generate the documentation for 1.7

author: Enrico Tassi 2018-04-20 10:54:22 +0200
committer: Enrico Tassi 2018-04-20 10:54:22 +0200
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+<title>mathcomp.algebra.poly</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library mathcomp.algebra.poly</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 a library for univariate polynomials over ring          
+ structures; it also provides an extended theory for polynomials whose      
+ coefficients range over commutative rings and integral domains.            
+
+<div class="paragraph"> </div>
+
+           {poly R} == the type of polynomials with coefficients of type R, 
+                       represented as lists with a non zero last element    
+                       (big endian representation); the coeficient type R   
+                       must have a canonical ringType structure cR. In fact 
+                       {poly R} denotes the concrete type polynomial cR; R  
+                       is just a phantom argument that lets type inference  
+                       reconstruct the (hidden) ringType structure cR.      
+          p : seq R == the big-endian sequence of coefficients of p, via    
+                       the coercion polyseq : polynomial &gt;-&gt; seq.           
+             Poly s == the polynomial with coefficient sequence s (ignoring 
+                       trailing zeroes).                                    
+ \poly(i &lt; n) E(i) == the polynomial of degree at most n - 1 whose         
+                       coefficients are given by the general term E(i)      
+  0, 1, - p, p + q, == the usual ring operations: {poly R} has a canonical  
+ p * q, p ^+ n, ...    ringType structure, which is commutative / integral  
+                       when R is commutative / integral, respectively.      
+      polyC c, c%:P == the constant polynomial c                            
+                 'X == the (unique) variable                                
+               'X^n == a power of 'X; 'X^0 is 1, 'X^1 is convertible to 'X  
+               p`<i>i == the coefficient of 'X^i in p; this is in fact just   
+                       the ring_scope notation generic seq-indexing using   
+                       nth 0%R, combined with the polyseq coercion.         
+            coefp i == the linear function p |-&gt; p`<i>i (self-exapanding).    
+             size p == 1 + the degree of p, or 0 if p = 0 (this is the      
+                       generic seq function combined with polyseq).         
+        lead_coef p == the coefficient of the highest monomial in p, or 0   
+                       if p = 0 (hence lead_coef p = 0 iff p = 0)           
+        p \is monic &lt;=&gt; lead_coef p == 1 (0 is not monic).                  
+ p \is a polyOver S &lt;=&gt; the coefficients of p satisfy S; S should have a    
+                        key that should be (at least) an addrPred.          
+             p. [x]  == the evaluation of a polynomial p at a point x using  
+                       the Horner scheme                                    
+<a name="lab1"></a><h3 class="section">The multi-rule hornerE (resp., hornerE_comm) unwinds</h3>
+
+                       horner evaluation of a polynomial expression (resp., 
+                       in a non commutative ring, with side conditions).    
+             p^`  == formal derivative of p                               
+             p^`(n) == formal n-derivative of p                             
+            p^`N(n) == formal n-derivative of p divided by n!               
+            p \Po q == polynomial composition; because this is naturally a  
+                       a linear morphism in the first argument, this        
+                       notation is transposed (q comes before p for redex   
+                       selection, etc).                                     
+                      := \sum(i &lt; size p) p`<i>i *: q ^+ i                    
+      comm_poly p x == x and p. [x] commute; this is a sufficient condition  
+                       for evaluating (q * p). [x] as q. [x] * p. [x] when R   
+                       is not commutative.                                  
+      comm_coef p x == x commutes with all the coefficients of p (clearly,  
+                       this implies comm_poly p x).                         
+           root p x == x is a root of p, i.e., p. [x] = 0                    
+    n.-unity_root x == x is an nth root of unity, i.e., a root of 'X^n - 1  
+ n.-primitive_root x == x is a primitive nth root of unity, i.e., n is the  
+                       least positive integer m &gt; 0 such that x ^+ m = 1.   
+<a name="lab2"></a><h3 class="section">The submodule poly.UnityRootTheory can be used to</h3>
+
+                       import selectively the part of the theory of roots   
+                       of unity that doesn't mention polynomials explicitly 
+       map_poly f p == the image of the polynomial by the function f (which 
+     (locally, p^f)    is usually a ring morphism).                         
+               p^:P == p lifted to {poly {poly R}} (:= map_poly polyC p).   
+   commr_rmorph f u == u commutes with the image of f (i.e., with all f x). 
+   horner_morph cfu == given cfu : commr_rmorph f u, the function mapping p 
+                       to the value of map_poly f p at u; this is a ring    
+                       morphism from {poly R} to the codomain of f when f   
+                       is a ring morphism.                                  
+      horner_eval u == the function mapping p to p. [u]; this function can   
+                       only be used for u in a commutative ring, so it is   
+                       always a linear ring morphism from {poly R} to R.    
+     diff_roots x y == x and y are distinct roots; if R is a field, this    
+                       just means x != y, but this concept is generalized   
+                       to the case where R is only a ring with units (i.e., 
+                       a unitRingType); in which case it means that x and y 
+                       commute, and that the difference x - y is a unit     
+                       (i.e., has a multiplicative inverse) in R.           
+                       to just x != y).                                     
+       uniq_roots s == s is a sequence or pairwise distinct roots, in the   
+                       sense of diff_roots p above.                         
+<a name="lab3"></a><h3 class="section">We only show that these operations and properties are transferred by</h3>
+
+       morphisms whose domain is a field (thus ensuring injectivity).       
+ We prove the factor_theorem, and the max_poly_roots inequality relating    
+ the number of distinct roots of a polynomial and its size.                 
+   The some polynomial lemmas use following suffix interpretation :         
+   C - constant polynomial (as in polyseqC : a%:P = nseq (a != 0) a).       
+   X - the polynomial variable 'X (as in coefX : 'X`<i>i = (i == 1%N)).       
+   Xn - power of 'X (as in monicXn : monic 'X^n).                            
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <span class="id" title="var">GRing.Theory</span>.<br/>
+<span class="id" title="keyword">Local Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;{ 'poly' T }" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "{ 'poly'  T }").<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;c %:P" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "c %:P").<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;p ^:P" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "p ^:P").<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;'X" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;''X^' n" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 3, <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''X^' n").<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;\poly_ ( i &lt; n ) E"<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 36, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 36, <span class="id" title="var">i</span>, <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50,<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="var">format</span> "\poly_ ( i  &lt;  n )  E").<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;p \Po q" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50).<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;p ^`N ( n )" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "p ^`N ( n )").<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;n .-unity_root" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "n .-unity_root").<br/>
+<span class="id" title="keyword">Reserved Notation</span> &quot;n .-primitive_root"<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "n .-primitive_root").<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="Polynomial"><span class="id" title="section">Polynomial</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="Polynomial.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Defines a polynomial as a sequence with &lt;&gt; 0 last element  
+</div>
+<div class="code">
+<span class="id" title="keyword">Record</span> <a name="polynomial"><span class="id" title="record">polynomial</span></a> := <a name="Polynomial"><span class="id" title="constructor">Polynomial</span></a> {<a name="polyseq"><span class="id" title="projection">polyseq</span></a> :&gt; <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#Polynomial.R"><span class="id" title="variable">R</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> 1 <a class="idref" href="mathcomp.algebra.poly.html#polyseq"><span class="id" title="method">polyseq</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">Canonical</span> <span class="id" title="var">polynomial_subType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#341c160c3e7b20d967b85d1852a7f89f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#341c160c3e7b20d967b85d1852a7f89f"><span class="id" title="notation">subType</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#341c160c3e7b20d967b85d1852a7f89f"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyseq"><span class="id" title="projection">polyseq</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#341c160c3e7b20d967b85d1852a7f89f"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="polynomial_eqMixin"><span class="id" title="definition">polynomial_eqMixin</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">eqMixin</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">&lt;:]</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_eqType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial_eqMixin"><span class="id" title="definition">polynomial_eqMixin</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="polynomial_choiceMixin"><span class="id" title="definition">polynomial_choiceMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">&lt;:]</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_choiceType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial_choiceMixin"><span class="id" title="definition">polynomial_choiceMixin</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="poly_inj"><span class="id" title="lemma">poly_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyseq"><span class="id" title="projection">polyseq</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="poly_of"><span class="id" title="definition">poly_of</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#phant"><span class="id" title="inductive">phant</span></a> <a class="idref" href="mathcomp.algebra.poly.html#Polynomial.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a>.<br/>
+<span class="id" title="keyword">Identity</span> <span class="id" title="keyword">Coercion</span> <span class="id" title="var">type_poly_of</span> : <span class="id" title="var">poly_of</span> &gt;-&gt; <span class="id" title="var">polynomial</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="coefp_head"><span class="id" title="definition">coefp_head</span></a> <span class="id" title="var">h</span> <span class="id" title="var">i</span> (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#poly_of"><span class="id" title="definition">poly_of</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <a class="idref" href="mathcomp.algebra.poly.html#Polynomial.R"><span class="id" title="variable">R</span></a>)) := <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#h"><span class="id" title="variable">h</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">End</span> <a class="idref" href="mathcomp.algebra.poly.html#Polynomial"><span class="id" title="section">Polynomial</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ We need to break off the section here to let the argument scope 
+ directives take effect.                                          
+</div>
+<div class="code">
+<span class="id" title="keyword">Notation</span> <a name="699040ddc0986f520cece215f531d947"><span class="id" title="notation">&quot;</span></a>{ 'poly' T }" := (<a class="idref" href="mathcomp.algebra.poly.html#poly_of"><span class="id" title="definition">poly_of</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <span class="id" title="var">T</span>)).<br/>
+<span class="id" title="keyword">Notation</span> <a name="coefp"><span class="id" title="abbreviation">coefp</span></a> <span class="id" title="var">i</span> := (<a class="idref" href="mathcomp.algebra.poly.html#coefp_head"><span class="id" title="definition">coefp_head</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> <span class="id" title="var">i</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PolynomialTheory"><span class="id" title="section">PolynomialTheory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="PolynomialTheory.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">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">d</span> : <a class="idref" href="mathcomp.algebra.poly.html#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#PolynomialTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_subType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#6645671e2203a23d135a621a3cf0157c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#6645671e2203a23d135a621a3cf0157c"><span class="id" title="notation">subType</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#6645671e2203a23d135a621a3cf0157c"><span class="id" title="notation">of</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#PolynomialTheory.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.ssreflect.eqtype.html#6645671e2203a23d135a621a3cf0157c"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_eqType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#cb062fd562aed512787df99359c6e3f2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#cb062fd562aed512787df99359c6e3f2"><span class="id" title="notation">eqType</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#cb062fd562aed512787df99359c6e3f2"><span class="id" title="notation">of</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#PolynomialTheory.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.ssreflect.eqtype.html#cb062fd562aed512787df99359c6e3f2"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_choiceType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">choiceType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">of</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#PolynomialTheory.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.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="lead_coef"><span class="id" title="definition">lead_coef</span></a> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_</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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a>.<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefE"><span class="id" title="lemma">lead_coefE</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.poly.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#p"><span class="id" title="variable">p</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">_</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.poly.html#p"><span class="id" title="variable">p</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">Definition</span> <a name="poly_nil"><span class="id" title="definition">poly_nil</span></a> := @<a class="idref" href="mathcomp.algebra.poly.html#Polynomial"><span class="id" title="constructor">Polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Theory.oner_neq0"><span class="id" title="definition">oner_neq0</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>).<br/>
+<span class="id" title="keyword">Definition</span> <a name="polyC"><span class="id" title="definition">polyC</span></a> <span class="id" title="var">c</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#PolynomialTheory.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.ssreflect.eqtype.html#insubd"><span class="id" title="definition">insubd</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_nil"><span class="id" title="definition">poly_nil</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Remember the boolean (c != 0) is coerced to 1 if true and 0 if false  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="polyseqC"><span class="id" title="lemma">polyseqC</span></a> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0) <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_polyC"><span class="id" title="lemma">size_polyC</span></a> <span class="id" title="var">c</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#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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="http://coq.inria.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#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefC"><span class="id" title="lemma">coefC</span></a> <span class="id" title="var">c</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyCK"><span class="id" title="lemma">polyCK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyC"><span class="id" title="definition">polyC</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#coefp"><span class="id" title="abbreviation">coefp</span></a> 0).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC_inj"><span class="id" title="lemma">polyC_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyC"><span class="id" title="definition">polyC</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefC"><span class="id" title="lemma">lead_coefC</span></a> <span class="id" title="var">c</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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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.poly.html#c"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ Extensional interpretation (poly &lt;=&gt; nat -&gt; R)  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="polyP"><span class="id" title="lemma">polyP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 <a class="idref" href="mathcomp.algebra.poly.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#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">↔</span></a> <a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size1_polyC"><span class="id" title="lemma">size1_polyC</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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.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#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_0</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Builds a polynomial by extension.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="cons_poly"><span class="id" title="definition">cons_poly</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.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;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="mathcomp.algebra.poly.html#Polynomial"><span class="id" title="constructor">Polynomial</span></a> ((<span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span>) <span class="id" title="keyword">as</span> <span class="id" title="var">s</span>) <span class="id" title="var">ns</span> <span class="id" title="keyword">then</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="mathcomp.algebra.poly.html#Polynomial"><span class="id" title="constructor">Polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">s</span>) <span class="id" title="var">ns</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseq_cons"><span class="id" title="lemma">polyseq_cons</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#cons_poly"><span class="id" title="definition">cons_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.poly.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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nilp"><span class="id" title="definition">nilp</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_cons_poly"><span class="id" title="lemma">size_cons_poly</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<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#cons_poly"><span class="id" title="definition">cons_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.poly.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nilp"><span class="id" title="definition">nilp</span></a> <a class="idref" href="mathcomp.algebra.poly.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_cons"><span class="id" title="lemma">coef_cons</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#cons_poly"><span class="id" title="definition">cons_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Build a polynomial directly from a list of coefficients.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="Poly"><span class="id" title="definition">Poly</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.algebra.poly.html#cons_poly"><span class="id" title="definition">cons_poly</span></a> 0<a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="PolyK"><span class="id" title="lemma">PolyK</span></a> <span class="id" title="var">c</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</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#Poly"><span class="id" title="definition">Poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseqK"><span class="id" title="lemma">polyseqK</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#Poly"><span class="id" title="definition">Poly</span></a> <a class="idref" href="mathcomp.algebra.poly.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#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_Poly"><span class="id" title="lemma">size_Poly</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#Poly"><span class="id" title="definition">Poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><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.poly.html#s"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_Poly"><span class="id" title="lemma">coef_Poly</span></a> <span class="id" title="var">s</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#Poly"><span class="id" title="definition">Poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</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="http://coq.inria.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#s"><span class="id" title="variable">s</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/>
+</div>
+
+<div class="doc">
+ Build a polynomial from an infinite sequence of coefficients and a bound.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="poly_expanded_def"><span class="id" title="definition">poly_expanded_def</span></a> <span class="id" title="var">n</span> <span class="id" title="var">E</span> := <a class="idref" href="mathcomp.algebra.poly.html#Poly"><span class="id" title="definition">Poly</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+<span class="id" title="keyword">Fact</span> <a name="poly_key"><span class="id" title="lemma">poly_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="poly"><span class="id" title="definition">poly</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.poly.html#poly_key"><span class="id" title="lemma">poly_key</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_expanded_def"><span class="id" title="definition">poly_expanded_def</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_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.poly.html#poly"><span class="id" title="definition">poly</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">Lemma</span> <a name="polyseq_poly"><span class="id" title="lemma">polyseq_poly</span></a> <span class="id" title="var">n</span> <span class="id" title="var">E</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</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#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3f6d75f5a9cb81b61e8423a7aac22056"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3f6d75f5a9cb81b61e8423a7aac22056"><span class="id" title="notation">eta</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3f6d75f5a9cb81b61e8423a7aac22056"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_poly"><span class="id" title="lemma">size_poly</span></a> <span class="id" title="var">n</span> <span class="id" title="var">E</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#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">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.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_poly_eq"><span class="id" title="lemma">size_poly_eq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_poly"><span class="id" title="lemma">coef_poly</span></a> <span class="id" title="var">n</span> <span class="id" title="var">E</span> <span class="id" title="var">k</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">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">_k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> 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"><span class="id" title="lemma">lead_coef_poly</span></a> <span class="id" title="var">n</span> <span class="id" title="var">E</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</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#lead_coef"><span class="id" title="definition">lead_coef</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</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="coefK"><span class="id" title="lemma">coefK</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="http://coq.inria.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#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Zmodule structure for polynomial  
