title="lemma">leq_gcdpl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.leq_gcdpr"><span class="id" title="lemma">leq_gcdpr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_gcd"><span class="id" title="lemma">dvdp_gcd</span></a> <span class="id" title="var">p</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdpC"><span class="id" title="lemma">gcdpC</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>, <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcd1p"><span class="id" title="lemma">gcd1p</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> 1 <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp1"><span class="id" title="lemma">gcdp1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 1 <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_addl_mul"><span class="id" title="lemma">gcdp_addl_mul</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_addl"><span class="id" title="lemma">gcdp_addl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_addr"><span class="id" title="lemma">gcdp_addr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_mull"><span class="id" title="lemma">gcdp_mull</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_mulr"><span class="id" title="lemma">gcdp_mulr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_scalel"><span class="id" title="lemma">gcdp_scalel</span></a> <span class="id" title="var">c</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_scaler"><span class="id" title="lemma">gcdp_scaler</span></a> <span class="id" title="var">c</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_gcd_idl"><span class="id" title="lemma">dvdp_gcd_idl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_gcd_idr"><span class="id" title="lemma">dvdp_gcd_idr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_exp"><span class="id" title="lemma">gcdp_exp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">k</span> <span class="id" title="var">l</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#minn"><span class="id" title="definition">minn</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#l"><span class="id" title="variable">l</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_eq0"><span class="id" title="lemma">gcdp_eq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_gcdr"><span class="id" title="lemma">eqp_gcdr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_gcdl"><span class="id" title="lemma">eqp_gcdl</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_gcd"><span class="id" title="lemma">eqp_gcd</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="var">q1</span> <span class="id" title="var">q2</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q2"><span class="id" title="variable">q2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q2"><span class="id" title="variable">q2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_rgcd_gcd"><span class="id" title="lemma">eqp_rgcd_gcd</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rgcdp"><span class="id" title="definition">rgcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_modr"><span class="id" title="lemma">gcdp_modr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_modl"><span class="id" title="lemma">gcdp_modl</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_def"><span class="id" title="lemma">gcdp_def</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">d'</span>, <a class="idref" href="mathcomp.algebra.polydiv.html#d'"><span class="id" title="variable">d'</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d'"><span class="id" title="variable">d'</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d'"><span class="id" title="variable">d'</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><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="Pdiv.CommonIdomain.coprimep_size_gcd"><span class="id" title="lemma">coprimep_size_gcd</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_def"><span class="id" title="lemma">coprimep_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_scalel"><span class="id" title="lemma">coprimep_scalel</span></a> <span class="id" title="var">c</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_scaler"><span class="id" title="lemma">coprimep_scaler</span></a> <span class="id" title="var">c</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span>:<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimepp"><span class="id" title="lemma">coprimepp</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_eqp1"><span class="id" title="lemma">gcdp_eqp1</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_sym"><span class="id" title="lemma">coprimep_sym</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprime1p"><span class="id" title="lemma">coprime1p</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> 1 <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep1"><span class="id" title="lemma">coprimep1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep0"><span class="id" title="lemma">coprimep0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprime0p"><span class="id" title="lemma">coprime0p</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> 0 <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is different from coprimeP in div. shall we keep this? -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimepP"><span class="id" title="lemma">coprimepP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">d</span>, <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimepPn"><span class="id" title="lemma">coprimepPn</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">d</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_dvdl"><span class="id" title="lemma">coprimep_dvdl</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_dvdr"><span class="id" title="lemma">coprimep_dvdr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_modl"><span class="id" title="lemma">coprimep_modl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_modr"><span class="id" title="lemma">coprimep_modr</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.rcoprimep_coprimep"><span class="id" title="lemma">rcoprimep_coprimep</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rcoprimep"><span class="id" title="definition">rcoprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_coprimepr"><span class="id" title="lemma">eqp_coprimepr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_coprimepl"><span class="id" title="lemma">eqp_coprimepl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This should be implemented with an extended remainder sequence -</div> -<div class="code"> -<span class="id" title="keyword">Fixpoint</span> <a name="Pdiv.CommonIdomain.egcdp_rec"><span class="id" title="definition">egcdp_rec</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">k</span> {<span class="id" title="keyword">struct</span> <span class="id" title="var">k</span>} : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">k'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a><br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">v</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> := <a class="idref" href="mathcomp.algebra.polydiv.html#egcdp_rec"><span