docs/htmldoc/mathcomp.algebra.polyXY.html


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<h1 class="libtitle">Library mathcomp.algebra.polyXY</h1>

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

<br/>
</div>

<div class="doc">
 This file provides additional primitives and theory for bivariate          
 polynomials (polynomials of two variables), represented as polynomials     
 with (univariate) polynomial coefficients :                                
            'Y == the (generic) second variable (:= 'X%:P).                 
          p^:P == the bivariate polynomial p['X], for p univariate.         
               := map_poly polyC p (this notation is defined in poly.v).    
      u. [x, y] == the bivariate polynomial u evaluated at 'X = x, 'Y = y.   
               := u. [x%:P]. [y].                                             
       sizeY u == the size of u in 'Y (1 + the 'Y-degree of u, if u != 0).  
               := \max</i>(i &lt; size u) size u`<i>i.                              
      swapXY u == the bivariate polynomial u['Y, 'X], for u bivariate.      
    poly_XaY p == the bivariate polynomial p['X + 'Y], for p univariate.    
               := p^:P \Po ('X + 'Y).                                       
    poly_XmY p == the bivariate polynomial p['X * 'Y], for p univariate.    
               := P^:P \Po ('X * 'Y).                                       
 sub_annihilant p q == for univariate p, q != 0, a nonzero polynomial whose 
                  roots include all the differences of roots of p and q, in 
                  all field extensions (:= resultant (poly_XaY p) q^:P).    
 div_annihilant p q == for polynomials p != 0, q with q. [0] != 0, a nonzero 
                  polynomial whose roots include all the quotients of roots 
                  of p by roots of q, in all field extensions               
                  (:= resultant (poly_XmY p) q^:P).                         
 The latter two "annhilants" provide uniform witnesses for an alternative   
 proof of the closure of the algebraicOver predicate (see mxpoly.v). The    
 fact that the annhilant does not depend on the particular choice of roots  
 of p and q is crucial for the proof of the Primitive Element Theorem (file 
 separable.v).                                                               
</div>
<div class="code">

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

<br/>
<span class="id" title="keyword">Local Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">GRing.Theory</span>.<br/>

<br/>

<br/>
<span class="id" title="keyword">Notation</span> <a name="123da97e9d81425c68e579737576ac03"><span class="id" title="notation">&quot;</span></a>'Y" := <a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a> : <span class="id" title="var">ring_scope</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">&quot;</span></a>p ^:P" := (<span class="id" title="var">p</span> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyC"><span class="id" title="definition">polyC</span></a>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "p ^:P") : <span class="id" title="var">ring_scope</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">&quot;</span></a>p .[ x , y ]" := (<span class="id" title="var">p</span><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><span class="id" title="var">x</span><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">].[</span></a><span class="id" title="var">y</span><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>, <span class="id" title="var">format</span> "p .[ x ,  y ]") : <span class="id" title="var">ring_scope</span>.<br/>

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

<br/>
<span class="id" title="keyword">Variable</span> <a name="PolyXY_Ring.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">u</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Ring.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a>) (<span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Ring.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Ring.R"><span class="id" title="variable">R</span></a>).<br/>