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="add_poly_def"><span class="id" title="definition">add_poly_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> := <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#maxn"><span class="id" title="definition">maxn</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.poly.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a>)<a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Fact</span> <a name="add_poly_key"><span class="id" title="lemma">add_poly_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="add_poly"><span class="id" title="definition">add_poly</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.poly.html#add_poly_key"><span class="id" title="lemma">add_poly_key</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_poly_def"><span class="id" title="definition">add_poly_def</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">add_poly_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.poly.html#add_poly"><span class="id" title="definition">add_poly</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="opp_poly_def"><span class="id" title="definition">opp_poly_def</span></a> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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/>
+<span class="id" title="keyword">Fact</span> <a name="opp_poly_key"><span class="id" title="lemma">opp_poly_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="opp_poly"><span class="id" title="definition">opp_poly</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.poly.html#opp_poly_key"><span class="id" title="lemma">opp_poly_key</span></a> <a class="idref" href="mathcomp.algebra.poly.html#opp_poly_def"><span class="id" title="definition">opp_poly_def</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">opp_poly_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.poly.html#opp_poly"><span class="id" title="definition">opp_poly</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">Fact</span> <a name="coef_add_poly"><span class="id" title="lemma">coef_add_poly</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#add_poly"><span class="id" title="definition">add_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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="http://coq.inria.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#p"><span class="id" title="variable">p</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#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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">Fact</span> <a name="coef_opp_poly"><span class="id" title="lemma">coef_opp_poly</span></a> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#opp_poly"><span class="id" title="definition">opp_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">Fact</span> <a name="add_polyA"><span class="id" title="lemma">add_polyA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_poly"><span class="id" title="definition">add_poly</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="add_polyC"><span class="id" title="lemma">add_polyC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_poly"><span class="id" title="definition">add_poly</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="add_poly0"><span class="id" title="lemma">add_poly0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 0<a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_poly"><span class="id" title="definition">add_poly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="add_polyN"><span class="id" title="lemma">add_polyN</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_inverse"><span class="id" title="definition">left_inverse</span></a> 0<a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#opp_poly"><span class="id" title="definition">opp_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_poly"><span class="id" title="definition">add_poly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="poly_zmodMixin"><span class="id" title="definition">poly_zmodMixin</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Exports.ZmodMixin"><span class="id" title="abbreviation">ZmodMixin</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_polyA"><span class="id" title="lemma">add_polyA</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_polyC"><span class="id" title="lemma">add_polyC</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_poly0"><span class="id" title="lemma">add_poly0</span></a> <a class="idref" href="mathcomp.algebra.poly.html#add_polyN"><span class="id" title="lemma">add_polyN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_zmodType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Exports.ZmodType"><span class="id" title="abbreviation">ZmodType</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#PolynomialTheory.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#poly_zmodMixin"><span class="id" title="definition">poly_zmodMixin</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_zmodType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Exports.ZmodType"><span class="id" title="abbreviation">ZmodType</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#poly_zmodMixin"><span class="id" title="definition">poly_zmodMixin</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Properties of the zero polynomial  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="polyC0"><span class="id" title="lemma">polyC0</span></a> : 0<a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</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#PolynomialTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseq0"><span class="id" title="lemma">polyseq0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><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#PolynomialTheory.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.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_poly0"><span class="id" title="lemma">size_poly0</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef0"><span class="id" title="lemma">coef0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><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#PolynomialTheory.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.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="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="lead_coef0"><span class="id" title="lemma">lead_coef0</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_poly_eq0"><span class="id" title="lemma">size_poly_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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.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%<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.poly.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_leq0"><span class="id" title="lemma">size_poly_leq0</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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><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.poly.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_leq0P"><span class="id" title="lemma">size_poly_leq0P</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.poly.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> 0) (<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#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> 0%<span class="id" title="var">N</span>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_poly_gt0"><span class="id" title="lemma">size_poly_gt0</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>0 <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.poly.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nil_poly"><span class="id" title="lemma">nil_poly</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nilp"><span class="id" title="definition">nilp</span></a> <a class="idref" href="mathcomp.algebra.poly.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="http://coq.inria.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#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="poly0Vpos"><span class="id" title="lemma">poly0Vpos</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.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> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polySpred"><span class="id" title="lemma">polySpred</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.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.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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coef_eq0"><span class="id" title="lemma">lead_coef_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.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.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="polyC_eq0"><span class="id" title="lemma">polyC_eq0</span></a> (<span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">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.poly.html#c"><span class="id" title="variable">c</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_poly1P"><span class="id" title="lemma">size_poly1P</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">c</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.poly.html#c"><span class="id" title="variable">c</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#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#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.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> 1%<span class="id" title="var">N</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="leq_sizeP"><span class="id" title="lemma">leq_sizeP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">j</span>, <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">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.poly.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</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> 0) (<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#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Size, leading coef, morphism properties of coef  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefD"><span class="id" title="lemma">coefD</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</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="http://coq.inria.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#p"><span class="id" title="variable">p</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#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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="coefN"><span class="id" title="lemma">coefN</span></a> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="coefB"><span class="id" title="lemma">coefB</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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="http://coq.inria.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#p"><span class="id" title="variable">p</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#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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">Canonical</span> <span class="id" title="var">coefp_additive</span> <span class="id" title="var">i</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">(</span></a><span class="id" title="keyword">fun</span> <span class="id" title="var">p</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#coefB"><span class="id" title="lemma">coefB</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</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.poly.html#coefp"><span class="id" title="abbreviation">coefp</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>)).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefMn"><span class="id" title="lemma">coefMn</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</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="http://coq.inria.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#p"><span class="id" title="variable">p</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#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefMNn"><span class="id" title="lemma">coefMNn</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3baee4193385688d2f1fcb170107cf5b"><span class="id" title="notation">*-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</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="http://coq.inria.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#p"><span class="id" title="variable">p</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#3baee4193385688d2f1fcb170107cf5b"><span class="id" title="notation">*-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_sum"><span class="id" title="lemma">coef_sum</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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">k</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">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">_k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">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">_k</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC_add"><span class="id" title="lemma">polyC_add</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.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#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC_opp"><span class="id" title="lemma">polyC_opp</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="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#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC_sub"><span class="id" title="lemma">polyC_sub</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.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#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyC_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyC_sub"><span class="id" title="lemma">polyC_sub</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC_muln"><span class="id" title="lemma">polyC_muln</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.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#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_opp"><span class="id" title="lemma">size_opp</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.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coef_opp"><span class="id" title="lemma">lead_coef_opp</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.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.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.ssralg.html#941c6d086004545bd62614d0213e75e5"><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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_add"><span class="id" title="lemma">size_add</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#maxn"><span class="id" title="definition">maxn</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.poly.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_addl"><span class="id" title="lemma">size_addl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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.poly.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> <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#p"><span class="id" title="variable">p</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#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#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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_sum"><span class="id" title="lemma">size_sum</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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;<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#598471fe11d752586b6a83b8a4d0f333"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#598471fe11d752586b6a83b8a4d0f333"><span class="id" title="notation">max_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#598471fe11d752586b6a83b8a4d0f333"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#598471fe11d752586b6a83b8a4d0f333"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#598471fe11d752586b6a83b8a4d0f333"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#598471fe11d752586b6a83b8a4d0f333"><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.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefDl"><span class="id" title="lemma">lead_coefDl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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.poly.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> <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.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Polynomial ring structure.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="mul_poly_def"><span class="id" title="definition">mul_poly_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><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.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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">_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#j"><span class="id" title="variable">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.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Fact</span> <a name="mul_poly_key"><span class="id" title="lemma">mul_poly_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="mul_poly"><span class="id" title="definition">mul_poly</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.poly.html#mul_poly_key"><span class="id" title="lemma">mul_poly_key</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_poly_def"><span class="id" title="definition">mul_poly_def</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">mul_poly_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.poly.html#mul_poly"><span class="id" title="definition">mul_poly</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">Fact</span> <a name="coef_mul_poly"><span class="id" title="lemma">coef_mul_poly</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">i</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#mul_poly"><span class="id" title="definition">mul_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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">_</span></a>(<a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#j"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="coef_mul_poly_rev"><span class="id" title="lemma">coef_mul_poly_rev</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">i</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#mul_poly"><span class="id" title="definition">mul_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_</span></a>(<a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#j"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="mul_polyA"><span class="id" title="lemma">mul_polyA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_poly"><span class="id" title="definition">mul_poly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="mul_1poly"><span class="id" title="lemma">mul_1poly</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 1<a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_poly"><span class="id" title="definition">mul_poly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="mul_poly1"><span class="id" title="lemma">mul_poly1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> 1<a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_poly"><span class="id" title="definition">mul_poly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="mul_polyDl"><span class="id" title="lemma">mul_polyDl</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_distributive"><span class="id" title="definition">left_distributive</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_poly"><span class="id" title="definition">mul_poly</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="mul_polyDr"><span class="id" title="lemma">mul_polyDr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_poly"><span class="id" title="definition">mul_poly</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="poly1_neq0"><span class="id" title="lemma">poly1_neq0</span></a> : 1<a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">:&gt;</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#PolynomialTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="poly_ringMixin"><span class="id" title="definition">poly_ringMixin</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.RingMixin"><span class="id" title="abbreviation">RingMixin</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_polyA"><span class="id" title="lemma">mul_polyA</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_1poly"><span class="id" title="lemma">mul_1poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_poly1"><span class="id" title="lemma">mul_poly1</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_polyDl"><span class="id" title="lemma">mul_polyDl</span></a> <a class="idref" href="mathcomp.algebra.poly.html#mul_polyDr"><span class="id" title="lemma">mul_polyDr</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly1_neq0"><span class="id" title="lemma">poly1_neq0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_ringType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.RingType"><span class="id" title="abbreviation">RingType</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#PolynomialTheory.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#poly_ringMixin"><span class="id" title="definition">poly_ringMixin</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_ringType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.RingType"><span class="id" title="abbreviation">RingType</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#poly_ringMixin"><span class="id" title="definition">poly_ringMixin</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC1"><span class="id" title="lemma">polyC1</span></a> : 1<a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</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#PolynomialTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseq1"><span class="id" title="lemma">polyseq1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><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#PolynomialTheory.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.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">[::</span></a> 1<a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_poly1"><span class="id" title="lemma">size_poly1</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef1"><span class="id" title="lemma">coef1</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coef1"><span class="id" title="lemma">lead_coef1</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefM"><span class="id" title="lemma">coefM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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">_</span></a>(<a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#j"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefMr"><span class="id" title="lemma">coefMr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_</span></a>(<a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#j"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_mul_leq"><span class="id" title="lemma">size_mul_leq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><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.poly.html#q"><span class="id" title="variable">q</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="mul_lead_coef"><span class="id" title="lemma">mul_lead_coef</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<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.poly.html#p"><span class="id" title="variable">p</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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</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">_</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#b870774a3786e6850cf468108b4e1ee5"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><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.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#b870774a3786e6850cf468108b4e1ee5"><span class="id" title="notation">).-2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_proper_mul"><span class="id" title="lemma">size_proper_mul</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<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.poly.html#p"><span class="id" title="variable">p</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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><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.poly.html#q"><span class="id" title="variable">q</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="lead_coef_proper_mul"><span class="id" title="lemma">lead_coef_proper_mul</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">c</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.poly.html#p"><span class="id" title="variable">p</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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</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#lead_coef"><span class="id" title="definition">lead_coef</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_prod_leq"><span class="id" title="lemma">size_prod_leq</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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;<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><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.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefCM"><span class="id" title="lemma">coefCM</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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#p"><span class="id" title="variable">p</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="http://coq.inria.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#c"><span class="id" title="variable">c</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#p"><span class="id" title="variable">p</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="coefMC"><span class="id" title="lemma">coefMC</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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="http://coq.inria.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#p"><span class="id" title="variable">p</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#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC_mul"><span class="id" title="lemma">polyC_mul</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.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#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="polyC_multiplicative"><span class="id" title="lemma">polyC_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.poly.html#polyC"><span class="id" title="definition">polyC</span></a>.<br/>