class="id" title="definition">egcdp_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <span class="id" title="var">k'</span> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <span class="id" title="var">v</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">u</span> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <span class="id" title="var">v</span> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonIdomain.egcdp"><span class="id" title="definition">egcdp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.egcdp_rec"><span class="id" title="definition">egcdp_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>)<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.egcdp_rec"><span class="id" title="definition">egcdp_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - No provable egcd0p -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.egcdp0"><span class="id" title="lemma">egcdp0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.egcdp"><span class="id" title="definition">egcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.egcdp_recP"><span class="id" title="lemma">egcdp_recP</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">k</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>, <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.egcdp_rec"><span class="id" title="definition">egcdp_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>) <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.egcdpP"><span class="id" title="lemma">egcdpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> (<span class="id" title="var">e</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.egcdp"><span class="id" title="definition">egcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>),<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.egcdpE"><span class="id" title="lemma">egcdpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> (<span class="id" title="var">e</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.egcdp"><span class="id" title="definition">egcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.Bezoutp"><span class="id" title="lemma">Bezoutp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.Bezout_coprimepP"><span class="id" title="lemma">Bezout_coprimepP</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>,<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_root"><span class="id" title="lemma">coprimep_root</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.Gauss_dvdpl"><span class="id" title="lemma">Gauss_dvdpl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">d</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.Gauss_dvdpr"><span class="id" title="lemma">Gauss_dvdpr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">d</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This could be simplified with the introduction of lcmp -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.Gauss_dvdp"><span class="id" title="lemma">Gauss_dvdp</span></a> <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="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.Gauss_gcdpr"><span class="id" title="lemma">Gauss_gcdpr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.Gauss_gcdpl"><span class="id" title="lemma">Gauss_gcdpl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_mulr"><span class="id" title="lemma">coprimep_mulr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_mull"><span class="id" title="lemma">coprimep_mull</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span>: <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp_coprime"><span class="id" title="lemma">modp_coprime</span></a> <span class="id" title="var">k</span> <span class="id" title="var">u</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_pexpl"><span class="id" title="lemma">coprimep_pexpl</span></a> <span class="id" title="var">k</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_pexpr"><span class="id" title="lemma">coprimep_pexpr</span></a> <span class="id" title="var">k</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : 0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_expl"><span class="id" title="lemma">coprimep_expl</span></a> <span class="id" title="var">k</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_expr"><span class="id" title="lemma">coprimep_expr</span></a> <span class="id" title="var">k</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_mul2l"><span class="id" title="lemma">gcdp_mul2l</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_mul2r"><span class="id" title="lemma">gcdp_mul2r</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.mulp_gcdr"><span class="id" title="lemma">mulp_gcdr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.mulp_gcdl"><span class="id" title="lemma">mulp_gcdl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_div_gcd"><span class="id" title="lemma">coprimep_div_gcd</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.divp_eq0"><span class="id" title="lemma">divp_eq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_div_eq0"><span class="id" title="lemma">dvdp_div_eq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.Bezout_coprimepPn"><span class="id" title="lemma">Bezout_coprimepPn</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">uv</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cf4676be165a6295cd8b63fc45b45d8a"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cf4676be165a6295cd8b63fc45b45d8a"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cf4676be165a6295cd8b63fc45b45d8a"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cf4676be165a6295cd8b63fc45b45d8a"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">&</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>)<br/> - (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_pexp2r"><span class="id" title="lemma">dvdp_pexp2r</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">k</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7f2a7ef2c63af7359b22787a9daf336e"><span class="id" title="notation">></span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.root_gcd"><span class="id" title="lemma">root_gcd</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</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.