<br/>
<span class="id" title="keyword">Fact</span> <a name="swapXY_key"><span class="id" title="lemma">swapXY_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>.  <br/>
<span class="id" title="keyword">Definition</span> <a name="swapXY_def"><span class="id" title="definition">swapXY_def</span></a> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Ring.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><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.poly.html#polyC"><span class="id" title="definition">polyC</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">Y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="swapXY"><span class="id" title="definition">swapXY</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY_key"><span class="id" title="lemma">swapXY_key</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY_def"><span class="id" title="definition">swapXY_def</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">swapXY_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">fun</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#58f94351327943cd874eb55da8e0ca14"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sizeY"><span class="id" title="definition">sizeY</span></a> <span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="mathcomp.ssreflect.bigop.html#2e8f487d341e1ab4c6af2ac15a318eda"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#2e8f487d341e1ab4c6af2ac15a318eda"><span class="id" title="notation">max_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#2e8f487d341e1ab4c6af2ac15a318eda"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#2e8f487d341e1ab4c6af2ac15a318eda"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#2e8f487d341e1ab4c6af2ac15a318eda"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#2e8f487d341e1ab4c6af2ac15a318eda"><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.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#2e8f487d341e1ab4c6af2ac15a318eda"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="poly_XaY"><span class="id" title="definition">poly_XaY</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Ring.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a> := <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">^:</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">Y</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="poly_XmY"><span class="id" title="definition">poly_XmY</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Ring.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a> := <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">^:</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">Y</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="sub_annihilant"><span class="id" title="definition">sub_annihilant</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="mathcomp.algebra.mxpoly.html#resultant"><span class="id" title="definition">resultant</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#poly_XaY"><span class="id" title="definition">poly_XaY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">^:</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">P</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="div_annihilant"><span class="id" title="definition">div_annihilant</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="mathcomp.algebra.mxpoly.html#resultant"><span class="id" title="definition">resultant</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#poly_XmY"><span class="id" title="definition">poly_XmY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">^:</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">P</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_polyC"><span class="id" title="lemma">swapXY_polyC</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">^:</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">P</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_X"><span class="id" title="lemma">swapXY_X</span></a> : <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">Y</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_Y"><span class="id" title="lemma">swapXY_Y</span></a> : <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#123da97e9d81425c68e579737576ac03"><span class="id" title="notation">Y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_is_additive"><span class="id" title="lemma">swapXY_is_additive</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.additive"><span class="id" title="abbreviation">additive</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a>.<br/>
 <span class="id" title="keyword">Canonical</span> <span class="id" title="var">swapXY_addf</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY_is_additive"><span class="id" title="lemma">swapXY_is_additive</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="coef_swapXY"><span class="id" title="lemma">coef_swapXY</span></a> <span class="id" title="var">u</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">)`</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">_j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">_j</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">_i</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXYK"><span class="id" title="lemma">swapXYK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#involutive"><span class="id" title="definition">involutive</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_map_polyC"><span class="id" title="lemma">swapXY_map_polyC</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">^:</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_eq0"><span class="id" title="lemma">swapXY_eq0</span></a> <span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_is_multiplicative"><span class="id" title="lemma">swapXY_is_multiplicative</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.multiplicative"><span class="id" title="abbreviation">multiplicative</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">swapXY_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.AddRMorphism"><span class="id" title="abbreviation">AddRMorphism</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY_is_multiplicative"><span class="id" title="lemma">swapXY_is_multiplicative</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_is_scalable"><span class="id" title="lemma">swapXY_is_scalable</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.scalable_for"><span class="id" title="abbreviation">scalable_for</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.poly.html#polyC"><span class="id" title="definition">polyC</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c42c5cb909c30537f9f6acfcf01cf7e1"><span class="id" title="notation">\;</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a>) <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a>.<br/>
 <span class="id" title="keyword">Canonical</span> <span class="id" title="var">swapXY_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.AddLinear"><span class="id" title="abbreviation">AddLinear</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY_is_scalable"><span class="id" title="lemma">swapXY_is_scalable</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">swapXY_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#8900f6ae77a86586561e15965d5870c7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8900f6ae77a86586561e15965d5870c7"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8900f6ae77a86586561e15965d5870c7"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#8900f6ae77a86586561e15965d5870c7"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_comp_poly"><span class="id" title="lemma">swapXY_comp_poly</span></a> <span class="id" title="var">p</span> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">^:</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">^:</span></a><a class="idref" href="mathcomp.algebra.poly.html#f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="max_size_coefXY"><span class="id" title="lemma">max_size_coefXY</span></a> <span class="id" title="var">u</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#sizeY"><span class="id" title="definition">sizeY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="max_size_lead_coefXY"><span class="id" title="lemma">max_size_lead_coefXY</span></a> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</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.polyXY.html#u"><span class="id" title="variable">u</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#sizeY"><span class="id" title="definition">sizeY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="max_size_evalX"><span class="id" title="lemma">max_size_evalX</span></a> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#sizeY"><span class="id" title="definition">sizeY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><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.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="max_size_evalC"><span class="id" title="lemma">max_size_evalC</span></a> <span class="id" title="var">u</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#sizeY"><span class="id" title="definition">sizeY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sizeYE"><span class="id" title="lemma">sizeYE</span></a> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#sizeY"><span class="id" title="definition">sizeY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sizeY_eq0"><span class="id" title="lemma">sizeY_eq0</span></a> <span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#sizeY"><span class="id" title="definition">sizeY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="sizeY_mulX"><span class="id" title="lemma">sizeY_mulX</span></a> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#sizeY"><span class="id" title="definition">sizeY</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#sizeY"><span class="id" title="definition">sizeY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_poly_XaY"><span class="id" title="lemma">swapXY_poly_XaY</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#poly_XaY"><span class="id" title="definition">poly_XaY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#poly_XaY"><span class="id" title="definition">poly_XaY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_poly_XmY"><span class="id" title="lemma">swapXY_poly_XmY</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#poly_XmY"><span class="id" title="definition">poly_XmY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#poly_XmY"><span class="id" title="definition">poly_XmY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="poly_XaY0"><span class="id" title="lemma">poly_XaY0</span></a> : <a class="idref" href="mathcomp.algebra.polyXY.html#poly_XaY"><span class="id" title="definition">poly_XaY</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="poly_XmY0"><span class="id" title="lemma">poly_XmY0</span></a> : <a class="idref" href="mathcomp.algebra.polyXY.html#poly_XmY"><span class="id" title="definition">poly_XmY</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Ring"><span class="id" title="section">PolyXY_Ring</span></a>.<br/>