+ <span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyC_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.poly.html#polyC_multiplicative"><span class="id" title="lemma">polyC_multiplicative</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC_exp"><span class="id" title="lemma">polyC_exp</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.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#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_exp_leq"><span class="id" title="lemma">size_exp_leq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_Msign"><span class="id" title="lemma">size_Msign</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</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#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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="coefp0_multiplicative"><span class="id" title="lemma">coefp0_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.poly.html#coefp"><span class="id" title="abbreviation">coefp</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">coefp0_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.poly.html#coefp0_multiplicative"><span class="id" title="lemma">coefp0_multiplicative</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Algebra structure of polynomials.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="scale_poly_def"><span class="id" title="definition">scale_poly_def</span></a> <span class="id" title="var">a</span> (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.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#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Fact</span> <a name="scale_poly_key"><span class="id" title="lemma">scale_poly_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="scale_poly"><span class="id" title="definition">scale_poly</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.poly.html#scale_poly_key"><span class="id" title="lemma">scale_poly_key</span></a> <a class="idref" href="mathcomp.algebra.poly.html#scale_poly_def"><span class="id" title="definition">scale_poly_def</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">scale_poly_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.poly.html#scale_poly"><span class="id" title="definition">scale_poly</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">Fact</span> <a name="scale_polyE"><span class="id" title="lemma">scale_polyE</span></a> <span class="id" title="var">a</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#scale_poly"><span class="id" title="definition">scale_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.poly.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#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="scale_polyA"><span class="id" title="lemma">scale_polyA</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#scale_poly"><span class="id" title="definition">scale_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#scale_poly"><span class="id" title="definition">scale_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.poly.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#scale_poly"><span class="id" title="definition">scale_poly</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b"><span class="id" title="variable">b</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="scale_1poly"><span class="id" title="lemma">scale_1poly</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 1 <a class="idref" href="mathcomp.algebra.poly.html#scale_poly"><span class="id" title="definition">scale_poly</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="scale_polyDr"><span class="id" title="lemma">scale_polyDr</span></a> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#scale_poly"><span class="id" title="definition">scale_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="scale_polyDl"><span class="id" title="lemma">scale_polyDl</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#scale_poly"><span class="id" title="definition">scale_poly</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="scale_polyAl"><span class="id" title="lemma">scale_polyAl</span></a> <span class="id" title="var">a</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#scale_poly"><span class="id" title="definition">scale_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#scale_poly"><span class="id" title="definition">scale_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="poly_lmodMixin"><span class="id" title="definition">poly_lmodMixin</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lmodule.Exports.LmodMixin"><span class="id" title="abbreviation">LmodMixin</span></a> <a class="idref" href="mathcomp.algebra.poly.html#scale_polyA"><span class="id" title="lemma">scale_polyA</span></a> <a class="idref" href="mathcomp.algebra.poly.html#scale_1poly"><span class="id" title="lemma">scale_1poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#scale_polyDr"><span class="id" title="lemma">scale_polyDr</span></a> <a class="idref" href="mathcomp.algebra.poly.html#scale_polyDl"><span class="id" title="lemma">scale_polyDl</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_lmodType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lmodule.Exports.LmodType"><span class="id" title="abbreviation">LmodType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.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#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.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#poly_lmodMixin"><span class="id" title="definition">poly_lmodMixin</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_lmodType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lmodule.Exports.LmodType"><span class="id" title="abbreviation">LmodType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#poly_lmodMixin"><span class="id" title="definition">poly_lmodMixin</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_lalgType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lalgebra.Exports.LalgType"><span class="id" title="abbreviation">LalgType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.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#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.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#scale_polyAl"><span class="id" title="lemma">scale_polyAl</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_lalgType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lalgebra.Exports.LalgType"><span class="id" title="abbreviation">LalgType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#scale_polyAl"><span class="id" title="lemma">scale_polyAl</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="mul_polyC"><span class="id" title="lemma">mul_polyC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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#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#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="alg_polyC"><span class="id" title="lemma">alg_polyC</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d54beaee78833d410cb3b1b3603748cc"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d54beaee78833d410cb3b1b3603748cc"><span class="id" title="notation">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</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#PolynomialTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefZ"><span class="id" title="lemma">coefZ</span></a> <span class="id" title="var">a</span> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="http://coq.inria.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#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="size_scale_leq"><span class="id" title="lemma">size_scale_leq</span></a> <span class="id" title="var">a</span> <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.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">coefp_linear</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#3caf1c46544edfa98868625c22bf2d5e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3caf1c46544edfa98868625c22bf2d5e"><span class="id" title="notation">scalar</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#PolynomialTheory.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.ssralg.html#3caf1c46544edfa98868625c22bf2d5e"><span class="id" title="notation">}</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.AddLinear"><span class="id" title="abbreviation">AddLinear</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">(</span></a><span class="id" title="keyword">fun</span> <span class="id" title="var">a</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#coefZ"><span class="id" title="lemma">coefZ</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</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.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.poly.html#coefp"><span class="id" title="abbreviation">coefp</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>)).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">coefp0_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.poly.html#coefp"><span class="id" title="abbreviation">coefp</span></a> 0<a class="idref" href="mathcomp.algebra.ssralg.html#8900f6ae77a86586561e15965d5870c7"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ The indeterminate, at last!  
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="polyX_def"><span class="id" title="definition">polyX_def</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#Poly"><span class="id" title="definition">Poly</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">[::</span></a> 0<a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">;</span></a> 1<a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Fact</span> <a name="polyX_key"><span class="id" title="lemma">polyX_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="polyX"><span class="id" title="definition">polyX</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#PolynomialTheory.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.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyX_key"><span class="id" title="lemma">polyX_key</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyX_def"><span class="id" title="definition">polyX_def</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyX_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d39a57968bbde827b22f32a7c98c9b3d"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d39a57968bbde827b22f32a7c98c9b3d"><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#d39a57968bbde827b22f32a7c98c9b3d"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyX"><span class="id" title="definition">polyX</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d39a57968bbde827b22f32a7c98c9b3d"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseqX"><span class="id" title="lemma">polyseqX</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">[::</span></a> 0<a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">;</span></a> 1<a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_polyX"><span class="id" title="lemma">size_polyX</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.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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> 2.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyX_eq0"><span class="id" title="lemma">polyX_eq0</span></a> : <a class="idref" href="http://coq.inria.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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefX"><span class="id" title="lemma">coefX</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefX"><span class="id" title="lemma">lead_coefX</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.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="commr_polyX"><span class="id" title="lemma">commr_polyX</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefMX"><span class="id" title="lemma">coefMX</span></a> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefXM"><span class="id" title="lemma">coefXM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#p"><span class="id" title="variable">p</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="cons_poly_def"><span class="id" title="lemma">cons_poly_def</span></a> <span class="id" title="var">p</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.poly.html#cons_poly"><span class="id" title="definition">cons_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.poly.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#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="poly_ind"><span class="id" title="lemma">poly_ind</span></a> (<span class="id" title="var">K</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#PolynomialTheory.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.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#K"><span class="id" title="variable">K</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">c</span>, <a class="idref" href="mathcomp.algebra.poly.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.algebra.poly.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> <a class="idref" href="mathcomp.algebra.poly.html#K"><span class="id" title="variable">K</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">p</span>, <a class="idref" href="mathcomp.algebra.poly.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.algebra.poly.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>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseqXsubC"><span class="id" title="lemma">polyseqXsubC</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">;</span></a> 1<a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_XsubC"><span class="id" title="lemma">size_XsubC</span></a> <span class="id" title="var">a</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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> 2%<span class="id" title="var">N</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_XaddC"><span class="id" title="lemma">size_XaddC</span></a> <span class="id" title="var">b</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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> 2.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefXsubC"><span class="id" title="lemma">lead_coefXsubC</span></a> <span class="id" title="var">a</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.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyXsubC_eq0"><span class="id" title="lemma">polyXsubC_eq0</span></a> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">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.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_MXaddC"><span class="id" title="lemma">size_MXaddC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> :<br/>
+&nbsp;&nbsp;<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#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseqMX"><span class="id" title="lemma">polyseqMX</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.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.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.algebra.poly.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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_mulX"><span class="id" title="lemma">size_mulX</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefMX"><span class="id" title="lemma">lead_coefMX</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.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_XmulC"><span class="id" title="lemma">size_XmulC</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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> 2.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefXn"><span class="id" title="lemma">coefXn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseqXn"><span class="id" title="lemma">polyseqXn</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> 0) 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_polyXn"><span class="id" title="lemma">size_polyXn</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="commr_polyXn"><span class="id" title="lemma">commr_polyXn</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefXn"><span class="id" title="lemma">lead_coefXn</span></a> <span class="id" title="var">n</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.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyseqMXn"><span class="id" title="lemma">polyseqMXn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.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.poly.html#p"><span class="id" title="variable">p</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#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> 0 <a class="idref" href="mathcomp.algebra.poly.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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefMXn"><span class="id" title="lemma">coefMXn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coefXnM"><span class="id" title="lemma">coefXnM</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</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#p"><span class="id" title="variable">p</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ Expansion of a polynomial as an indexed sum  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="poly_def"><span class="id" title="lemma">poly_def</span></a> <span class="id" title="var">n</span> <span class="id" title="var">E</span> : <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Monic predicate  
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="monic"><span class="id" title="definition">monic</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#65976b52191df85d9b4b677c3ff31c49"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#65976b52191df85d9b4b677c3ff31c49"><span class="id" title="notation">qualify</span></a> <span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#65976b52191df85d9b4b677c3ff31c49"><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.poly.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> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#65976b52191df85d9b4b677c3ff31c49"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Fact</span> <a name="monic_key"><span class="id" title="lemma">monic_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_key"><span class="id" title="inductive">pred_key</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.  <br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">monic_keyed</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#KeyedQualifier"><span class="id" title="definition">KeyedQualifier</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic_key"><span class="id" title="lemma">monic_key</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monicE"><span class="id" title="lemma">monicE</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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.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> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a name="monicP"><span class="id" title="lemma">monicP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</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.poly.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> 1) (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monic1"><span class="id" title="lemma">monic1</span></a> : 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a name="monicX"><span class="id" title="lemma">monicX</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.  <br/>
+<span class="id" title="keyword">Lemma</span> <a name="monicXn"><span class="id" title="lemma">monicXn</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monic_neq0"><span class="id" title="lemma">monic_neq0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.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.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coef_monicM"><span class="id" title="lemma">lead_coef_monicM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#q"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coef_Mmonic"><span class="id" title="lemma">lead_coef_Mmonic</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_monicM"><span class="id" title="lemma">size_monicM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><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.poly.html#q"><span class="id" title="variable">q</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="size_Mmonic"><span class="id" title="lemma">size_Mmonic</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.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.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><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.poly.html#q"><span class="id" title="variable">q</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="monicMl"><span class="id" title="lemma">monicMl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monicMr"><span class="id" title="lemma">monicMr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="monic_mulr_closed"><span class="id" title="lemma">monic_mulr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.mulr_closed"><span class="id" title="abbreviation">mulr_closed</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
+ <span class="id" title="keyword">Canonical</span> <span class="id" title="var">monic_mulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.MulrPred"><span class="id" title="definition">MulrPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic_mulr_closed"><span class="id" title="lemma">monic_mulr_closed</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monic_exp"><span class="id" title="lemma">monic_exp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monic_prod"><span class="id" title="lemma">monic_prod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">rI</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rI"><span class="id" title="variable">rI</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monicXsubC"><span class="id" title="lemma">monicXsubC</span></a> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monic_prod_XsubC"><span class="id" title="lemma">monic_prod_XsubC</span></a> <span class="id" title="var">I</span> <span class="id" title="var">rI</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rI"><span class="id" title="variable">rI</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_prod_XsubC"><span class="id" title="lemma">size_prod_XsubC</span></a> <span class="id" title="var">I</span> <span class="id" title="var">rI</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rI"><span class="id" title="variable">rI</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.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.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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.poly.html#rI"><span class="id" title="variable">rI</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_exp_XsubC"><span class="id" title="lemma">size_exp_XsubC</span></a> <span class="id" title="var">n</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ Some facts about regular elements.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lreg_lead"><span class="id" title="lemma">lreg_lead</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.lreg"><span class="id" title="definition">GRing.lreg</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.poly.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> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.lreg"><span class="id" title="definition">GRing.lreg</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rreg_lead"><span class="id" title="lemma">rreg_lead</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.rreg"><span class="id" title="definition">GRing.rreg</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.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> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.rreg"><span class="id" title="definition">GRing.rreg</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lreg_lead0"><span class="id" title="lemma">lreg_lead0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.lreg"><span class="id" title="definition">GRing.lreg</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.poly.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> <a class="idref" href="mathcomp.algebra.poly.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.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rreg_lead0"><span class="id" title="lemma">rreg_lead0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.rreg"><span class="id" title="definition">GRing.rreg</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.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> <a class="idref" href="mathcomp.algebra.poly.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.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lreg_size"><span class="id" title="lemma">lreg_size</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.lreg"><span class="id" title="definition">GRing.lreg</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lreg_polyZ_eq0"><span class="id" title="lemma">lreg_polyZ_eq0</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.lreg"><span class="id" title="definition">GRing.lreg</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.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_lreg"><span class="id" title="lemma">lead_coef_lreg</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.lreg"><span class="id" title="definition">GRing.lreg</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.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#c"><span class="id" title="variable">c</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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rreg_size"><span class="id" title="lemma">rreg_size</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.rreg"><span class="id" title="definition">GRing.rreg</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rreg_polyMC_eq0"><span class="id" title="lemma">rreg_polyMC_eq0</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.rreg"><span class="id" title="definition">GRing.rreg</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">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.poly.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="rreg_div0"><span class="id" title="lemma">rreg_div0</span></a> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">d</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.rreg"><span class="id" title="definition">GRing.rreg</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#d"><span class="id" title="variable">d</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</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.poly.html#d"><span class="id" title="variable">d</span></a> <a 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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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#d"><span class="id" title="variable">d</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#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monic_comreg"><span class="id" title="lemma">monic_comreg</span></a> <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><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#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.rreg"><span class="id" title="definition">GRing.rreg</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ Horner evaluation of polynomials  
+</div>
+<div class="code">
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">s</span> <span class="id" title="var">rs</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+<span class="id" title="keyword">Fixpoint</span> <a name="horner_rec"><span class="id" title="definition">horner_rec</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.algebra.poly.html#horner_rec"><span class="id" title="definition">horner_rec</span></a> <span class="id" title="var">s'</span> <a class="idref" href="mathcomp.algebra.poly.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.poly.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> <span class="id" title="var">a</span> <span class="id" title="keyword">else</span> 0.<br/>