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.root_biggcd"><span class="id" title="lemma">root_biggcd</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> (<span class="id" title="var">ps</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>),<br/> - <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">/</span></a>0<a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">(</span></a><span class="id" title="var">p</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#ps"><span class="id" title="variable">ps</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">p</span> ⇒ <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#ps"><span class="id" title="variable">ps</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - "gdcop Q P" is the Greatest Divisor of P which is coprime to Q - if P null, we pose that gdcop returns 1 if Q null, 0 otherwise -</div> -<div class="code"> -<span class="id" title="keyword">Fixpoint</span> <a name="Pdiv.CommonIdomain.gdcop_rec"><span class="id" title="definition">gdcop_rec</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> <span class="id" title="var">k</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">m</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#gdcop_rec"><span class="id" title="definition">gdcop_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.divp"><span class="id" title="definition">divp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>)) <span class="id" title="var">m</span><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonIdomain.gdcop"><span class="id" title="definition">gdcop</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop_rec"><span class="id" title="definition">gdcop_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="Pdiv.CommonIdomain.gdcop_spec"><span class="id" title="inductive">gdcop_spec</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <a name="Pdiv.CommonIdomain.GdcopSpec"><span class="id" title="constructor">GdcopSpec</span></a> <span class="id" title="var">r</span> <span class="id" title="keyword">of</span> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.dvdp"><span class="id" title="definition">dvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) & (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>)<br/> - & (<span class="id" title="keyword">∀</span> <span class="id" title="var">d</span>, <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.dvdp"><span class="id" title="definition">dvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.dvdp"><span class="id" title="definition">dvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>)<br/> - : <a class="idref" href="mathcomp.algebra.polydiv.html#gdcop_spec"><span class="id" title="inductive">gdcop_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gdcop0"><span class="id" title="lemma">gdcop0</span></a> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gdcop_recP"><span class="id" title="lemma">gdcop_recP</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">q</span> <span class="id" title="var">p</span> <span class="id" title="var">k</span>,<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop_spec"><span class="id" title="inductive">gdcop_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop_rec"><span class="id" title="definition">gdcop_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#k"><span class="id" title="variable">k</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gdcopP"><span class="id" title="lemma">gdcopP</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop_spec"><span class="id" title="inductive">gdcop_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_gdco"><span class="id" title="lemma">coprimep_gdco</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0)%<span class="id" title="var">B</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.size2_dvdp_gdco"><span class="id" title="lemma">size2_dvdp_gdco</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_gdco"><span class="id" title="lemma">dvdp_gdco</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.root_gdco"><span class="id" title="lemma">root_gdco</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_comp_poly"><span class="id" title="lemma">dvdp_comp_poly</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.gcdp_comp_poly"><span class="id" title="lemma">gcdp_comp_poly</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.gcdp"><span class="id" title="definition">gcdp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_comp_poly"><span class="id" title="lemma">coprimep_comp_poly</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#011952147f33cb889a964c228c4ebadd"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_addl_mul"><span class="id" title="lemma">coprimep_addl_mul</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonIdomain.irreducible_poly"><span class="id" title="definition">irreducible_poly</span></a> <span class="id" title="var">p</span> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7f2a7ef2c63af7359b22787a9daf336e"><span class="id" title="notation">></span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">q</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 1%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#17ff6c89d595412214a9dd9e39474c36"><span class="id" title="notation">Prop</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.irredp_neq0"><span class="id" title="lemma">irredp_neq0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.irreducible_poly"><span class="id" title="definition">irreducible_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Pdiv.CommonIdomain.apply_irredp"><span class="id" title="definition">apply_irredp</span></a> <span class="id" title="var">p</span> (<span class="id" title="var">irr_p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.irreducible_poly"><span class="id" title="definition">irreducible_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) := <a class="idref" href="mathcomp.algebra.polydiv.html#irr_p"><span class="id" title="variable">irr_p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a>.<br/> -<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.apply_irredp"><span class="id" title="definition">apply_irredp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.apply_irredp"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.apply_irredp"><span class="id" title="definition">irreducible_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.apply_irredp"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.apply_irredp"><span class="id" title="definition">Funclass</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.modp_XsubC"><span class="id" title="lemma">modp_XsubC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimep_XsubC"><span class="id" title="lemma">coprimep_XsubC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.coprimepX"><span class="id" title="lemma">coprimepX</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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.polydiv.html#p"><span class="id" title="variable">p</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.eqp_monic"><span class="id" title="lemma">eqp_monic</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</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/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_mul_XsubC"><span class="id" title="lemma">dvdp_mul_XsubC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">c</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.dvdp_prod_XsubC"><span class="id" title="lemma">dvdp_prod_XsubC</span></a> (<span class="id" title="var">I</span> : <span class="id" title="keyword">Type</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.polydiv.