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diff --git a/docs/htmldoc/mathcomp.algebra.zmodp.html b/docs/htmldoc/mathcomp.algebra.zmodp.html new file mode 100644 index 0000000..f4141db --- /dev/null +++ b/docs/htmldoc/mathcomp.algebra.zmodp.html @@ -0,0 +1,422 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml"> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> +<link href="coqdoc.css" rel="stylesheet" type="text/css" /> +<title>mathcomp.algebra.zmodp</title> +</head> + +<body> + +<div id="page"> + +<div id="header"> +</div> + +<div id="main"> + +<h1 class="libtitle">Library mathcomp.algebra.zmodp</h1> + +<div class="code"> +<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> + Distributed under the terms of CeCILL-B. *)</span><br/> +<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Definition of the additive group and ring Zp, represented as 'I_p +<div class="paragraph"> </div> + + Definitions: + From fintype.v: + 'I_p == the subtype of integers less than p, taken here as the type of + the integers mod p. + This file: + inZp == the natural projection from nat into the integers mod p, + represented as 'I_p. Here p is implicit, but MUST be of the + form n.+1. + The operations: + Zp0 == the identity element for addition + Zp1 == the identity element for multiplication, and a generator of + additive group + Zp_opp == inverse function for addition + Zp_add == addition + Zp_mul == multiplication + Zp_inv == inverse function for multiplication + Note that while 'I_n.+1 has canonical finZmodType and finGroupType + structures, only 'I_n.+2 has a canonical ring structure (it has, in fact, + a canonical finComUnitRing structure), and hence an associated + multiplicative unit finGroupType. To mitigate the issues caused by the + trivial "ring" (which is, indeed is NOT a ring in the ssralg/finalg + formalization), we define additional notation: + 'Z_p == the type of integers mod (max p 2); this is always a proper + ring, by constructions. Note that 'Z_p is provably equal to + 'I_p if p > 1, and convertible to 'I_p if p is of the form + n.+2. + Zp p == the subgroup of integers mod (max p 1) in 'Z_p; this is thus + is thus all of 'Z_p if p > 1, and else the trivial group. + units_Zp p == the group of all units of 'Z_p -- i.e., the group of + (multiplicative) automorphisms of Zp p. + We show that Zp and units_Zp are abelian, and compute their orders. + We use a similar technique to represent the prime fields: + 'F_p == the finite field of integers mod the first prime divisor of + maxn p 2. This is provably equal to 'Z_p and 'I_p if p is + provably prime, and indeed convertible to the above if p is + a concrete prime such as 2, 5 or 23. + Note finally that due to the canonical structures it is possible to use + 0%R instead of Zp0, and 1%R instead of Zp1 (for the latter, p must be of + the form n.+2, and 1%R : nat will simplify to 1%N). +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> + +<br/> +<span class="id" title="keyword">Local Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ZpDef"><span class="id" title="section">ZpDef</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + +<div class="paragraph"> </div> + + Mod p arithmetic on the finite set {0, 1, 2, ..., p - 1} + +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="ZpDef.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#nat"><span class="id" title="inductive">nat</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.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_p</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Standard injection; val (inZp i) = i %% p +</div> +<div class="code"> +<span class="id" title="keyword">Definition</span> <a name="inZp"><span class="id" title="definition">inZp</span></a> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#Ordinal"><span class="id" title="constructor">Ordinal</span></a> (<a class="idref" href="mathcomp.ssreflect.div.html#ltn_pmod"><span class="id" title="lemma">ltn_pmod</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#ltn0Sn"><span class="id" title="lemma">ltn0Sn</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ZpDef.p'"><span class="id" title="variable">p'</span></a>)).<br/> +<span class="id" title="keyword">Lemma</span> <a name="modZp"><span class="id" title="lemma">modZp</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#2179ac53e82aa7c0b2f2f5a16b5510ea"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="abbreviation">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.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="valZpK"><span class="id" title="lemma">valZpK</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Operations +</div> +<div class="code"> +<span class="id" title="keyword">Definition</span> <a name="Zp0"><span class="id" title="definition">Zp0</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_p</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#ord0"><span class="id" title="definition">ord0</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp1"><span class="id" title="definition">Zp1</span></a> := <a class="idref" href="mathcomp.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> 1.<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp_opp"><span class="id" title="definition">Zp_opp</span></a> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="abbreviation">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.zmodp.html#x"><span class="id" title="variable">x</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp_add"><span class="id" title="definition">Zp_add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> := <a class="idref" href="mathcomp.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#y"><span class="id" title="variable">y</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp_mul"><span class="id" title="definition">Zp_mul</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> := <a class="idref" href="mathcomp.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#y"><span class="id" title="variable">y</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp_inv"><span class="id" title="definition">Zp_inv</span></a> <span class="id" title="var">x</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</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.zmodp.html#p"><span class="id" title="abbreviation">p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.div.html#egcdn"><span class="id" title="definition">egcdn</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="abbreviation">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">).1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Additive group structure. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_add0z"><span class="id" title="lemma">Zp_add0z</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> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp0"><span class="id" title="definition">Zp0</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_add"><span class="id" title="definition">Zp_add</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_addNz"><span class="id" title="lemma">Zp_addNz</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> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp0"><span class="id" title="definition">Zp0</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_opp"><span class="id" title="definition">Zp_opp</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_add"><span class="id" title="definition">Zp_add</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_addA"><span class="id" title="lemma">Zp_addA</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.zmodp.html#Zp_add"><span class="id" title="definition">Zp_add</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_addC"><span class="id" title="lemma">Zp_addC</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.zmodp.html#Zp_add"><span class="id" title="definition">Zp_add</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp_zmodMixin"><span class="id" title="definition">Zp_zmodMixin</span></a> := <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.zmodp.html#Zp_addA"><span class="id" title="lemma">Zp_addA</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_addC"><span class="id" title="lemma">Zp_addC</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_add0z"><span class="id" title="lemma">Zp_add0z</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_addNz"><span class="id" title="lemma">Zp_addNz</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_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.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_p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_zmodMixin"><span class="id" title="definition">Zp_zmodMixin</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_finZmodType</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.finalg.html#2980bb304205aec85bc1eeb5d0a573a5"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#2980bb304205aec85bc1eeb5d0a573a5"><span class="id" title="notation">finZmodType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#2980bb304205aec85bc1eeb5d0a573a5"><span class="id" title="notation">of</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_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#2980bb304205aec85bc1eeb5d0a573a5"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_baseFinGroupType</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.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">baseFinGroupType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">of</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_p</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_finGroupType</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.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">finGroupType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">of</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_p</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Ring operations +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mul1z"><span class="id" title="lemma">Zp_mul1z</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> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mulC"><span class="id" title="lemma">Zp_mulC</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.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mulz1"><span class="id" title="lemma">Zp_mulz1</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> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mulA"><span class="id" title="lemma">Zp_mulA</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.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mul_addr"><span class="id" title="lemma">Zp_mul_addr</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.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_add"><span class="id" title="definition">Zp_add</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mul_addl"><span class="id" title="lemma">Zp_mul_addl</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.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_add"><span class="id" title="definition">Zp_add</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mulVz"><span class="id" title="lemma">Zp_mulVz</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="abbreviation">p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_inv"><span class="id" title="definition">Zp_inv</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mulzV"><span class="id" title="lemma">Zp_mulzV</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="abbreviation">p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_inv"><span class="id" title="definition">Zp_inv</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_intro_unit"><span class="id" title="lemma">Zp_intro_unit</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_mul"><span class="id" title="definition">Zp_mul</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a> <a class="idref" href="http://coq.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.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="abbreviation">p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_inv_out"><span class="id" title="lemma">Zp_inv_out</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.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="abbreviation">p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#Zp_inv"><span class="id" title="definition">Zp_inv</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mulrn"><span class="id" title="lemma">Zp_mulrn</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#891e51846c7d1d63a9cb5458374cf308"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.zmodp.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.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#n"><span class="id" title="variable">n</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_mulgC"><span class="id" title="lemma">Zp_mulgC</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.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_p</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#mulg"><span class="id" title="definition">mulg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_abelian"><span class="id" title="lemma">Zp_abelian</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><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_p</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_expg"><span class="id" title="lemma">Zp_expg</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#86a04fb564fb97d388cad84a3a204260"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.zmodp.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.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#n"><span class="id" title="variable">n</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp1_expgz"><span class="id" title="lemma">Zp1_expgz</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#86a04fb564fb97d388cad84a3a204260"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span 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.