<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="swapXY_map"><span class="id" title="lemma">swapXY_map</span></a> (<span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#6566b94c06c342b0768c3d2d73badf6e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6566b94c06c342b0768c3d2d73badf6e"><span class="id" title="notation">additive</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6566b94c06c342b0768c3d2d73badf6e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">u</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><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.polyXY.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><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.polyXY.html#f"><span class="id" title="variable">f</span></a>.<br/>

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

<br/>
<span class="id" title="keyword">Variable</span> <a name="PolyXY_ComRing.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/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_ComRing.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a>) (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_ComRing.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_ComRing.R"><span class="id" title="variable">R</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="horner_swapXY"><span class="id" title="lemma">horner_swapXY</span></a> <span class="id" title="var">u</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#eval"><span class="id" title="abbreviation">eval</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="horner_polyC"><span class="id" title="lemma">horner_polyC</span></a> <span class="id" title="var">u</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#eval"><span class="id" title="abbreviation">eval</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="horner2_swapXY"><span class="id" title="lemma">horner2_swapXY</span></a> <span class="id" title="var">u</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#swapXY"><span class="id" title="definition">swapXY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">]</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="horner_poly_XaY"><span class="id" title="lemma">horner_poly_XaY</span></a> <span class="id" title="var">p</span> <span class="id" title="var">v</span> : <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#poly_XaY"><span class="id" title="definition">poly_XaY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="horner_poly_XmY"><span class="id" title="lemma">horner_poly_XmY</span></a> <span class="id" title="var">p</span> <span class="id" title="var">v</span> : <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#poly_XmY"><span class="id" title="definition">poly_XmY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_ComRing"><span class="id" title="section">PolyXY_ComRing</span></a>.<br/>