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">p</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#20f16c1d55d1e4ca9bb0e0513dd4b06a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.irredp_XsubC"><span class="id" title="lemma">irredp_XsubC</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.irreducible_poly"><span class="id" title="definition">irreducible_poly</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#dc2ed3a32abac1baa27cfc93ddc4e844"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#8b14e41ab5fcce2460b8672da1456d67"><span class="id" title="notation">P</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.CommonIdomain.irredp_XsubCP"><span class="id" title="lemma">irredp_XsubCP</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.irreducible_poly"><span class="id" title="definition">irreducible_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain.IDomainPseudoDivision"><span class="id" title="section">IDomainPseudoDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">eqpxx</span> <span class="id" title="var">divp0</span> <span class="id" title="var">divp1</span> <span class="id" title="var">mod0p</span> <span class="id" title="var">modp0</span> <span class="id" title="var">modp1</span> <span class="id" title="var">dvdp_mull</span> <span class="id" title="var">dvdp_mulr</span> <span class="id" title="var">dvdpp</span> : <span class="id" title="var">core</span>.<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">dvdp0</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain"><span class="id" title="module">CommonIdomain</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.Idomain"><span class="id" title="module">Idomain</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Include</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs"><span class="id" title="module">IdomainDefs</span></a>.<br/> -<span class="id" title="keyword">Export</span> <span class="id" title="var">IdomainDefs</span>.<br/> -<span class="id" title="keyword">Include</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.WeakIdomain"><span class="id" title="module">WeakIdomain</span></a>.<br/> -<span class="id" title="keyword">Include</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain"><span class="id" title="module">CommonIdomain</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Idomain"><span class="id" title="module">Idomain</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.IdomainMonic"><span class="id" title="module">IdomainMonic</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Ring</span> <span class="id" title="var">ComRing</span> <span class="id" title="var">UnitRing</span> <span class="id" title="var">IdomainDefs</span> <span class="id" title="var">Idomain</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.IdomainMonic.MonicDivisor"><span class="id" title="section">MonicDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.IdomainMonic.MonicDivisor.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>.<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Pdiv.IdomainMonic.MonicDivisor.monq"><span class="id" title="variable">monq</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c94c2df86ca03f22f8f8b739cd7e1e88"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</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> <span class="id" title="var">d</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.divpE"><span class="id" title="lemma">divpE</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.modpE"><span class="id" title="lemma">modpE</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.scalpE"><span class="id" title="lemma">scalpE</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.divp_eq"><span class="id" title="lemma">divp_eq</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.divpp"><span class="id" title="lemma">divpp</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.dvdp_eq"><span class="id" title="lemma">dvdp_eq</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.dvdpP"><span class="id" title="lemma">dvdpP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">qq</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.mulpK"><span class="id" title="lemma">mulpK</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainMonic.mulKp"><span class="id" title="lemma">mulKp</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor.q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic.MonicDivisor"><span class="id" title="section">MonicDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainMonic"><span class="id" title="module">IdomainMonic</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.IdomainUnit"><span class="id" title="module">IdomainUnit</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Ring</span> <span class="id" title="var">ComRing</span> <span class="id" title="var">UnitRing</span> <span class="id" title="var">IdomainDefs</span> <span class="id" title="var">Idomain</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.IdomainUnit.UnitDivisor"><span class="id" title="section">UnitDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.IdomainUnit.UnitDivisor.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>.<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Pdiv.IdomainUnit.UnitDivisor.ulcd"><span class="id" title="variable">ulcd</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</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> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_eq"><span class="id" title="lemma">divp_eq</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.edivpP"><span class="id" title="lemma">edivpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divpP"><span class="id" title="lemma">divpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.modpP"><span class="id" title="lemma">modpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.ulc_eqpP"><span class="id" title="lemma">ulc_eqpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.dvdp_eq"><span class="id" title="lemma">dvdp_eq</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.ucl_eqp_eq"><span class="id" title="lemma">ucl_eqp_eq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><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/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#69c431a9c94f6f30a655bd7ddb59037b"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.modp_scalel"><span class="id" title="lemma">modp_scalel</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_scalel"><span class="id" title="lemma">divp_scalel</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.eqp_modpl"><span class="id" title="lemma">eqp_modpl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.eqp_divl"><span class="id" title="lemma">eqp_divl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.modp_opp"><span class="id" title="lemma">modp_opp</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_opp"><span class="id" title="lemma">divp_opp</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.modp_add"><span class="id" title="lemma">modp_add</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_add"><span class="id" title="lemma">divp_add</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.mulpK"><span class="id" title="lemma">mulpK</span></a> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.mulKp"><span class="id" title="lemma">mulKp</span></a> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_addl_mul_small"><span class="id" title="lemma">divp_addl_mul_small</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.modp_addl_mul_small"><span