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_cycle"><span class="id" title="lemma">Zp_cycle</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation"><[</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">]></span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="order_Zp1"><span class="id" title="lemma">order_Zp1</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#89402f0d9375903caa99ad84144160d5"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#89402f0d9375903caa99ad84144160d5"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="abbreviation">p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.zmodp.html#ZpDef"><span class="id" title="section">ZpDef</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ord1"><span class="id" title="lemma">ord1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#all_equal_to"><span class="id" title="definition">all_equal_to</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.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_1</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="lshift0"><span class="id" title="lemma">lshift0</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#lshift"><span class="id" title="definition">lshift</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#m"><span class="id" title="variable">m</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.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.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> <a 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.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><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_</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.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.zmodp.html#m"><span class="id" title="variable">m</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="rshift1"><span class="id" title="lemma">rshift1</span></a> <span class="id" title="var">n</span> : @<a class="idref" href="mathcomp.ssreflect.fintype.html#rshift"><span class="id" title="definition">rshift</span></a> 1 <a class="idref" href="mathcomp.algebra.zmodp.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#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#lift"><span class="id" title="definition">lift</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.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.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="split1"><span class="id" title="lemma">split1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">i</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.fintype.html#split"><span class="id" title="definition">split</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.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="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_</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#9de6d53cccc27f521f3ab56b38159140"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#b3eea360671e1b32b18a26e15b3aace3"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#9de6d53cccc27f521f3ab56b38159140"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.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.ssrfun.html#oapp"><span class="id" title="abbreviation">oapp</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">_</span> 0) (<a class="idref" href="mathcomp.ssreflect.fintype.html#unlift"><span class="id" title="definition">unlift</span></a> 0 <a class="idref" href="mathcomp.algebra.zmodp.html#i"><span class="id" title="variable">i</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="big_ord1"><span class="id" title="lemma">big_ord1</span></a> <span class="id" title="var">R</span> <span class="id" title="var">idx</span> (<span class="id" title="var">op</span> : @<a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.law"><span class="id" title="record">Monoid.law</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#idx"><span class="id" title="variable">idx</span></a>) <span class="id" title="var">F</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#op"><span class="id" title="variable">op</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#idx"><span class="id" title="variable">idx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation"><</span></a> 1<a class="idref" href="mathcomp.ssreflect.bigop.html#567079cee6eb2eba482323c7e8d08df5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.zmodp.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.zmodp.html#F"><span class="id" title="variable">F</span></a> 0.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="big_ord1_cond"><span class="id" title="lemma">big_ord1_cond</span></a> <span class="id" title="var">R</span> <span class="id" title="var">idx</span> (<span class="id" title="var">op</span> : @<a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.law"><span class="id" title="record">Monoid.law</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#idx"><span class="id" title="variable">idx</span></a>) <span class="id" title="var">P</span> <span class="id" title="var">F</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#op"><span class="id" title="variable">op</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#idx"><span class="id" title="variable">idx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation"><</span></a> 1 <a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#dc42c7ad0ea9096c0f795649807315df"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.algebra.zmodp.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.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#P"><span class="id" title="variable">P</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.algebra.zmodp.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.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#idx"><span class="id" title="variable">idx</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ZpRing"><span class="id" title="section">ZpRing</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="ZpRing.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#nat"><span class="id" title="inductive">nat</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_nontrivial"><span class="id" title="lemma">Zp_nontrivial</span></a> : <a class="idref" href="mathcomp.algebra.zmodp.html#Zp1"><span class="id" title="definition">Zp1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.