+<span class="id" title="keyword">Definition</span> <a name="horner"><span class="id" title="definition">horner</span></a> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.poly.html#horner_rec"><span class="id" title="definition">horner_rec</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner0"><span class="id" title="lemma">horner0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><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#PolynomialTheory.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#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a 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="hornerC"><span class="id" title="lemma">hornerC</span></a> <span class="id" title="var">c</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#c"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerX"><span class="id" title="lemma">hornerX</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_cons"><span class="id" title="lemma">horner_cons</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#cons_poly"><span class="id" title="definition">cons_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><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.poly.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.poly.html#c"><span class="id" title="variable">c</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_coef0"><span class="id" title="lemma">horner_coef0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a>0<a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</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">_0</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerMXaddC"><span class="id" title="lemma">hornerMXaddC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><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.poly.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.poly.html#c"><span class="id" title="variable">c</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerMX"><span class="id" title="lemma">hornerMX</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><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.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_Poly"><span class="id" title="lemma">horner_Poly</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#Poly"><span class="id" title="definition">Poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#horner_rec"><span class="id" title="definition">horner_rec</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_coef"><span class="id" title="lemma">horner_coef</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_coef_wide"><span class="id" title="lemma">horner_coef_wide</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;<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#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_poly"><span class="id" title="lemma">horner_poly</span></a> <span class="id" title="var">n</span> <span class="id" title="var">E</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerN"><span class="id" title="lemma">hornerN</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerD"><span class="id" title="lemma">hornerD</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><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.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerXsubC"><span class="id" title="lemma">hornerXsubC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#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.poly.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_sum"><span class="id" title="lemma">horner_sum</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">F</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cbc2f2ab11c1c376b5c4511d28b14d74"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerCM"><span class="id" title="lemma">hornerCM</span></a> <span class="id" title="var">a</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerZ"><span class="id" title="lemma">hornerZ</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#c"><span class="id" title="variable">c</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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerMn"><span class="id" title="lemma">hornerMn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="comm_coef"><span class="id" title="definition">comm_coef</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> := <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.html#p"><span class="id" title="variable">p</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">Definition</span> <a name="comm_poly"><span class="id" title="definition">comm_poly</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.algebra.poly.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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><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.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comm_coef_poly"><span class="id" title="lemma">comm_coef_poly</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_coef"><span class="id" title="definition">comm_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span 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.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comm_poly0"><span class="id" title="lemma">comm_poly0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> 0 <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comm_poly1"><span class="id" title="lemma">comm_poly1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> 1 <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comm_polyX"><span class="id" title="lemma">comm_polyX</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerM_comm"><span class="id" title="lemma">hornerM_comm</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span 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.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><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.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_exp_comm"><span class="id" title="lemma">horner_exp_comm</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span 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.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerXn"><span class="id" title="lemma">hornerXn</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.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#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="hornerE_comm"><span class="id" title="definition">hornerE_comm</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#hornerD"><span class="id" title="lemma">hornerD</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerN"><span class="id" title="lemma">hornerN</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerX"><span class="id" title="lemma">hornerX</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerC"><span class="id" title="lemma">hornerC</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#horner_cons"><span class="id" title="lemma">horner_cons</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#simp"><span class="id" title="abbreviation">simp</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerCM"><span class="id" title="lemma">hornerCM</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerZ"><span class="id" title="lemma">hornerZ</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="keyword">fun</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.algebra.poly.html#hornerM_comm"><span class="id" title="lemma">hornerM_comm</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#comm_polyX"><span class="id" title="lemma">comm_polyX</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">))</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="root"><span class="id" title="definition">root</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a> := <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="mem_root"><span class="id" title="lemma">mem_root</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.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="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rootE"><span class="id" title="lemma">rootE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rootP"><span class="id" title="lemma">rootP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rootPt"><span class="id" title="lemma">rootPt</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0) (<a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rootPf"><span class="id" title="lemma">rootPf</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rootC"><span class="id" title="lemma">rootC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="root0"><span class="id" title="lemma">root0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> 0 <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="root1"><span class="id" title="lemma">root1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> 1 <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rootX"><span class="id" title="lemma">rootX</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rootN"><span class="id" title="lemma">rootN</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="root_size_gt1"><span class="id" title="lemma">root_size_gt1</span></a> <span class="id" title="var">a</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.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.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="root_XsubC"><span class="id" title="lemma">root_XsubC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="root_XaddC"><span class="id" title="lemma">root_XaddC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a name="factor_theorem"><span class="id" title="lemma">factor_theorem</span></a> <span class="id" title="var">p</span> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="multiplicity_XsubC"><span class="id" title="lemma">multiplicity_XsubC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <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">q</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3b17cb5f3a16fa64a62421f68786f750"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Roots of unity.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_Xn_sub_1"><span class="id" title="lemma">size_Xn_sub_1</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialTheory.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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monic_Xn_sub_1"><span class="id" title="lemma">monic_Xn_sub_1</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="root_of_unity"><span class="id" title="definition">root_of_unity</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</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.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> 1).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="unity_rootE"><span class="id" title="lemma">unity_rootE</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#7f47f838360ae58c7843275633f83d07"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#7f47f838360ae58c7843275633f83d07"><span class="id" title="notation">unity_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="unity_rootP"><span class="id" title="lemma">unity_rootP</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1) (<a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#7f47f838360ae58c7843275633f83d07"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#7f47f838360ae58c7843275633f83d07"><span class="id" title="notation">unity_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="primitive_root_of_unity"><span class="id" title="definition">primitive_root_of_unity</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#924e46d1120f21a5b355c376b609abe3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#924e46d1120f21a5b355c376b609abe3"><span class="id" title="notation">∀</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#924e46d1120f21a5b355c376b609abe3"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#9de6d53cccc27f521f3ab56b38159140"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#9de6d53cccc27f521f3ab56b38159140"><span class="id" title="notation">I_n</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.poly.html#7f47f838360ae58c7843275633f83d07"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#7f47f838360ae58c7843275633f83d07"><span class="id" title="notation">unity_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#924e46d1120f21a5b355c376b609abe3"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="prim_order_exists"><span class="id" title="lemma">prim_order_exists</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#aa34fd1c61c5cf0a3356b624a5d2afed"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">)}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PolynomialTheory.OnePrimitive"><span class="id" title="section">OnePrimitive</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="PolynomialTheory.OnePrimitive.prim_z"><span class="id" title="variable">prim_z</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="prim_order_gt0"><span class="id" title="lemma">prim_order_gt0</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</span></a> 0.  <br/>
+<span class="id" title="keyword">Let</span> <a name="PolynomialTheory.OnePrimitive.n_gt0"><span class="id" title="variable">n_gt0</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#prim_order_gt0"><span class="id" title="lemma">prim_order_gt0</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="prim_expr_order"><span class="id" title="lemma">prim_expr_order</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="prim_expr_mod"><span class="id" title="lemma">prim_expr_mod</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#2179ac53e82aa7c0b2f2f5a16b5510ea"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="prim_order_dvd"><span class="id" title="lemma">prim_order_dvd</span></a> <span class="id" title="var">i</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.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#aa34fd1c61c5cf0a3356b624a5d2afed"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="eq_prim_root_expr"><span class="id" title="lemma">eq_prim_root_expr</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#j"><span class="id" title="variable">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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#29294f431c8c9e3d170b3ccfa621d03f"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.poly.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#29294f431c8c9e3d170b3ccfa621d03f"><span class="id" title="notation">%[</span></a><a class="idref" href="mathcomp.ssreflect.div.html#29294f431c8c9e3d170b3ccfa621d03f"><span class="id" title="notation">mod</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.div.html#29294f431c8c9e3d170b3ccfa621d03f"><span class="id" title="notation">]</span></a><a 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="exp_prim_root"><span class="id" title="lemma">exp_prim_root</span></a> <span class="id" title="var">k</span> : <a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#df17451da28eb630dbb51b12706ba39e"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#gcdn"><span class="id" title="definition">gcdn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">).-</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">primitive_root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="dvdn_prim_root"><span class="id" title="lemma">dvdn_prim_root</span></a> <span class="id" title="var">m</span> : (<a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#aa34fd1c61c5cf0a3356b624a5d2afed"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">primitive_root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#df17451da28eb630dbb51b12706ba39e"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.OnePrimitive"><span class="id" title="section">OnePrimitive</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="prim_root_exp_coprime"><span class="id" title="lemma">prim_root_exp_coprime</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> <span class="id" title="var">k</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#2bd58b7fc104e5befe2ca1fecac7c623"><span class="id" title="notation">primitive_root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Lifting a ring predicate to polynomials.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="polyOver"><span class="id" title="definition">polyOver</span></a> (<span class="id" title="var">S</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_class"><span class="id" title="abbreviation">pred_class</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3838d61fb3e8125493e649946f677b04"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3838d61fb3e8125493e649946f677b04"><span class="id" title="notation">qualify</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3838d61fb3e8125493e649946f677b04"><span class="id" title="notation">a</span></a> <span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3838d61fb3e8125493e649946f677b04"><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#PolynomialTheory.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.ssr.ssrbool.html#3838d61fb3e8125493e649946f677b04"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#3838d61fb3e8125493e649946f677b04"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="polyOver_key"><span class="id" title="lemma">polyOver_key</span></a> <span class="id" title="var">S</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_key"><span class="id" title="inductive">pred_key</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>).  <br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyOver_keyed</span> <span class="id" title="var">S</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#KeyedQualifier"><span class="id" title="definition">KeyedQualifier</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polyOver_key"><span class="id" title="lemma">polyOver_key</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOverS"><span class="id" title="lemma">polyOverS</span></a> (<span class="id" title="var">S1</span> <span class="id" title="var">S2</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_class"><span class="id" title="abbreviation">pred_class</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S2"><span class="id" title="variable">S2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S2"><span class="id" title="variable">S2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOver0"><span class="id" title="lemma">polyOver0</span></a> <span class="id" title="var">S</span> : 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOver_poly"><span class="id" title="lemma">polyOver_poly</span></a> (<span class="id" title="var">S</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_class"><span class="id" title="abbreviation">pred_class</span></a>) <span class="id" title="var">n</span> <span class="id" title="var">E</span> :<br/>
+&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">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PolynomialTheory.PolyOverAdd"><span class="id" title="section">PolyOverAdd</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="PolynomialTheory.PolyOverAdd.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#predPredType"><span class="id" title="definition">predPredType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) (<a name="PolynomialTheory.PolyOverAdd.addS"><span class="id" title="variable">addS</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.addrPred"><span class="id" title="abbreviation">addrPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) (<a name="PolynomialTheory.PolyOverAdd.kS"><span class="id" title="variable">kS</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#keyed_pred"><span class="id" title="record">keyed_pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#addS"><span class="id" title="variable">addS</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOverP"><span class="id" title="lemma">polyOverP</span></a> {<span class="id" title="var">p</span>} : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverAdd.kS"><span class="id" title="variable">kS</span></a>) (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverAdd.kS"><span class="id" title="variable">kS</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOverC"><span class="id" title="lemma">polyOverC</span></a> <span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverAdd.kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverAdd.kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="polyOver_addr_closed"><span class="id" title="lemma">polyOver_addr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.addr_closed"><span class="id" title="abbreviation">addr_closed</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverAdd.kS"><span class="id" title="variable">kS</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyOver_addrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.AddrPred"><span class="id" title="definition">AddrPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver_addr_closed"><span class="id" title="lemma">polyOver_addr_closed</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverAdd"><span class="id" title="section">PolyOverAdd</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="polyOverNr"><span class="id" title="lemma">polyOverNr</span></a> <span class="id" title="var">S</span> (<span class="id" title="var">addS</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.zmodPred"><span class="id" title="abbreviation">zmodPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">kS</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#keyed_pred"><span class="id" title="record">keyed_pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#addS"><span class="id" title="variable">addS</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.oppr_closed"><span class="id" title="abbreviation">oppr_closed</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyOver_opprPred</span> <span class="id" title="var">S</span> <span class="id" title="var">addS</span> <span class="id" title="var">kS</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.OpprPred"><span class="id" title="definition">OpprPred</span></a> (@<a class="idref" href="mathcomp.algebra.poly.html#polyOverNr"><span class="id" title="lemma">polyOverNr</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.algebra.poly.html#addS"><span class="id" title="variable">addS</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyOver_zmodPred</span> <span class="id" title="var">S</span> <span class="id" title="var">addS</span> <span class="id" title="var">kS</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.ZmodPred"><span class="id" title="definition">ZmodPred</span></a> (@<a class="idref" href="mathcomp.algebra.poly.html#polyOverNr"><span class="id" title="lemma">polyOverNr</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.algebra.poly.html#addS"><span class="id" title="variable">addS</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PolynomialTheory.PolyOverSemiring"><span class="id" title="section">PolyOverSemiring</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Context</span> (<span class="id" title="var">S</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_class"><span class="id" title="abbreviation">pred_class</span></a>) (<span class="id" title="var">ringS</span> : @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.semiringPred"><span class="id" title="abbreviation">semiringPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">kS</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#keyed_pred"><span class="id" title="record">keyed_pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ringS"><span class="id" title="variable">ringS</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="polyOver_mulr_closed"><span class="id" title="lemma">polyOver_mulr_closed</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.mulr_closed"><span class="id" title="abbreviation">mulr_closed</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring.kS"><span class="id" title="variable">kS</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyOver_mulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.MulrPred"><span class="id" title="definition">MulrPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver_mulr_closed"><span class="id" title="lemma">polyOver_mulr_closed</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyOver_semiringPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SemiringPred"><span class="id" title="definition">SemiringPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver_mulr_closed"><span class="id" title="lemma">polyOver_mulr_closed</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOverZ"><span class="id" title="lemma">polyOverZ</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring.kS"><span class="id" title="variable">kS</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring.kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">c</span> <span class="id" title="var">p</span>, <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring.kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOverX"><span class="id" title="lemma">polyOverX</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring.kS"><span class="id" title="variable">kS</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rpred_horner"><span class="id" title="lemma">rpred_horner</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring.kS"><span class="id" title="variable">kS</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring.kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring.kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverSemiring"><span class="id" title="section">PolyOverSemiring</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PolynomialTheory.PolyOverRing"><span class="id" title="section">PolyOverRing</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Context</span> (<span class="id" title="var">S</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_class"><span class="id" title="abbreviation">pred_class</span></a>) (<span class="id" title="var">ringS</span> : @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.subringPred"><span class="id" title="abbreviation">subringPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">kS</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#keyed_pred"><span class="id" title="record">keyed_pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ringS"><span class="id" title="variable">ringS</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyOver_smulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SmulrPred"><span class="id" title="definition">SmulrPred</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polyOver_mulr_closed"><span class="id" title="lemma">polyOver_mulr_closed</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverRing.kS"><span class="id" title="variable">kS</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polyOver_subringPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SubringPred"><span class="id" title="definition">SubringPred</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polyOver_mulr_closed"><span class="id" title="lemma">polyOver_mulr_closed</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverRing.kS"><span class="id" title="variable">kS</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOverXsubC"><span class="id" title="lemma">polyOverXsubC</span></a> <span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverRing.kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverRing.kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.PolyOverRing"><span class="id" title="section">PolyOverRing</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Single derivative.