class="id" title="lemma">modp_addl_mul_small</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_addl_mul"><span class="id" title="lemma">divp_addl_mul</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divpp"><span class="id" title="lemma">divpp</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.leq_trunc_divp"><span class="id" title="lemma">leq_trunc_divp</span></a> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.dvdpP"><span class="id" title="lemma">dvdpP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divpK"><span class="id" title="lemma">divpK</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divpKC"><span class="id" title="lemma">divpKC</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.dvdp_eq_div"><span class="id" title="lemma">dvdp_eq_div</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.dvdp_eq_mul"><span class="id" title="lemma">dvdp_eq_mul</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_mulA"><span class="id" title="lemma">divp_mulA</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_mulAC"><span class="id" title="lemma">divp_mulAC</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_mulCA"><span class="id" title="lemma">divp_mulCA</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.modp_mul"><span class="id" title="lemma">modp_mul</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.UnitDivisor"><span class="id" title="section">UnitDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.IdomainUnit.MoreUnitDivisor"><span class="id" title="section">MoreUnitDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.IdomainUnit.MoreUnitDivisor.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>.<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Pdiv.IdomainUnit.MoreUnitDivisor.ulcd"><span class="id" title="variable">ulcd</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.expp_sub"><span class="id" title="lemma">expp_sub</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_pmul2l"><span class="id" title="lemma">divp_pmul2l</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_pmul2r"><span class="id" title="lemma">divp_pmul2r</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_divl"><span class="id" title="lemma">divp_divl</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divpAC"><span class="id" title="lemma">divpAC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.modp_scaler"><span class="id" title="lemma">modp_scaler</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.IdomainUnit.divp_scaler"><span class="id" title="lemma">divp_scaler</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor.d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit.MoreUnitDivisor"><span class="id" title="section">MoreUnitDivisor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainUnit"><span class="id" title="module">IdomainUnit</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.Field"><span class="id" title="module">Field</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Ring</span> <span class="id" title="var">ComRing</span> <span class="id" title="var">UnitRing</span>.<br/> -<span class="id" title="keyword">Include</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs"><span class="id" title="module">IdomainDefs</span></a>.<br/> -<span class="id" title="keyword">Export</span> <span class="id" title="var">IdomainDefs</span>.<br/> -<span class="id" title="keyword">Include</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.CommonIdomain"><span class="id" title="module">CommonIdomain</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.Field.FieldDivision"><span class="id" title="section">FieldDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</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> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_eq"><span class="id" title="lemma">divp_eq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_modpP"><span class="id" title="lemma">divp_modpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">d</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divpP"><span class="id" title="lemma">divpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">d</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modpP"><span class="id" title="lemma">modpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">d</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqpfP"><span class="id" title="lemma">eqpfP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#69c431a9c94f6f30a655bd7ddb59037b"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.dvdp_eq"><span class="id" title="lemma">dvdp_eq</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqpf_eq"><span class="id" title="lemma">eqpf_eq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modp_scalel"><span class="id" title="lemma">modp_scalel</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.mulpK"><span class="id" title="lemma">mulpK</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.mulKp"><span class="id" title="lemma">mulKp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_scalel"><span class="id" title="lemma">divp_scalel</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modp_scaler"><span class="id" title="lemma">modp_scaler</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_scaler"><span class="id" title="lemma">divp_scaler</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_modpl"><span class="id" title="lemma">eqp_modpl</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_divl"><span class="id" title="lemma">eqp_divl</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_modpr"><span class="id" title="lemma">eqp_modpr</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_mod"><span class="id" title="lemma">eqp_mod</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="var">q1</span> <span class="id" title="var">q2</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q2"><span class="id" title="variable">q2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q2"><span class="id" title="variable">q2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_divr"><span class="id" title="lemma">eqp_divr</span></a> (<span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_div"><span class="id" title="lemma">eqp_div</span></a> <span class="id" title="var">p1</span> <span class="id" title="var">p2</span> <span class="id" title="var">q1</span> <span class="id" title="var">q2</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q2"><span class="id" title="variable">q2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p1"><span class="id" title="variable">p1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q1"><span class="id" title="variable">q1</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p2"><span class="id" title="variable">p2</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q2"><span class="id" title="variable">q2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_gdcor"><span class="id" title="lemma">eqp_gdcor</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_gdcol"><span class="id" title="lemma">eqp_gdcol</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_rgdco_gdco"><span class="id" title="lemma">eqp_rgdco_gdco</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rgdcop"><span