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_p</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp_ringMixin"><span class="id" title="definition">Zp_ringMixin</span></a> :=<br/> + <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Exports.ComRingMixin"><span class="id" title="abbreviation">ComRingMixin</span></a> (@<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_mulA"><span class="id" title="lemma">Zp_mulA</span></a> <span class="id" title="var">_</span>) (@<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_mulC"><span class="id" title="lemma">Zp_mulC</span></a> <span class="id" title="var">_</span>) (@<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_mul1z"><span class="id" title="lemma">Zp_mul1z</span></a> <span class="id" title="var">_</span>) (@<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_mul_addl"><span class="id" title="lemma">Zp_mul_addl</span></a> <span class="id" title="var">_</span>)<br/> + <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_nontrivial"><span class="id" title="lemma">Zp_nontrivial</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_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.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_p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_ringMixin"><span class="id" title="definition">Zp_ringMixin</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_finRingType</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.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">finRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">of</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_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_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.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_p</span></a> (@<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_mulC"><span class="id" title="lemma">Zp_mulC</span></a> <span class="id" title="var">_</span>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_finComRingType</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.finalg.html#381777e14bce98b548cb274563c7fc56"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#381777e14bce98b548cb274563c7fc56"><span class="id" title="notation">finComRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#381777e14bce98b548cb274563c7fc56"><span class="id" title="notation">of</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_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#381777e14bce98b548cb274563c7fc56"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp_unitRingMixin"><span class="id" title="definition">Zp_unitRingMixin</span></a> :=<br/> + <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.zmodp.html#Zp_mulVz"><span class="id" title="lemma">Zp_mulVz</span></a> <span class="id" title="var">_</span>) (@<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_intro_unit"><span class="id" title="lemma">Zp_intro_unit</span></a> <span class="id" title="var">_</span>) (@<a class="idref" href="mathcomp.algebra.zmodp.html#Zp_inv_out"><span class="id" title="lemma">Zp_inv_out</span></a> <span class="id" title="var">_</span>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_unitRingType</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.UnitRing.Exports.UnitRingType"><span class="id" title="abbreviation">UnitRingType</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_p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_unitRingMixin"><span class="id" title="definition">Zp_unitRingMixin</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_finUnitRingType</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.finalg.html#7f21453830587186138043335ab91dd1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#7f21453830587186138043335ab91dd1"><span class="id" title="notation">finUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#7f21453830587186138043335ab91dd1"><span class="id" title="notation">of</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_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#7f21453830587186138043335ab91dd1"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_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.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_p</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">Zp_finComUnitRingType</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.finalg.html#f0aa4fcf143660f4378ecfead8f3fdda"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f0aa4fcf143660f4378ecfead8f3fdda"><span class="id" title="notation">finComUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#f0aa4fcf143660f4378ecfead8f3fdda"><span class="id" title="notation">of</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_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f0aa4fcf143660f4378ecfead8f3fdda"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_nat"><span class="id" title="lemma">Zp_nat</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> <a class="idref" href="mathcomp.algebra.zmodp.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.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_p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="natr_Zp"><span class="id" title="lemma">natr_Zp</span></a> (<span class="id" title="var">x</span> : <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_p</span></a>) : <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="natr_negZp"><span class="id" title="lemma">natr_negZp</span></a> (<span class="id" title="var">x</span> : <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_p</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#941c6d086004545bd62614d0213e75e5"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.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.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="unit_Zp_mulgC"><span class="id" title="lemma">unit_Zp_mulgC</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.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">unit</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_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">}</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#mulg"><span class="id" title="definition">mulg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="unit_Zp_expg"><span class="id" title="lemma">unit_Zp_expg</span></a> (<span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">unit</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_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">}</span></a>) <span class="id" title="var">n</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> (<a class="idref" href="mathcomp.algebra.zmodp.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#86a04fb564fb97d388cad84a3a204260"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.zmodp.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.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.zmodp.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.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_p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.zmodp.html#ZpRing"><span class="id" title="section">ZpRing</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp_trunc"><span class="id" title="definition">Zp_trunc</span></a> <span class="id" title="var">p</span> := <a class="idref" href="mathcomp.algebra.zmodp.html#p"><span class="id" title="variable">p</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">Notation</span> <a name="9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">"</span></a>''Z_' p" := <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_</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#Zp_trunc"><span class="id" title="definition">Zp_trunc</span></a> <span class="id" title="var">p</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">).