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="deriv"><span class="id" title="definition">deriv</span></a> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">&lt;</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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_deriv"><span class="id" title="lemma">coef_deriv</span></a> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</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="http://coq.inria.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#p"><span class="id" title="variable">p</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#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOver_deriv"><span class="id" title="lemma">polyOver_deriv</span></a> <span class="id" title="var">S</span> (<span class="id" title="var">ringS</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.semiringPred"><span class="id" title="abbreviation">semiringPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">kS</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#keyed_pred"><span class="id" title="record">keyed_pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ringS"><span class="id" title="variable">ringS</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span>, <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivC"><span class="id" title="lemma">derivC</span></a> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a> <a 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="derivX"><span class="id" title="lemma">derivX</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivXn"><span class="id" title="lemma">derivXn</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a> <a class="idref" href="http://coq.inria.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#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="deriv_is_linear"><span class="id" title="lemma">deriv_is_linear</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.linear"><span class="id" title="abbreviation">linear</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv"><span class="id" title="definition">deriv</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">deriv_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv_is_linear"><span class="id" title="lemma">deriv_is_linear</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">deriv_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.Linear"><span class="id" title="abbreviation">Linear</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv_is_linear"><span class="id" title="lemma">deriv_is_linear</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="deriv0"><span class="id" title="lemma">deriv0</span></a> : 0<a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a> <a 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="derivD"><span class="id" title="lemma">derivD</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv"><span class="id" title="definition">deriv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivN"><span class="id" title="lemma">derivN</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv"><span class="id" title="definition">deriv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivB"><span class="id" title="lemma">derivB</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv"><span class="id" title="definition">deriv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivXsubC"><span class="id" title="lemma">derivXsubC</span></a> (<span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">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.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivMn"><span class="id" title="lemma">derivMn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivMNn"><span class="id" title="lemma">derivMNn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3baee4193385688d2f1fcb170107cf5b"><span class="id" title="notation">*-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3baee4193385688d2f1fcb170107cf5b"><span class="id" title="notation">*-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivZ"><span class="id" title="lemma">derivZ</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.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#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="deriv_mulC"><span class="id" title="lemma">deriv_mulC</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.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#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivMXaddC"><span class="id" title="lemma">derivMXaddC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><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.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivM"><span class="id" title="lemma">derivM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><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.poly.html#q"><span class="id" title="variable">q</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#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="derivE"><span class="id" title="definition">derivE</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">lazy</span> <span class="id" title="keyword">beta</span> <span class="id" title="keyword">delta</span> [<span class="id" title="var">morphism_2</span> <span class="id" title="var">morphism_1</span>] <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#derivZ"><span class="id" title="lemma">derivZ</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv_mulC"><span class="id" title="lemma">deriv_mulC</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivC"><span class="id" title="lemma">derivC</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivX"><span class="id" title="lemma">derivX</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivMXaddC"><span class="id" title="lemma">derivMXaddC</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivXsubC"><span class="id" title="lemma">derivXsubC</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivM"><span class="id" title="lemma">derivM</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivB"><span class="id" title="lemma">derivB</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#derivD"><span class="id" title="lemma">derivD</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivN"><span class="id" title="lemma">derivN</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivXn"><span class="id" title="lemma">derivXn</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivM"><span class="id" title="lemma">derivM</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivMn"><span class="id" title="lemma">derivMn</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Iterated derivative.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="derivn"><span class="id" title="definition">derivn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv"><span class="id" title="definition">deriv</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivn0"><span class="id" title="lemma">derivn0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a>0<a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivn1"><span class="id" title="lemma">derivn1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnS"><span class="id" title="lemma">derivnS</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivSn"><span class="id" title="lemma">derivSn</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_derivn"><span class="id" title="lemma">coef_derivn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</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="http://coq.inria.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#p"><span class="id" title="variable">p</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">_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">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#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#ee4a5b37d4e0ed25312d896332934f0f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#ee4a5b37d4e0ed25312d896332934f0f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#ee4a5b37d4e0ed25312d896332934f0f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#ee4a5b37d4e0ed25312d896332934f0f"><span class="id" title="notation">_</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOver_derivn"><span class="id" title="lemma">polyOver_derivn</span></a> <span class="id" title="var">S</span> (<span class="id" title="var">ringS</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.semiringPred"><span class="id" title="abbreviation">semiringPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">kS</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#keyed_pred"><span class="id" title="record">keyed_pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ringS"><span class="id" title="variable">ringS</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">n</span>, <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="derivn_is_linear"><span class="id" title="lemma">derivn_is_linear</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.linear"><span class="id" title="abbreviation">linear</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#derivn"><span class="id" title="definition">derivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+ <span class="id" title="keyword">Canonical</span> <span class="id" title="var">derivn_additive</span> <span class="id" title="var">n</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.poly.html#derivn_is_linear"><span class="id" title="lemma">derivn_is_linear</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">derivn_linear</span> <span class="id" title="var">n</span> :=  <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.Linear"><span class="id" title="abbreviation">Linear</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#derivn_is_linear"><span class="id" title="lemma">derivn_is_linear</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnC"><span class="id" title="lemma">derivnC</span></a> <span class="id" title="var">c</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnD"><span class="id" title="lemma">derivnD</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivn"><span class="id" title="definition">derivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivn_sub"><span class="id" title="lemma">derivn_sub</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivn"><span class="id" title="definition">derivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnMn"><span class="id" title="lemma">derivnMn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnMNn"><span class="id" title="lemma">derivnMNn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3baee4193385688d2f1fcb170107cf5b"><span class="id" title="notation">*-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3baee4193385688d2f1fcb170107cf5b"><span class="id" title="notation">*-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnN"><span class="id" title="lemma">derivnN</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#derivn"><span class="id" title="definition">derivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnZ"><span class="id" title="lemma">derivnZ</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.scalable"><span class="id" title="abbreviation">scalable</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#derivn"><span class="id" title="definition">derivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnXn"><span class="id" title="lemma">derivnXn</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^(</span></a><a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#ee4a5b37d4e0ed25312d896332934f0f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#ee4a5b37d4e0ed25312d896332934f0f"><span class="id" title="notation">_</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivnMXaddC"><span class="id" title="lemma">derivnMXaddC</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">c</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>  <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><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.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivn_poly0"><span class="id" title="lemma">derivn_poly0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a 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="lt_size_deriv"><span class="id" title="lemma">lt_size_deriv</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#PolynomialTheory.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#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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ A normalising version of derivation to get the division by n! in Taylor  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="nderivn"><span class="id" title="definition">nderivn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">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#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a>  <a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.poly.html#b1bd1c5077c681a8a37f97ae835f4bf6"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_nderivn"><span class="id" title="lemma">coef_nderivn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</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="http://coq.inria.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#p"><span class="id" title="variable">p</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">_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">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#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a>  <a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Here is the division by n!  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="nderivn_def"><span class="id" title="lemma">nderivn_def</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#d70b9939cc44180cb082f293ad21429e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f2f331192ff43772ca561a371dde9740"><span class="id" title="notation">`!</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOver_nderivn"><span class="id" title="lemma">polyOver_nderivn</span></a> <span class="id" title="var">S</span> (<span class="id" title="var">ringS</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.semiringPred"><span class="id" title="abbreviation">semiringPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">kS</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#keyed_pred"><span class="id" title="record">keyed_pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ringS"><span class="id" title="variable">ringS</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">n</span>, <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivn0"><span class="id" title="lemma">nderivn0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a>0<a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivn1"><span class="id" title="lemma">nderivn1</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#9caef92b6a6aef95cfbc4574952f8622"><span class="id" title="notation">^`</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnC"><span class="id" title="lemma">nderivnC</span></a> <span class="id" title="var">c</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> 0.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnXn"><span class="id" title="lemma">nderivnXn</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#5342abeecce868539daa879464131e00"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">^(</span></a><a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#5342abeecce868539daa879464131e00"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.binomial.html#f55f24aacb42fe0283014d29bcccb8c2"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="nderivn_is_linear"><span class="id" title="lemma">nderivn_is_linear</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.linear"><span class="id" title="abbreviation">linear</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#nderivn"><span class="id" title="definition">nderivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nderivn_additive</span> <span class="id" title="var">n</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.poly.html#nderivn_is_linear"><span class="id" title="lemma">nderivn_is_linear</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nderivn_linear</span> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.Linear"><span class="id" title="abbreviation">Linear</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#nderivn_is_linear"><span class="id" title="lemma">nderivn_is_linear</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnD"><span class="id" title="lemma">nderivnD</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#nderivn"><span class="id" title="definition">nderivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnB"><span class="id" title="lemma">nderivnB</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#nderivn"><span class="id" title="definition">nderivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnMn"><span class="id" title="lemma">nderivnMn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnMNn"><span class="id" title="lemma">nderivnMNn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">m</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3baee4193385688d2f1fcb170107cf5b"><span class="id" title="notation">*-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3baee4193385688d2f1fcb170107cf5b"><span class="id" title="notation">*-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#m"><span class="id" title="variable">m</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnN"><span class="id" title="lemma">nderivnN</span></a> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#nderivn"><span class="id" title="definition">nderivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnZ"><span class="id" title="lemma">nderivnZ</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.scalable"><span class="id" title="abbreviation">scalable</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#nderivn"><span class="id" title="definition">nderivn</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivnMXaddC"><span class="id" title="lemma">nderivnMXaddC</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">c</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><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.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#4a33fe2dad4a62417624cbe36418a1fe"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><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.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#a1cd5f9ec97ed6af469b2ab6da5bc5f9"><span class="id" title="notation">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivn_poly0"><span class="id" title="lemma">nderivn_poly0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a> <a 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="nderiv_taylor"><span class="id" title="lemma">nderiv_taylor</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">h</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.poly.html#h"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.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.poly.html#h"><span class="id" title="variable">h</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><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.poly.html#h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderiv_taylor_wide"><span class="id" title="lemma">nderiv_taylor_wide</span></a> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">h</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.poly.html#h"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.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.poly.html#h"><span class="id" title="variable">h</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.poly.html#a942f71ef6e79d81ec1823e85631f18b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#baba25a9b511327d163a4abdecb45e2a"><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.poly.html#h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialTheory"><span class="id" title="section">PolynomialTheory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="ae876944c1ac880ac0794dfc7e1a5c1d"><span class="id" title="notation">&quot;</span></a>\poly_ ( i &lt; n ) E" := (<a class="idref" href="mathcomp.algebra.poly.html#poly"><span class="id" title="definition">poly</span></a> <span class="id" title="var">n</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> ⇒ <span class="id" title="var">E</span>)) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">&quot;</span></a>c %:P" := (<a class="idref" href="mathcomp.algebra.poly.html#polyC"><span class="id" title="definition">polyC</span></a> <span class="id" title="var">c</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">&quot;</span></a>'X" := (<a class="idref" href="mathcomp.algebra.poly.html#polyX"><span class="id" title="definition">polyX</span></a> <span class="id" title="var">_</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">&quot;</span></a>''X^' n" := (<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#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <span class="id" title="var">n</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">&quot;</span></a>p .[ x ]" := (<a class="idref" href="mathcomp.algebra.poly.html#horner"><span class="id" title="definition">horner</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">&quot;</span></a>n .-unity_root" := (<a class="idref" href="mathcomp.algebra.poly.html#root_of_unity"><span class="id" title="definition">root_of_unity</span></a> <span class="id" title="var">n</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">&quot;</span></a>n .-primitive_root" := (<a class="idref" href="mathcomp.algebra.poly.html#primitive_root_of_unity"><span class="id" title="definition">primitive_root_of_unity</span></a> <span class="id" title="var">n</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">&quot;</span></a>a ^` " := (<a class="idref" href="mathcomp.algebra.poly.html#deriv"><span class="id" title="definition">deriv</span></a> <span class="id" title="var">a</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="52c8dee738284046bb7e0fb3f2a04f1e"><span class="id" title="notation">&quot;</span></a>a ^` ( n )" := (<a class="idref" href="mathcomp.algebra.poly.html#derivn"><span class="id" title="definition">derivn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">a</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">&quot;</span></a>a ^`N ( n )" := (<a class="idref" href="mathcomp.algebra.poly.html#nderivn"><span class="id" title="definition">nderivn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">a</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Container morphism.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Section</span> <a name="MapPoly"><span class="id" title="section">MapPoly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MapPoly.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MapPoly.Definitions.aR"><span class="id" title="variable">aR</span></a> <a name="MapPoly.Definitions.rR"><span class="id" title="variable">rR</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>) (<a name="MapPoly.Definitions.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#aR"><span class="id" title="variable">aR</span></a> <a class="idref" href="http://coq.inria.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#rR"><span class="id" title="variable">rR</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="map_poly"><span class="id" title="definition">map_poly</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#MapPoly.Definitions.aR"><span class="id" title="variable">aR</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#ae876944c1ac880ac0794dfc7e1a5c1d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#ae876944c1ac880ac0794dfc7e1a5c1d"><span class="id" title="notation">poly_</span></a><a class="idref" href="mathcomp.algebra.poly.html#ae876944c1ac880ac0794dfc7e1a5c1d"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.poly.html#ae876944c1ac880ac0794dfc7e1a5c1d"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#ae876944c1ac880ac0794dfc7e1a5c1d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Definitions.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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/>
+</div>
+
+<div class="doc">
+ Alternative definition; the one above is more convenient because it lets 
+ us use the lemmas on \poly, e.g., size (map_poly p) &lt;= size p is an      
+ instance of size_poly.                                                    