class="id" title="definition">rgdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modp_opp"><span class="id" title="lemma">modp_opp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_opp"><span class="id" title="lemma">divp_opp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modp_add"><span class="id" title="lemma">modp_add</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modNp"><span class="id" title="lemma">modNp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_add"><span class="id" title="lemma">divp_add</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_addl_mul_small"><span class="id" title="lemma">divp_addl_mul_small</span></a> <span class="id" title="var">d</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modp_addl_mul_small"><span class="id" title="lemma">modp_addl_mul_small</span></a> <span class="id" title="var">d</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_addl_mul"><span class="id" title="lemma">divp_addl_mul</span></a> <span class="id" title="var">d</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divpp"><span class="id" title="lemma">divpp</span></a> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.leq_trunc_divp"><span class="id" title="lemma">leq_trunc_divp</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divpK"><span class="id" title="lemma">divpK</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divpKC"><span class="id" title="lemma">divpKC</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.dvdp_eq_div"><span class="id" title="lemma">dvdp_eq_div</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.dvdp_eq_mul"><span class="id" title="lemma">dvdp_eq_mul</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_mulA"><span class="id" title="lemma">divp_mulA</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_mulAC"><span class="id" title="lemma">divp_mulAC</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_mulCA"><span class="id" title="lemma">divp_mulCA</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.expp_sub"><span class="id" title="lemma">expp_sub</span></a> <span class="id" title="var">d</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#73030c22bc0b1fa771c65aa5414c65f9"><span class="id" title="notation">≥</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_pmul2l"><span class="id" title="lemma">divp_pmul2l</span></a> <span class="id" title="var">d</span> <span class="id" title="var">q</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_pmul2r"><span class="id" title="lemma">divp_pmul2r</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divp_divl"><span class="id" title="lemma">divp_divl</span></a> <span class="id" title="var">r</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divpAC"><span class="id" title="lemma">divpAC</span></a> <span class="id" title="var">d</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.edivp_def"><span class="id" title="lemma">edivp_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.divpE"><span class="id" title="lemma">divpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)^-(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modpE"><span class="id" title="lemma">modpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)^-(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2cbbcf28cb71296a00bdaede8cf3ea56"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.scalpE"><span class="id" title="lemma">scalpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span>.<br/> - -<br/> -</div> - -<div class="doc"> - Just to have it without importing the weak theory -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.dvdpE"><span class="id" title="lemma">dvdpE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdvdp"><span class="id" title="definition">rdvdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="Pdiv.Field.edivp_spec"><span class="id" title="inductive">edivp_spec</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - <a name="Pdiv.Field.EdivpSpec"><span class="id" title="constructor">EdivpSpec</span></a> <span class="id" title="var">n</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="keyword">of</span><br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> & <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">==></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">)</span></a> : <a class="idref" href="mathcomp.algebra.polydiv.html#edivp_spec"><span class="id" title="inductive">edivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.edivpP"><span class="id" title="lemma">edivpP</span></a> <span class="id" title="var">m</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.edivp_spec"><span class="id" title="inductive">edivp_spec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.edivp_eq"><span class="id" title="lemma">edivp_eq</span></a> <span class="id" title="var">d</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.algebra.polydiv.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.modp_mul"><span class="id" title="lemma">modp_mul</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.dvdpP"><span class="id" title="lemma">dvdpP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">qq</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#qq"><span class="id" title="variable">qq</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>) (<a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.Bezout_eq1_coprimepP"><span class="id" title="lemma">Bezout_eq1_coprimepP</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>,<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.dvdp_gdcor"><span class="id" title="lemma">dvdp_gdcor</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.reducible_cubic_root"><span class="id" title="lemma">reducible_cubic_root</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 4 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cf4676be165a6295cd8b63fc45b45d8a"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cf4676be165a6295cd8b63fc45b45d8a"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><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.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.cubic_irreducible"><span class="id" title="lemma">cubic_irreducible</span></a> <span class="id" title="var">p</span> :<br/> - 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#a4a996d4bc3c0cd413f31230c87b2a7a"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#a4a996d4bc3c0cd413f31230c87b2a7a"><span class="id" title="notation">≤</span></a> 4 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.irreducible_poly"><span class="id" title="definition">irreducible_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.Field.FieldDivision.FieldRingMap"><span class="id" title="section">FieldRingMap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.Field.FieldDivision.FieldRingMap.rR"><span class="id" title="variable">rR</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.Field.FieldDivision.FieldRingMap.