+2</span></a><br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">p</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''Z_' p") : <span class="id" title="var">type_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">"</span></a>''F_' p" := <a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">Z_</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#pdiv"><span class="id" title="definition">pdiv</span></a> <span class="id" title="var">p</span><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">)</span></a><br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">p</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''F_' p") : <span class="id" title="var">type_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Groups"><span class="id" title="section">Groups</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Groups.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#nat"><span class="id" title="inductive">nat</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Zp"><span class="id" title="definition">Zp</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.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><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#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">Z_p</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><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">else</span></a> 1%<span class="id" title="var">g</span>.<br/> +<span class="id" title="keyword">Definition</span> <a name="units_Zp"><span class="id" title="definition">units_Zp</span></a> := <a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">unit</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">Z_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">}</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_cast"><span class="id" title="lemma">Zp_cast</span></a> : <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><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.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#Zp_trunc"><span class="id" title="definition">Zp_trunc</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">).+2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="val_Zp_nat"><span class="id" title="lemma">val_Zp_nat</span></a> (<span class="id" title="var">p_gt1</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 1) <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">Z_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="mathcomp.algebra.zmodp.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#2179ac53e82aa7c0b2f2f5a16b5510ea"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></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="Zp_nat_mod"><span class="id" title="lemma">Zp_nat_mod</span></a> (<span class="id" title="var">p_gt1</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 1)<span class="id" title="var">m</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#2179ac53e82aa7c0b2f2f5a16b5510ea"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="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.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">Z_p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_Zp"><span class="id" title="lemma">char_Zp</span></a> : <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><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.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">Z_p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="unitZpE"><span class="id" title="lemma">unitZpE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="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="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <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.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#9daeb9ead3dc7cfd1f9338b8de9c8c09"><span class="id" title="notation">Z_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">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.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.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Zp_group_set"><span class="id" title="lemma">Zp_group_set</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp"><span class="id" title="definition">Zp</span></a>.<br/> + <span class="id" title="keyword">Canonical</span> <span class="id" title="var">Zp_group</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#Group"><span class="id" title="constructor">Group</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Zp_group_set"><span class="id" title="lemma">Zp_group_set</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="card_Zp"><span class="id" title="lemma">card_Zp</span></a> : <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span 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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#Zp"><span class="id" title="definition">Zp</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mem_Zp"><span class="id" title="lemma">mem_Zp</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><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.zmodp.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.zmodp.html#Zp"><span class="id" title="definition">Zp</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">units_Zp_group</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#units_Zp"><span class="id" title="definition">units_Zp</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="card_units_Zp"><span class="id" title="lemma">card_units_Zp</span></a> : <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span 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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#units_Zp"><span class="id" title="definition">units_Zp</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.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.prime.html#totient"><span class="id" title="definition">totient</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Groups.p"><span class="id" title="variable">p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="units_Zp_abelian"><span class="id" title="lemma">units_Zp_abelian</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#units_Zp"><span class="id" title="definition">units_Zp</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.zmodp.html#Groups"><span class="id" title="section">Groups</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Field structure for primes. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="PrimeField"><span class="id" title="section">PrimeField</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="PrimeField.