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="map_polyE"><span class="id" title="lemma">map_polyE</span></a> <span class="id" title="var">p</span> : <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#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#Poly"><span class="id" title="definition">Poly</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Definitions.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="commr_rmorph"><span class="id" title="definition">commr_rmorph</span></a> <span class="id" title="var">u</span> := <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.comm"><span class="id" title="definition">GRing.comm</span></a> <a class="idref" href="mathcomp.algebra.poly.html#u"><span class="id" title="variable">u</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Definitions.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="horner_morph"><span class="id" title="definition">horner_morph</span></a> <span class="id" title="var">u</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.algebra.poly.html#commr_rmorph"><span class="id" title="definition">commr_rmorph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#u"><span class="id" title="variable">u</span></a> := <span class="id" title="keyword">fun</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.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.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>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> <a name="MapPoly.aR"><span class="id" title="variable">aR</span></a> <a name="MapPoly.rR"><span class="id" title="variable">rR</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MapPoly.Combinatorial"><span class="id" title="section">Combinatorial</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MapPoly.Combinatorial.iR"><span class="id" title="variable">iR</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>) (<a name="MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.aR"><span class="id" title="variable">aR</span></a> <a class="idref" href="http://coq.inria.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#MapPoly.rR"><span class="id" title="variable">rR</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_poly0"><span class="id" title="lemma">map_poly0</span></a> : 0<a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">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> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="eq_map_poly"><span class="id" title="lemma">eq_map_poly</span></a> (<span class="id" title="var">g</span> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.aR"><span class="id" title="variable">aR</span></a> <a class="idref" href="http://coq.inria.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#MapPoly.rR"><span class="id" title="variable">rR</span></a>) : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.algebra.poly.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</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#g"><span class="id" title="variable">g</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_poly_id"><span class="id" title="lemma">map_poly_id</span></a> <span class="id" title="var">g</span> (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.iR"><span class="id" title="variable">iR</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.iR"><span class="id" title="variable">iR</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">),</span></a> <a class="idref" href="mathcomp.algebra.poly.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <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#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.algebra.poly.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#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_map_id0"><span class="id" title="lemma">coef_map_id0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#9625b440a0052f6dbfd015f5bb8b5125"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">f</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="http://coq.inria.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#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="map_Poly_id0"><span class="id" title="lemma">map_Poly_id0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.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#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#Poly"><span class="id" title="definition">Poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">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.poly.html#Poly"><span class="id" title="definition">Poly</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_poly_comp_id0"><span class="id" title="lemma">map_poly_comp_id0</span></a> (<span class="id" title="var">g</span> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.iR"><span class="id" title="variable">iR</span></a> <a class="idref" href="http://coq.inria.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#MapPoly.aR"><span class="id" title="variable">aR</span></a>) <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.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#map_poly"><span class="id" title="definition">map_poly</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.algebra.poly.html#g"><span class="id" title="variable">g</span></a>) <a class="idref" href="mathcomp.algebra.poly.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#655a5cbe5192b668144a8107711db834"><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#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_map_poly_id0"><span class="id" title="lemma">size_map_poly_id0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</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.poly.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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_poly_eq0_id0"><span class="id" title="lemma">map_poly_eq0_id0</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</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.poly.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">f</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.poly.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_map_id0"><span class="id" title="lemma">lead_coef_map_id0</span></a> <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.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#MapPoly.Combinatorial.f"><span class="id" title="variable">f</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.poly.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.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">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.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</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.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Hypotheses</span> (<a name="MapPoly.Combinatorial.inj_f"><span class="id" title="variable">inj_f</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a>) (<a name="MapPoly.Combinatorial.f_0"><span class="id" title="variable">f_0</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</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="size_map_inj_poly"><span class="id" title="lemma">size_map_inj_poly</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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_inj_poly"><span class="id" title="lemma">map_inj_poly</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coef_map_inj"><span class="id" title="lemma">lead_coef_map_inj</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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#655a5cbe5192b668144a8107711db834"><span class="id" title="notation">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.poly.html#MapPoly.Combinatorial.f"><span class="id" title="variable">f</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.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Combinatorial"><span class="id" title="section">Combinatorial</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_polyK"><span class="id" title="lemma">map_polyK</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.aR"><span class="id" title="variable">aR</span></a> <a class="idref" href="http://coq.inria.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#MapPoly.rR"><span class="id" title="variable">rR</span></a>) <span class="id" title="var">g</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.algebra.poly.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.html#f"><span class="id" title="variable">f</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#g"><span class="id" title="variable">g</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#f"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MapPoly.Additive"><span class="id" title="section">Additive</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MapPoly.Additive.iR"><span class="id" title="variable">iR</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>) (<a name="MapPoly.Additive.f"><span class="id" title="variable">f</span></a> : <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.poly.html#MapPoly.aR"><span class="id" title="variable">aR</span></a> <a class="idref" href="http://coq.inria.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#MapPoly.rR"><span class="id" title="variable">rR</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6566b94c06c342b0768c3d2d73badf6e"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="coef_map"><span class="id" title="lemma">coef_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">f</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="http://coq.inria.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#MapPoly.Additive.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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="map_Poly"><span class="id" title="lemma">map_Poly</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#Poly"><span class="id" title="definition">Poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">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.poly.html#Poly"><span class="id" title="definition">Poly</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Additive.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_poly_comp"><span class="id" title="lemma">map_poly_comp</span></a> (<span class="id" title="var">g</span> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Additive.iR"><span class="id" title="variable">iR</span></a> <a class="idref" href="http://coq.inria.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#MapPoly.aR"><span class="id" title="variable">aR</span></a>) <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<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#MapPoly.Additive.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.algebra.poly.html#g"><span class="id" title="variable">g</span></a>) <a class="idref" href="mathcomp.algebra.poly.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#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Additive.f"><span class="id" title="variable">f</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#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="map_poly_is_additive"><span class="id" title="lemma">map_poly_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.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Additive.f"><span class="id" title="variable">f</span></a>).<br/>
+ <span class="id" title="keyword">Canonical</span> <span class="id" title="var">map_poly_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> <a class="idref" href="mathcomp.algebra.poly.html#map_poly_is_additive"><span class="id" title="lemma">map_poly_is_additive</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_polyC"><span class="id" title="lemma">map_polyC</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">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.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Additive.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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="lead_coef_map_eq"><span class="id" title="lemma">lead_coef_map_eq</span></a> <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Additive.f"><span class="id" title="variable">f</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.poly.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.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#7d2b3e616181246d58c271ab24bc9d56"><span class="id" title="notation">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.poly.html#MapPoly.Additive.f"><span class="id" title="variable">f</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.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.Additive"><span class="id" title="section">Additive</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="MapPoly.f"><span class="id" title="variable">f</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.poly.html#MapPoly.aR"><span class="id" title="variable">aR</span></a> <a class="idref" href="http://coq.inria.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#MapPoly.rR"><span class="id" title="variable">rR</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.aR"><span class="id" title="variable">aR</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="map_poly_is_rmorphism"><span class="id" title="lemma">map_poly_is_rmorphism</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.rmorphism"><span class="id" title="abbreviation">rmorphism</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#MapPoly.f"><span class="id" title="variable">f</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">map_poly_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.RMorphism"><span class="id" title="abbreviation">RMorphism</span></a> <a class="idref" href="mathcomp.algebra.poly.html#map_poly_is_rmorphism"><span class="id" title="lemma">map_poly_is_rmorphism</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_polyZ"><span class="id" title="lemma">map_polyZ</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">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.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a>.<br/>
+ <span class="id" title="keyword">Canonical</span> <span class="id" title="var">map_poly_linear</span> :=<br/>
+&nbsp;&nbsp;<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.poly.html#map_polyZ"><span class="id" title="lemma">map_polyZ</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.scalable_for"><span class="id" title="abbreviation">scalable_for</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c42c5cb909c30537f9f6acfcf01cf7e1"><span class="id" title="notation">\;</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#75c106c115e1ca6097cf58b45ce663bd"><span class="id" title="notation">*:%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#75c106c115e1ca6097cf58b45ce663bd"><span class="id" title="notation">R</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#MapPoly.f"><span class="id" title="variable">f</span></a>)).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">map_poly_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.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</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="map_polyX"><span class="id" title="lemma">map_polyX</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><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#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">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.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="map_polyXn"><span class="id" title="lemma">map_polyXn</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">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.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="monic_map"><span class="id" title="lemma">monic_map</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_map"><span class="id" title="lemma">horner_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</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#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.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> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.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>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_comm_poly"><span class="id" title="lemma">map_comm_poly</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span 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.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_comm_coef"><span class="id" title="lemma">map_comm_coef</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_coef"><span class="id" title="definition">comm_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span 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.poly.html#comm_coef"><span class="id" title="definition">comm_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rmorph_root"><span class="id" title="lemma">rmorph_root</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span 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.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rmorph_unity_root"><span class="id" title="lemma">rmorph_unity_root</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">unity_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">unity_root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MapPoly.HornerMorph"><span class="id" title="section">HornerMorph</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="MapPoly.HornerMorph.u"><span class="id" title="variable">u</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.rR"><span class="id" title="variable">rR</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="MapPoly.HornerMorph.cfu"><span class="id" title="variable">cfu</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#commr_rmorph"><span class="id" title="definition">commr_rmorph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.HornerMorph.u"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_morphC"><span class="id" title="lemma">horner_morphC</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.poly.html#horner_morph"><span class="id" title="definition">horner_morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.HornerMorph.cfu"><span class="id" title="variable">cfu</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_morphX"><span class="id" title="lemma">horner_morphX</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#horner_morph"><span class="id" title="definition">horner_morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.HornerMorph.cfu"><span class="id" title="variable">cfu</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.poly.html#MapPoly.HornerMorph.u"><span class="id" title="variable">u</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="horner_is_lrmorphism"><span class="id" title="lemma">horner_is_lrmorphism</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.LRMorphism.Exports.lrmorphism_for"><span class="id" title="abbreviation">lrmorphism_for</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#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.poly.html#horner_morph"><span class="id" title="definition">horner_morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.HornerMorph.cfu"><span class="id" title="variable">cfu</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">horner_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> <a class="idref" href="mathcomp.algebra.poly.html#horner_is_lrmorphism"><span class="id" title="lemma">horner_is_lrmorphism</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">horner_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.RMorphism"><span class="id" title="abbreviation">RMorphism</span></a> <a class="idref" href="mathcomp.algebra.poly.html#horner_is_lrmorphism"><span class="id" title="lemma">horner_is_lrmorphism</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">horner_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.poly.html#horner_is_lrmorphism"><span class="id" title="lemma">horner_is_lrmorphism</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">horner_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.poly.html#horner_morph"><span class="id" title="definition">horner_morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.HornerMorph.cfu"><span class="id" title="variable">cfu</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">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly.HornerMorph"><span class="id" title="section">HornerMorph</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="deriv_map"><span class="id" title="lemma">deriv_map</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">^`</span></a> <a class="idref" href="http://coq.inria.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#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="derivn_map"><span class="id" title="lemma">derivn_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.poly.html#52c8dee738284046bb7e0fb3f2a04f1e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#52c8dee738284046bb7e0fb3f2a04f1e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#52c8dee738284046bb7e0fb3f2a04f1e"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#52c8dee738284046bb7e0fb3f2a04f1e"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="nderivn_map"><span class="id" title="lemma">nderivn_map</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a><a class="idref" href="mathcomp.algebra.poly.html#a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">^`</span></a><a class="idref" href="mathcomp.algebra.poly.html#a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.algebra.poly.html#a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#a686d72f245928a3479b06c25ed0c2e0"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.poly.html#b0a4b98f65b912dc010655ba49aed12f"><span class="id" title="notation">f</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MapPoly"><span class="id" title="section">MapPoly</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Morphisms from the polynomial ring, and the initiality of polynomials  
+ with respect to these.                                                  
+</div>
+<div class="code">
+<span class="id" title="keyword">Section</span> <a name="MorphPoly"><span class="id" title="section">MorphPoly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> (<a name="MorphPoly.aR"><span class="id" title="variable">aR</span></a> <a name="MorphPoly.rR"><span class="id" title="variable">rR</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>) (<a name="MorphPoly.pf"><span class="id" title="variable">pf</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.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#aR"><span class="id" title="variable">aR</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rR"><span class="id" title="variable">rR</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="poly_morphX_comm"><span class="id" title="lemma">poly_morphX_comm</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#commr_rmorph"><span class="id" title="definition">commr_rmorph</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MorphPoly.pf"><span class="id" title="variable">pf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</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#MorphPoly.pf"><span class="id" title="variable">pf</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="poly_initial"><span class="id" title="lemma">poly_initial</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#MorphPoly.pf"><span class="id" title="variable">pf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.algebra.poly.html#horner_morph"><span class="id" title="definition">horner_morph</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_morphX_comm"><span class="id" title="lemma">poly_morphX_comm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MorphPoly"><span class="id" title="section">MorphPoly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="f734b1360ed3aede5055701a3bc465fd"><span class="id" title="notation">&quot;</span></a>p ^:P" := (<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> <span class="id" title="var">p</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PolyCompose"><span class="id" title="section">PolyCompose</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="PolyCompose.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <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#PolyCompose.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="comp_poly"><span class="id" title="definition">comp_poly</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.poly.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#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.algebra.poly.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>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_map_polyC"><span class="id" title="lemma">size_map_polyC</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.poly.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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_polyC_eq0"><span class="id" title="lemma">map_polyC_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.poly.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.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.poly.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="root_polyC"><span class="id" title="lemma">root_polyC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.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#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="http://coq.inria.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#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_polyE"><span class="id" title="lemma">comp_polyE</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b2bfc5b99c28e2c89b336d5f86347706"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a><a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyOver_comp"><span class="id" title="lemma">polyOver_comp</span></a> <span class="id" title="var">S</span> (<span class="id" title="var">ringS</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.semiringPred"><span class="id" title="abbreviation">semiringPred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">kS</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#keyed_pred"><span class="id" title="record">keyed_pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ringS"><span class="id" title="variable">ringS</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span>, <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polyOver"><span class="id" title="definition">polyOver</span></a> <a class="idref" href="mathcomp.algebra.poly.html#kS"><span class="id" title="variable">kS</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_polyCr"><span class="id" title="lemma">comp_polyCr</span></a> <span class="id" title="var">p</span> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</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.poly.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.poly.html#c"><span class="id" title="variable">c</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#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="comp_poly0r"><span class="id" title="lemma">comp_poly0r</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_0</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="comp_polyC"><span class="id" title="lemma">comp_polyC</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</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#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.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#c"><span class="id" title="variable">c</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">Fact</span> <a name="comp_poly_is_linear"><span class="id" title="lemma">comp_poly_is_linear</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.linear"><span class="id" title="abbreviation">linear</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#comp_poly"><span class="id" title="definition">comp_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comp_poly_additive</span> <span class="id" title="var">p</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.poly.html#comp_poly_is_linear"><span class="id" title="lemma">comp_poly_is_linear</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comp_poly_linear</span> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.Linear"><span class="id" title="abbreviation">Linear</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#comp_poly_is_linear"><span class="id" title="lemma">comp_poly_is_linear</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_poly0"><span class="id" title="lemma">comp_poly0</span></a> <span class="id" title="var">p</span> : 0 <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.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> 0.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_polyD"><span class="id" title="lemma">comp_polyD</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_polyB"><span class="id" title="lemma">comp_polyB</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_polyZ"><span class="id" title="lemma">comp_polyZ</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_polyXr"><span class="id" title="lemma">comp_polyXr</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_polyX"><span class="id" title="lemma">comp_polyX</span></a> <span class="id" title="var">p</span> : <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#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.