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

<br/>
<span class="id" title="keyword">Variable</span> <a name="PolyXY_Idomain.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">Types</span> (<span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Idomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Idomain.R"><span class="id" title="variable">R</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="size_poly_XaY"><span class="id" title="lemma">size_poly_XaY</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.polyXY.html#poly_XaY"><span class="id" title="definition">poly_XaY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="poly_XaY_eq0"><span class="id" title="lemma">poly_XaY_eq0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#poly_XaY"><span class="id" title="definition">poly_XaY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="size_poly_XmY"><span class="id" title="lemma">size_poly_XmY</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.polyXY.html#poly_XmY"><span class="id" title="definition">poly_XmY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="poly_XmY_eq0"><span class="id" title="lemma">poly_XmY_eq0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#poly_XmY"><span class="id" title="definition">poly_XmY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="lead_coef_poly_XaY"><span class="id" title="lemma">lead_coef_poly_XaY</span></a> <span class="id" title="var">p</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.polyXY.html#poly_XaY"><span class="id" title="definition">poly_XaY</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><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.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sub_annihilant_in_ideal"><span class="id" title="lemma">sub_annihilant_in_ideal</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">uv</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Idomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Idomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a><br/>
&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><br/>
&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">&amp;</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#sub_annihilant"><span class="id" title="definition">sub_annihilant</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sub_annihilantP"><span class="id" title="lemma">sub_annihilantP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#sub_annihilant"><span class="id" title="definition">sub_annihilant</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sub_annihilant_neq0"><span class="id" title="lemma">sub_annihilant_neq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#sub_annihilant"><span class="id" title="definition">sub_annihilant</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="div_annihilant_in_ideal"><span class="id" title="lemma">div_annihilant_in_ideal</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">uv</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Idomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Idomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}}</span></a><br/>
&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><br/>
&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">&amp;</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#div_annihilant"><span class="id" title="definition">div_annihilant</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#uv"><span class="id" title="variable">uv</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#7fe0b4b004984abb7cb175c362f61dea"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="div_annihilant_neq0"><span class="id" title="lemma">div_annihilant_neq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a>0<a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#div_annihilant"><span class="id" title="definition">div_annihilant</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Idomain"><span class="id" title="section">PolyXY_Idomain</span></a>.<br/>

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

<br/>
<span class="id" title="keyword">Variables</span> (<a name="PolyXY_Field.F"><span class="id" title="variable">F</span></a> <a name="PolyXY_Field.E"><span class="id" title="variable">E</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) (<a name="PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a>).<br/>

<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="div_annihilantP"><span class="id" title="lemma">div_annihilantP</span></a> (<span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.E"><span class="id" title="variable">E</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#div_annihilant"><span class="id" title="definition">div_annihilant</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="map_sub_annihilantP"><span class="id" title="lemma">map_sub_annihilantP</span></a> (<span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.E"><span class="id" title="variable">E</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#sub_annihilant"><span class="id" title="definition">sub_annihilant</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="map_div_annihilantP"><span class="id" title="lemma">map_div_annihilantP</span></a> (<span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.E"><span class="id" title="variable">E</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#div_annihilant"><span class="id" title="definition">div_annihilant</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4fa85b0aa898c2a7e18c3b076438c2e7"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="root_annihilant"><span class="id" title="lemma">root_annihilant</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> (<span class="id" title="var">pEx</span> := <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#pFtoE"><span class="id" title="abbreviation">pFtoE</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.polyXY.html#pEx"><span class="id" title="variable">pEx</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxpoly.html#algebraicOver"><span class="id" title="definition">algebraicOver</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">&amp;</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#pEx"><span class="id" title="variable">pEx</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.polyXY.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a>) <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="algebraic_root_polyXY"><span class="id" title="lemma">algebraic_root_polyXY</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">let</span> <span class="id" title="var">pEx</span> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#b033a3d34e421a2439563f5ffdab0b9b"><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.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#pEx"><span class="id" title="variable">pEx</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</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.polyXY.html#pEx"><span class="id" title="variable">pEx</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.mxpoly.html#algebraicOver"><span class="id" title="definition">algebraicOver</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxpoly.html#algebraicOver"><span class="id" title="definition">algebraicOver</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field.FtoE"><span class="id" title="variable">FtoE</span></a> <a class="idref" href="mathcomp.algebra.polyXY.html#y"><span class="id" title="variable">y</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.polyXY.html#PolyXY_Field"><span class="id" title="section">PolyXY_Field</span></a>.<br/>
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