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#nat"><span class="id" title="inductive">nat</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="PrimeField.F_prime"><span class="id" title="section">F_prime</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="PrimeField.F_prime.p_pr"><span class="id" title="variable">p_pr</span></a> : <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Fp_Zcast"><span class="id" title="lemma">Fp_Zcast</span></a> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#Zp_trunc"><span class="id" title="definition">Zp_trunc</span></a> (<a class="idref" href="mathcomp.ssreflect.prime.html#pdiv"><span class="id" title="definition">pdiv</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a>)<a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">).+2</span></a> <a class="idref" href="http://coq.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#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#Zp_trunc"><span class="id" title="definition">Zp_trunc</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">).+2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Fp_cast"><span class="id" title="lemma">Fp_cast</span></a> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#Zp_trunc"><span class="id" title="definition">Zp_trunc</span></a> (<a class="idref" href="mathcomp.ssreflect.prime.html#pdiv"><span class="id" title="definition">pdiv</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a>)<a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">).+2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="card_Fp"><span class="id" title="lemma">card_Fp</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="val_Fp_nat"><span class="id" title="lemma">val_Fp_nat</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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_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="mathcomp.algebra.zmodp.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#2179ac53e82aa7c0b2f2f5a16b5510ea"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></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="Fp_nat_mod"><span class="id" title="lemma">Fp_nat_mod</span></a> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#2179ac53e82aa7c0b2f2f5a16b5510ea"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="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.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_Fp"><span class="id" title="lemma">char_Fp</span></a> : <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.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#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_Fp_0"><span class="id" title="lemma">char_Fp_0</span></a> : <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="unitFpE"><span class="id" title="lemma">unitFpE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="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="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <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.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_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">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#1e40fee506a85b20590ef299005b003d"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.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.zmodp.html#PrimeField.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField.F_prime"><span class="id" title="section">F_prime</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Fp_fieldMixin"><span class="id" title="lemma">Fp_fieldMixin</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.mixin_of"><span class="id" title="definition">GRing.Field.mixin_of</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#582a1127c53482b5000949a35912f7c9"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#582a1127c53482b5000949a35912f7c9"><span class="id" title="notation">the</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Exports.unitRingType"><span class="id" title="abbreviation">unitRingType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#582a1127c53482b5000949a35912f7c9"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#582a1127c53482b5000949a35912f7c9"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Fp_idomainMixin"><span class="id" title="definition">Fp_idomainMixin</span></a> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.FieldIdomainMixin"><span class="id" title="abbreviation">FieldIdomainMixin</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Fp_fieldMixin"><span class="id" title="lemma">Fp_fieldMixin</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fp_idomainType</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.IntegralDomain.Exports.IdomainType"><span class="id" title="abbreviation">IdomainType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Fp_idomainMixin"><span class="id" title="definition">Fp_idomainMixin</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fp_finIdomainType</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.finalg.html#6c49b73b4d6aa1a932fafe7684bba39c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#6c49b73b4d6aa1a932fafe7684bba39c"><span class="id" title="notation">finIdomainType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#6c49b73b4d6aa1a932fafe7684bba39c"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#6c49b73b4d6aa1a932fafe7684bba39c"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fp_fieldType</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.Field.Exports.FieldType"><span class="id" title="abbreviation">FieldType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Fp_fieldMixin"><span class="id" title="lemma">Fp_fieldMixin</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fp_finFieldType</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.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">finFieldType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a><a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fp_decFieldType</span> :=<br/> + <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.algebra.ssralg.html#571e046df0f3cfb95cda10363e01c19e"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#571e046df0f3cfb95cda10363e01c19e"><span class="id" title="notation">decFieldType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#571e046df0f3cfb95cda10363e01c19e"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#571e046df0f3cfb95cda10363e01c19e"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#Fp_finFieldType"><span class="id" title="definition">Fp_finFieldType</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#571e046df0f3cfb95cda10363e01c19e"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.algebra.zmodp.html#PrimeField"><span class="id" title="section">PrimeField</span></a>.<br/> +</div> +</div> + +<div id="footer"> +<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> +</div> + +</div> + +</body> +</html>
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