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#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_poly_MXaddC"><span class="id" title="lemma">comp_poly_MXaddC</span></a> <span class="id" title="var">c</span> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</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#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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#c"><span class="id" title="variable">c</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="comp_polyXaddC_K"><span class="id" title="lemma">comp_polyXaddC_K</span></a> <span class="id" title="var">p</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><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.poly.html#z"><span class="id" title="variable">z</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#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">))</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><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#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</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#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_comp_poly_leq"><span class="id" title="lemma">size_comp_poly_leq</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<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#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.poly.html#3963cf6cfb5e8b54483fc37af1a6db2d"><span class="id" title="notation">Po</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><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.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PolyCompose"><span class="id" title="section">PolyCompose</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">&quot;</span></a>p \Po q" := (<a class="idref" href="mathcomp.algebra.poly.html#comp_poly"><span class="id" title="definition">comp_poly</span></a> <span class="id" title="var">q</span> <span class="id" title="var">p</span>) : <span class="id" title="var">ring_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_comp_poly"><span class="id" title="lemma">map_comp_poly</span></a> (<span class="id" title="var">aR</span> <span class="id" title="var">rR</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#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.poly.html#aR"><span class="id" title="variable">aR</span></a> <a class="idref" href="http://coq.inria.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#rR"><span class="id" title="variable">rR</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a>) <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<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#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.algebra.poly.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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.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#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PolynomialComRing"><span class="id" title="section">PolynomialComRing</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="PolynomialComRing.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">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.poly.html#PolynomialComRing.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="poly_mul_comm"><span class="id" title="lemma">poly_mul_comm</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_comRingType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Exports.ComRingType"><span class="id" title="abbreviation">ComRingType</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#PolynomialComRing.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#poly_mul_comm"><span class="id" title="lemma">poly_mul_comm</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_comRingType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Exports.ComRingType"><span class="id" title="abbreviation">ComRingType</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialComRing.R"><span class="id" title="variable">R</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#poly_mul_comm"><span class="id" title="lemma">poly_mul_comm</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_algType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Algebra.Exports.CommAlgType"><span class="id" title="abbreviation">CommAlgType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialComRing.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#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialComRing.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/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_algType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#30ca49fca582d0576271da5ba1a53c8c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#30ca49fca582d0576271da5ba1a53c8c"><span class="id" title="notation">algType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialComRing.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#30ca49fca582d0576271da5ba1a53c8c"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialComRing.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#30ca49fca582d0576271da5ba1a53c8c"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_algType"><span class="id" title="definition">poly_algType</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#30ca49fca582d0576271da5ba1a53c8c"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="hornerM"><span class="id" title="lemma">hornerM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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.poly.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> <a class="idref" href="mathcomp.algebra.poly.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.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.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>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_exp"><span class="id" title="lemma">horner_exp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</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#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> <a class="idref" href="mathcomp.algebra.poly.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.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_prod"><span class="id" title="lemma">horner_prod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialComRing.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> :<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.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</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#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> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#358fca18835530a08faf9e0f246b584a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</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#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>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="hornerE"><span class="id" title="definition">hornerE</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#hornerD"><span class="id" title="lemma">hornerD</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerN"><span class="id" title="lemma">hornerN</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerX"><span class="id" title="lemma">hornerX</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerC"><span class="id" title="lemma">hornerC</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#horner_cons"><span class="id" title="lemma">horner_cons</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#simp"><span class="id" title="abbreviation">simp</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerCM"><span class="id" title="lemma">hornerCM</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerZ"><span class="id" title="lemma">hornerZ</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#hornerM"><span class="id" title="lemma">hornerM</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="horner_eval"><span class="id" title="definition">horner_eval</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialComRing.R"><span class="id" title="variable">R</span></a>) := <a class="idref" href="mathcomp.algebra.poly.html#horner"><span class="id" title="definition">horner</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_evalE"><span class="id" title="lemma">horner_evalE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#horner_eval"><span class="id" title="definition">horner_eval</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.poly.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#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.poly.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>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="horner_eval_is_lrmorphism"><span class="id" title="lemma">horner_eval_is_lrmorphism</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.LRMorphism.Exports.lrmorphism_for"><span class="id" title="abbreviation">lrmorphism_for</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.poly.html#horner_eval"><span class="id" title="definition">horner_eval</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">horner_eval_additive</span> <span class="id" title="var">x</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.poly.html#horner_eval_is_lrmorphism"><span class="id" title="lemma">horner_eval_is_lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">horner_eval_rmorphism</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.RMorphism"><span class="id" title="abbreviation">RMorphism</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#horner_eval_is_lrmorphism"><span class="id" title="lemma">horner_eval_is_lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">horner_eval_linear</span> <span class="id" title="var">x</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.poly.html#horner_eval_is_lrmorphism"><span class="id" title="lemma">horner_eval_is_lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">horner_eval_lrmorphism</span> <span class="id" title="var">x</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.poly.html#horner_eval"><span class="id" title="definition">horner_eval</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</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">Fact</span> <a name="comp_poly_multiplicative"><span class="id" title="lemma">comp_poly_multiplicative</span></a> <span class="id" title="var">q</span> : <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.poly.html#comp_poly"><span class="id" title="definition">comp_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comp_poly_rmorphism</span> <span class="id" title="var">q</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.poly.html#comp_poly_multiplicative"><span class="id" title="lemma">comp_poly_multiplicative</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comp_poly_lrmorphism</span> <span class="id" title="var">q</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.poly.html#comp_poly"><span class="id" title="definition">comp_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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="comp_polyM"><span class="id" title="lemma">comp_polyM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</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">\</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#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_polyA"><span class="id" title="lemma">comp_polyA</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="mathcomp.algebra.poly.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.poly.html#q"><span class="id" title="variable">q</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#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.poly.html#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#cad3d28b9cdd7490720002e244568365"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</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">\</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#r"><span class="id" title="variable">r</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="horner_comp"><span class="id" title="lemma">horner_comp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#9956cd3926e9966aa6979e465e39d037"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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#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.poly.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> <a class="idref" href="mathcomp.algebra.poly.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.poly.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.poly.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>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="root_comp"><span class="id" title="lemma">root_comp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.algebra.poly.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.poly.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>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="deriv_comp"><span class="id" title="lemma">deriv_comp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">^`</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">^`</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#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">^`</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="deriv_exp"><span class="id" title="lemma">deriv_exp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><span class="id" title="notation">)^`</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#03bdc25c1ca3719165e27fca9d5b98af"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="derivCE"><span class="id" title="definition">derivCE</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#derivE"><span class="id" title="definition">derivE</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#deriv_exp"><span class="id" title="lemma">deriv_exp</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialComRing"><span class="id" title="section">PolynomialComRing</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PolynomialIdomain"><span class="id" title="section">PolynomialIdomain</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Integral domain structure on poly  
+</div>
+<div class="code">
+<span class="id" title="keyword">Variable</span> <a name="PolynomialIdomain.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/>
+
+<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.poly.html#PolynomialIdomain.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> <span class="id" title="var">m</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#PolynomialIdomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_mul"><span class="id" title="lemma">size_mul</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.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.poly.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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><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.poly.html#q"><span class="id" title="variable">q</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">Fact</span> <a name="poly_idomainAxiom"><span class="id" title="lemma">poly_idomainAxiom</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="poly_unit"><span class="id" title="definition">poly_unit</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialIdomain.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;<span class="id" title="keyword">fun</span> <span class="id" title="var">p</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><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.poly.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> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="poly_inv"><span class="id" title="definition">poly_inv</span></a> <span class="id" title="var">p</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_unit"><span class="id" title="definition">poly_unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_0</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">)^-1</span></a><a class="idref" href="mathcomp.algebra.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.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="poly_mulVp"><span class="id" title="lemma">poly_mulVp</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_unit"><span class="id" title="definition">poly_unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_inverse"><span class="id" title="definition">left_inverse</span></a> 1 <a class="idref" href="mathcomp.algebra.poly.html#poly_inv"><span class="id" title="definition">poly_inv</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="poly_intro_unit"><span class="id" title="lemma">poly_intro_unit</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</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#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> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_unit"><span class="id" title="definition">poly_unit</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fact</span> <a name="poly_inv_out"><span class="id" title="lemma">poly_inv_out</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c2f58fba484177bda65c2ab1289a6fe6"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c2f58fba484177bda65c2ab1289a6fe6"><span class="id" title="notation">predC</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_unit"><span class="id" title="definition">poly_unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c2f58fba484177bda65c2ab1289a6fe6"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_inv"><span class="id" title="definition">poly_inv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="poly_comUnitMixin"><span class="id" title="definition">poly_comUnitMixin</span></a> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Exports.ComUnitRingMixin"><span class="id" title="abbreviation">ComUnitRingMixin</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_mulVp"><span class="id" title="lemma">poly_mulVp</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_intro_unit"><span class="id" title="lemma">poly_intro_unit</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_inv_out"><span class="id" title="lemma">poly_inv_out</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_unitRingType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Exports.UnitRingType"><span class="id" title="abbreviation">UnitRingType</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#PolynomialIdomain.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#poly_comUnitMixin"><span class="id" title="definition">poly_comUnitMixin</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_unitRingType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#cb745c487a899dce62ab9ce5330f227e"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cb745c487a899dce62ab9ce5330f227e"><span class="id" title="notation">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cb745c487a899dce62ab9ce5330f227e"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialIdomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#cb745c487a899dce62ab9ce5330f227e"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_unitRingType"><span class="id" title="definition">poly_unitRingType</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#cb745c487a899dce62ab9ce5330f227e"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_unitAlgType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">unitAlgType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialIdomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">of</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#PolynomialIdomain.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.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_unitAlgType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">unitAlgType</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialIdomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialIdomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_comUnitRingType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">of</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#PolynomialIdomain.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.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">]</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_comUnitRingType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialIdomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">poly_idomainType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Exports.IdomainType"><span class="id" title="abbreviation">IdomainType</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#PolynomialIdomain.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#poly_idomainAxiom"><span class="id" title="lemma">poly_idomainAxiom</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">polynomial_idomainType</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#29ac7480dfde2720a0c36d25103fa4a7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#29ac7480dfde2720a0c36d25103fa4a7"><span class="id" title="notation">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#29ac7480dfde2720a0c36d25103fa4a7"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.poly.html#polynomial"><span class="id" title="record">polynomial</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialIdomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#29ac7480dfde2720a0c36d25103fa4a7"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.poly.html#poly_idomainType"><span class="id" title="definition">poly_idomainType</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#29ac7480dfde2720a0c36d25103fa4a7"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="poly_unitE"><span class="id" title="lemma">poly_unitE</span></a> <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.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> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="poly_invE"><span class="id" title="lemma">poly_invE</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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">_0</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">)^-1</span></a><a class="idref" href="mathcomp.algebra.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.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="polyC_inv"><span class="id" title="lemma">polyC_inv</span></a> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</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.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f3016d4e55aa553d3e912592ec65e342"><span class="id" title="notation">^-1</span></a><a class="idref" href="mathcomp.algebra.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="rootM"><span class="id" title="lemma">rootM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#q"><span class="id" title="variable">q</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rootZ"><span class="id" title="lemma">rootZ</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_scale"><span class="id" title="lemma">size_scale</span></a> <span class="id" title="var">a</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_Cmul"><span class="id" title="lemma">size_Cmul</span></a> <span class="id" title="var">a</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefM"><span class="id" title="lemma">lead_coefM</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coefZ"><span class="id" title="lemma">lead_coefZ</span></a> <span class="id" title="var">a</span> <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.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.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#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="scale_poly_eq0"><span class="id" title="lemma">scale_poly_eq0</span></a> <span class="id" title="var">a</span> <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.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.poly.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.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_prod"><span class="id" title="lemma">size_prod</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.poly.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.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#PolynomialIdomain.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;&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">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b29cd8e479370273da36336a1ca6eca7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#fc4dacaf383de64a30be0f81bf2f232f"><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.poly.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9482aae3d3b06e249765c1225dbb8cbb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.poly.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_exp"><span class="id" title="lemma">size_exp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>)<a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coef_exp"><span class="id" title="lemma">lead_coef_exp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</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.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="root_prod_XsubC"><span class="id" title="lemma">root_prod_XsubC</span></a> <span class="id" title="var">rs</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><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#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a><a 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="root_exp_XsubC"><span class="id" title="lemma">root_exp_XsubC</span></a> <span class="id" title="var">n</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><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#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</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.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.poly.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_comp_poly"><span class="id" title="lemma">size_comp_poly</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<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.poly.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#q"><span class="id" title="variable">q</span></a>)<a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><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.poly.html#q"><span class="id" title="variable">q</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">).-1</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_comp_poly2"><span class="id" title="lemma">size_comp_poly2</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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.poly.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#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#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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_poly2_eq0"><span class="id" title="lemma">comp_poly2_eq0</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</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.poly.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_comp"><span class="id" title="lemma">lead_coef_comp</span></a> <span class="id" title="var">p</span> <span class="id" title="var">q</span> :<br/>