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.FieldRingMap.rR"><span class="id" title="variable">rR</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><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">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.redivp_map"><span class="id" title="lemma">redivp_map</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.redivp"><span class="id" title="definition">redivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rscalp"><span class="id" title="definition">rscalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rdivp"><span class="id" title="definition">rdivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Ring.rmodp"><span class="id" title="definition">rmodp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#7ccad1b5013efc6ffe407f7b77347c15"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.FieldRingMap"><span class="id" title="section">FieldRingMap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.Field.FieldDivision.FieldMap"><span class="id" title="section">FieldMap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.Field.FieldDivision.FieldMap.rR"><span class="id" title="variable">rR</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Exports.idomainType"><span class="id" title="abbreviation">idomainType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.Field.FieldDivision.FieldMap.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.FieldMap.rR"><span class="id" title="variable">rR</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><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">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.edivp_map"><span class="id" title="lemma">edivp_map</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.edivp"><span class="id" title="definition">edivp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.scalp_map"><span class="id" title="lemma">scalp_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.IdomainDefs.scalp"><span class="id" title="definition">scalp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.map_divp"><span class="id" title="lemma">map_divp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#72a0c853cc9a32bb5fdc8a920a96e7c6"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.map_modp"><span class="id" title="lemma">map_modp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#d8832071e7663562cc14f17c6edf99dc"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.egcdp_map"><span class="id" title="lemma">egcdp_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/> - <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.egcdp"><span class="id" title="definition">egcdp</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.FieldMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.FieldMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>)<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.FieldMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.egcdp"><span class="id" title="definition">egcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">).1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.FieldMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.egcdp"><span class="id" title="definition">egcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">).2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.dvdp_map"><span class="id" title="lemma">dvdp_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#64fc6df2b95b79b2107dd5d7f2014b97"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.eqp_map"><span class="id" title="lemma">eqp_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#952776a2e27e0a80427a97e8cd81c9aa"><span class="id" title="notation">%=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.gcdp_map"><span class="id" title="lemma">gcdp_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gcdp"><span class="id" title="definition">gcdp</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.coprimep_map"><span class="id" title="lemma">coprimep_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.gdcop_rec_map"><span class="id" title="lemma">gdcop_rec_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop_rec"><span class="id" title="definition">gdcop_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop_rec"><span class="id" title="definition">gdcop_rec</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.Field.gdcop_map"><span class="id" title="lemma">gdcop_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.gdcop"><span class="id" title="definition">gdcop</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#ffd296f893410e849a35ec5356a13636"><span class="id" title="notation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision.FieldMap"><span class="id" title="section">FieldMap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.FieldDivision"><span class="id" title="section">FieldDivision</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field"><span class="id" title="module">Field</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Module</span> <a name="Pdiv.ClosedField"><span class="id" title="module">ClosedField</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">Field</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pdiv.ClosedField.closed"><span class="id" title="section">closed</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Pdiv.ClosedField.closed.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ClosedField.Exports.closedFieldType"><span class="id" title="abbreviation">closedFieldType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.ClosedField.root_coprimep"><span class="id" title="lemma">root_coprimep</span></a> (<span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ClosedField.closed.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>):<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><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.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Pdiv.ClosedField.coprimepP"><span class="id" title="lemma">coprimepP</span></a> (<span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ClosedField.closed.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#c2ef4fdf7ae62c36654f85f0d2a6c874"><span class="id" title="notation">}</span></a>):<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</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.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polydiv.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#e4361ce58e4de0a4b9786d0011b61316"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0) (<a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.Field.coprimep"><span class="id" title="definition">coprimep</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polydiv.html#q"><span class="id" title="variable">q</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ClosedField.closed"><span class="id" title="section">closed</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv.ClosedField"><span class="id" title="module">ClosedField</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polydiv.html#Pdiv"><span class="id" title="module">Pdiv</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Export</span> <span class="id" title="var">Pdiv.Field</span>.<br/> -</div> -</div> - -<div id="footer"> -<hr/><a href="index.html">Index</a><hr/>This page has been 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