+&nbsp;&nbsp;<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#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#lead_coef"><span class="id" title="definition">lead_coef</span></a> (<a class="idref" href="mathcomp.algebra.poly.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#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#1c39bf18749e5cc609e83c0a0ba5a372"><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.poly.html#p"><span class="id" title="variable">p</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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#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.poly.html#p"><span class="id" title="variable">p</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">Theorem</span> <a name="max_poly_roots"><span class="id" title="lemma">max_poly_roots</span></a> <span class="id" title="var">p</span> <span class="id" title="var">rs</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.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.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PolynomialIdomain"><span class="id" title="section">PolynomialIdomain</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MapFieldPoly"><span class="id" title="section">MapFieldPoly</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MapFieldPoly.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) (<a name="MapFieldPoly.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>) (<a name="MapFieldPoly.f"><span class="id" title="variable">f</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.poly.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.poly.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="size_map_poly"><span class="id" title="lemma">size_map_poly</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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">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.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="lead_coef_map"><span class="id" title="lemma">lead_coef_map</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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">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.poly.html#MapFieldPoly.f"><span class="id" title="variable">f</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.poly.html#p"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_poly_eq0"><span class="id" title="lemma">map_poly_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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">f</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.poly.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="map_poly_inj"><span class="id" title="lemma">map_poly_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapFieldPoly.f"><span class="id" title="variable">f</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_monic"><span class="id" title="lemma">map_monic</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.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.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#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_poly_com"><span class="id" title="lemma">map_poly_com</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#comm_poly"><span class="id" title="definition">comm_poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">f</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapFieldPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="fmorph_root"><span class="id" title="lemma">fmorph_root</span></a> <span class="id" title="var">p</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#6808a1f3c4c980ac2ccb289ca5e44206"><span class="id" title="notation">f</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapFieldPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="fmorph_unity_root"><span class="id" title="lemma">fmorph_unity_root</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">unity_root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapFieldPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">unity_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="fmorph_primitive_root"><span class="id" title="lemma">fmorph_primitive_root</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapFieldPoly.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MapFieldPoly"><span class="id" title="section">MapFieldPoly</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MaxRoots"><span class="id" title="section">MaxRoots</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="MaxRoots.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Exports.unitRingType"><span class="id" title="abbreviation">unitRingType</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.poly.html#MaxRoots.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">rs</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MaxRoots.R"><span class="id" title="variable">R</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#MaxRoots.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="diff_roots"><span class="id" title="definition">diff_roots</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.poly.html#MaxRoots.R"><span class="id" title="variable">R</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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.poly.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.poly.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Fixpoint</span> <a name="uniq_roots"><span class="id" title="definition">uniq_roots</span></a> <span class="id" title="var">rs</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">rs'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#diff_roots"><span class="id" title="definition">diff_roots</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">rs'</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#uniq_roots"><span class="id" title="definition">uniq_roots</span></a> <span class="id" title="var">rs'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="uniq_roots_prod_XsubC"><span class="id" title="lemma">uniq_roots_prod_XsubC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">rs</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.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#uniq_roots"><span class="id" title="definition">uniq_roots</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a 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#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">q</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><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#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</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.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a name="max_ring_poly_roots"><span class="id" title="lemma">max_ring_poly_roots</span></a> <span class="id" title="var">p</span> <span class="id" title="var">rs</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.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.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.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#uniq_roots"><span class="id" title="definition">uniq_roots</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</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.poly.html#p"><span class="id" title="variable">p</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="all_roots_prod_XsubC"><span class="id" title="lemma">all_roots_prod_XsubC</span></a> <span class="id" title="var">p</span> <span class="id" title="var">rs</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<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#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.ssrnat.html#361454269931ea8643f7b402f2ab7222"><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.poly.html#rs"><span class="id" title="variable">rs</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.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#uniq_roots"><span class="id" title="definition">uniq_roots</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a 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#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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><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#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</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.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MaxRoots"><span class="id" title="section">MaxRoots</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="FieldRoots"><span class="id" title="section">FieldRoots</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="FieldRoots.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.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">rs</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.F"><span class="id" title="variable">F</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="poly2_root"><span class="id" title="lemma">poly2_root</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.poly.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> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.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.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="uniq_rootsE"><span class="id" title="lemma">uniq_rootsE</span></a> <span class="id" title="var">rs</span> : <a class="idref" href="mathcomp.algebra.poly.html#uniq_roots"><span class="id" title="definition">uniq_roots</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.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#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="FieldRoots.UnityRoots"><span class="id" title="section">UnityRoots</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="max_unity_roots"><span class="id" title="lemma">max_unity_roots</span></a> <span class="id" title="var">rs</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">unity_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="mem_unity_roots"><span class="id" title="lemma">mem_unity_roots</span></a> <span class="id" title="var">rs</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">unity_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a> <a class="idref" href="http://coq.inria.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#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#da242fdb489aa84109b1eed313339a83"><span class="id" title="notation">unity_root</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#20bf07099d6d8cf369383b22fd37862e"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.algebra.poly.html#rs"><span class="id" title="variable">rs</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Showing the existence of a primitive root requires the theory in cyclic.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="FieldRoots.UnityRoots.z"><span class="id" title="variable">z</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.F"><span class="id" title="variable">F</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="FieldRoots.UnityRoots.prim_z"><span class="id" title="variable">prim_z</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.z"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a name="FieldRoots.UnityRoots.zn"><span class="id" title="variable">zn</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#b7adbae1ad6b5f8e6d4ef64ae286f319"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#b7adbae1ad6b5f8e6d4ef64ae286f319"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b7adbae1ad6b5f8e6d4ef64ae286f319"><span class="id" title="notation">|</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#b7adbae1ad6b5f8e6d4ef64ae286f319"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#index_iota"><span class="id" title="definition">index_iota</span></a> 0 <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#b7adbae1ad6b5f8e6d4ef64ae286f319"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="factor_Xn_sub_1"><span class="id" title="lemma">factor_Xn_sub_1</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">≤</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><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#d70623330b2787db6b196e37db7d8f45"><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#FieldRoots.UnityRoots.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</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.ssralg.html#792454e85a3eb4835c0ee22a75118f16"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.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#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="prim_rootP"><span class="id" title="lemma">prim_rootP</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">{</span></a><span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#9de6d53cccc27f521f3ab56b38159140"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#9de6d53cccc27f521f3ab56b38159140"><span class="id" title="notation">I_n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots.UnityRoots"><span class="id" title="section">UnityRoots</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#FieldRoots"><span class="id" title="section">FieldRoots</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MapPolyRoots"><span class="id" title="section">MapPolyRoots</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MapPolyRoots.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) (<a name="MapPolyRoots.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Exports.unitRingType"><span class="id" title="abbreviation">unitRingType</span></a>) (<a name="MapPolyRoots.f"><span class="id" title="variable">f</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.poly.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.poly.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_diff_roots"><span class="id" title="lemma">map_diff_roots</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.poly.html#diff_roots"><span class="id" title="definition">diff_roots</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#MapPolyRoots.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.poly.html#MapPolyRoots.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="map_uniq_roots"><span class="id" title="lemma">map_uniq_roots</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.algebra.poly.html#uniq_roots"><span class="id" title="definition">uniq_roots</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.algebra.poly.html#MapPolyRoots.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.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#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#MapPolyRoots"><span class="id" title="section">MapPolyRoots</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="AutPolyRoot"><span class="id" title="section">AutPolyRoot</span></a>.<br/>
+</div>
+
+<div class="doc">
+ The action of automorphisms on roots of unity.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="AutPolyRoot.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span> : <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.poly.html#AutPolyRoot.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.poly.html#AutPolyRoot.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0c709ebe43ddbd7719f75250a7b916d9"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="aut_prim_rootP"><span class="id" title="lemma">aut_prim_rootP</span></a> <span class="id" title="var">u</span> <span class="id" title="var">z</span> <span class="id" title="var">n</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">k</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="aut_unity_rootP"><span class="id" title="lemma">aut_unity_rootP</span></a> <span class="id" title="var">u</span> <span class="id" title="var">z</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">k</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="aut_unity_rootC"><span class="id" title="lemma">aut_unity_rootC</span></a> <span class="id" title="var">u</span> <span class="id" title="var">v</span> <span class="id" title="var">z</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">&gt;</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#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.poly.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#u"><span class="id" title="variable">u</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.poly.html#v"><span class="id" title="variable">v</span></a> (<a class="idref" href="mathcomp.algebra.poly.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#AutPolyRoot"><span class="id" title="section">AutPolyRoot</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a name="UnityRootTheory"><span class="id" title="module">UnityRootTheory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="c2b9762e4ad4f6f454c3d376a8ee5788"><span class="id" title="notation">&quot;</span></a>n .-unity_root" := (<a class="idref" href="mathcomp.algebra.poly.html#root_of_unity"><span class="id" title="definition">root_of_unity</span></a> <span class="id" title="var">n</span>) : <span class="id" title="var">unity_root_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="a3c684da3cc96e1b7db1e9715f63113c"><span class="id" title="notation">&quot;</span></a>n .-primitive_root" := (<a class="idref" href="mathcomp.algebra.poly.html#primitive_root_of_unity"><span class="id" title="definition">primitive_root_of_unity</span></a> <span class="id" title="var">n</span>) : <span class="id" title="var">unity_root_scope</span>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">unity_root_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.unity_rootE"><span class="id" title="definition">unity_rootE</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#unity_rootE"><span class="id" title="lemma">unity_rootE</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.unity_rootP"><span class="id" title="definition">unity_rootP</span></a> := @<a class="idref" href="mathcomp.algebra.poly.html#unity_rootP"><span class="id" title="lemma">unity_rootP</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.prim_order_exists"><span class="id" title="definition">prim_order_exists</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#prim_order_exists"><span class="id" title="lemma">prim_order_exists</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="UnityRootTheory.prim_order_gt0"><span class="id" title="abbreviation">prim_order_gt0</span></a> :=  <a class="idref" href="mathcomp.algebra.poly.html#prim_order_gt0"><span class="id" title="lemma">prim_order_gt0</span></a>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="UnityRootTheory.prim_expr_order"><span class="id" title="abbreviation">prim_expr_order</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#prim_expr_order"><span class="id" title="lemma">prim_expr_order</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.prim_expr_mod"><span class="id" title="definition">prim_expr_mod</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#prim_expr_mod"><span class="id" title="lemma">prim_expr_mod</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.prim_order_dvd"><span class="id" title="definition">prim_order_dvd</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#prim_order_dvd"><span class="id" title="lemma">prim_order_dvd</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.eq_prim_root_expr"><span class="id" title="definition">eq_prim_root_expr</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#eq_prim_root_expr"><span class="id" title="lemma">eq_prim_root_expr</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.rmorph_unity_root"><span class="id" title="definition">rmorph_unity_root</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#rmorph_unity_root"><span class="id" title="lemma">rmorph_unity_root</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.fmorph_unity_root"><span class="id" title="definition">fmorph_unity_root</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#fmorph_unity_root"><span class="id" title="lemma">fmorph_unity_root</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.fmorph_primitive_root"><span class="id" title="definition">fmorph_primitive_root</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#fmorph_primitive_root"><span class="id" title="lemma">fmorph_primitive_root</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.max_unity_roots"><span class="id" title="definition">max_unity_roots</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#max_unity_roots"><span class="id" title="lemma">max_unity_roots</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.mem_unity_roots"><span class="id" title="definition">mem_unity_roots</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#mem_unity_roots"><span class="id" title="lemma">mem_unity_roots</span></a>.<br/>
+<span class="id" title="keyword">Definition</span> <a name="UnityRootTheory.prim_rootP"><span class="id" title="definition">prim_rootP</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#prim_rootP"><span class="id" title="lemma">prim_rootP</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#UnityRootTheory"><span class="id" title="module">UnityRootTheory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="DecField"><span class="id" title="section">DecField</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="DecField.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.DecidableField.Exports.decFieldType"><span class="id" title="abbreviation">decFieldType</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="dec_factor_theorem"><span class="id" title="lemma">dec_factor_theorem</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#DecField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">{</span></a><span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#DecField.F"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">{</span></a><span class="id" title="var">q</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><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#DecField.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.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.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#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><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#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.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.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a><br/>
+&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;<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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.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> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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.poly.html#q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#DecField"><span class="id" title="section">DecField</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a name="PreClosedField"><span class="id" title="module">PreClosedField</span></a>.<br/>
+<span class="id" title="keyword">Section</span> <a name="PreClosedField.UseAxiom"><span class="id" title="section">UseAxiom</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="PreClosedField.UseAxiom.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>.<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="PreClosedField.UseAxiom.closedF"><span class="id" title="variable">closedF</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ClosedField.axiom"><span class="id" title="definition">GRing.ClosedField.axiom</span></a> <a class="idref" href="mathcomp.algebra.poly.html#PreClosedField.UseAxiom.F"><span class="id" title="variable">F</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.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#PreClosedField.UseAxiom.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="PreClosedField.closed_rootP"><span class="id" title="lemma">closed_rootP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</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.poly.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> 1%<span class="id" title="var">N</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="PreClosedField.closed_nonrootP"><span class="id" title="lemma">closed_nonrootP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.poly.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).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PreClosedField.UseAxiom"><span class="id" title="section">UseAxiom</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#PreClosedField"><span class="id" title="module">PreClosedField</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ClosedField"><span class="id" title="section">ClosedField</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="ClosedField.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ClosedField.Exports.closedFieldType"><span class="id" title="abbreviation">closedFieldType</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ClosedField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a name="ClosedField.closedF"><span class="id" title="variable">closedF</span></a> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Theory.solve_monicpoly"><span class="id" title="definition">solve_monicpoly</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ClosedField.F"><span class="id" title="variable">F</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="closed_rootP"><span class="id" title="lemma">closed_rootP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</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.poly.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> 1%<span class="id" title="var">N</span>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="closed_nonrootP"><span class="id" title="lemma">closed_nonrootP</span></a> <span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.poly.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.algebra.poly.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).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="closed_field_poly_normal"><span class="id" title="lemma">closed_field_poly_normal</span></a> <span class="id" title="var">p</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">{</span></a><span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ClosedField.F"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.poly.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#lead_coef"><span class="id" title="definition">lead_coef</span></a> <a class="idref" href="mathcomp.algebra.poly.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#81f8078534dcbb7e13a32d292f766525"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">&lt;-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><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#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#z"><span class="id" title="variable">z</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.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.poly.html#ClosedField"><span class="id" title="section">ClosedField</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
+\ No newline at end of file
author	Enrico Tassi	2018-04-20 10:54:22 +0200
committer	Enrico Tassi	2018-04-20 10:54:22 +0200
commit	ed05182cece6bb3706e09b2ce14af4a41a2e8141 (patch)
tree	e850d7314b6372d0476cf2ffaf7d3830721db7b1 /docs/htmldoc/mathcomp.algebra.poly.html
parent	3d196f44681fb3b23ff8a79fbd44e12308680531 (diff)