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diff --git a/docs/htmldoc/mathcomp.solvable.nilpotent.html b/docs/htmldoc/mathcomp.solvable.nilpotent.html deleted file mode 100644 index dbee59d..0000000 --- a/docs/htmldoc/mathcomp.solvable.nilpotent.html +++ /dev/null @@ -1,564 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" -"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml"> -<head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> -<link href="coqdoc.css" rel="stylesheet" type="text/css" /> -<title>mathcomp.solvable.nilpotent</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.solvable.nilpotent</h1> - -<div class="code"> -<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> - Distributed under the terms of CeCILL-B. *)</span><br/> - -<br/> -</div> - -<div class="doc"> - This file defines nilpotent and solvable groups, and give some of their - elementary properties; more will be added later (e.g., the nilpotence of - p-groups in sylow.v, or the fact that minimal normal subgroups of solvable - groups are elementary abelian in maximal.v). This file defines: - nilpotent G == G is nilpotent, i.e., [~: H, G] is a proper subgroup of H - for all nontrivial H <| G. - solvable G == G is solvable, i.e., H^`(1) is a proper subgroup of H for - all nontrivial subgroups H of G. - 'L_n(G) == the nth term of the lower central series, namely - [~: G, ..., G] (n Gs) if n > 0, with 'L_0(G) = G. - G is nilpotent iff 'L_n(G) = 1 for some n. - 'Z_n(G) == the nth term of the upper central series, i.e., - with 'Z_0(G) = 1, 'Z_n.+1(G) / 'Z_n(G) = 'Z(G / 'Z_n(G)). - nil_class G == the nilpotence class of G, i.e., the least n such that - 'L_n.+1(G) = 1 (or, equivalently, 'Z_n(G) = G), if G is - nilpotent; we take nil_class G = #|G| when G is not - nilpotent, so nil_class G < #|G| iff G is nilpotent. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="SeriesDefs"><span class="id" title="section">SeriesDefs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="SeriesDefs.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="SeriesDefs.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="SeriesDefs.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="lower_central_at_rec"><span class="id" title="definition">lower_central_at_rec</span></a> := <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#SeriesDefs.n"><span class="id" title="variable">n</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#SeriesDefs.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a>) <a class="idref" href="mathcomp.solvable.nilpotent.html#SeriesDefs.A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="upper_central_at_rec"><span class="id" title="definition">upper_central_at_rec</span></a> := <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#SeriesDefs.n"><span class="id" title="variable">n</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">B</span> ⇒ <a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#320f70d30c9a649ec82642b364681418"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#SeriesDefs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>) 1.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#SeriesDefs"><span class="id" title="section">SeriesDefs</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - By convention, the lower central series starts at 1 while the upper series - starts at 0 (sic). -</div> -<div class="code"> -<span class="id" title="keyword">Definition</span> <a name="lower_central_at"><span class="id" title="definition">lower_central_at</span></a> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.solvable.nilpotent.html#lower_central_at_rec"><span class="id" title="definition">lower_central_at_rec</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">.-1</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Note: 'nosimpl' MUST be used outside of a section -- the end of section - "cooking" destroys it. -</div> -<div class="code"> -<span class="id" title="keyword">Definition</span> <a name="upper_central_at"><span class="id" title="definition">upper_central_at</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#nosimpl"><span class="id" title="abbreviation">nosimpl</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#upper_central_at_rec"><span class="id" title="definition">upper_central_at_rec</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">"</span></a>''L_' n ( G )" := (<a class="idref" href="mathcomp.solvable.nilpotent.html#lower_central_at"><span class="id" title="definition">lower_central_at</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''L_' n ( G )") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">"</span></a>''Z_' n ( G )" := (<a class="idref" href="mathcomp.solvable.nilpotent.html#upper_central_at"><span class="id" title="definition">upper_central_at</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''Z_' n ( G )") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="PropertiesDefs"><span class="id" title="section">PropertiesDefs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="PropertiesDefs.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="PropertiesDefs.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="nilpotent"><span class="id" title="definition">nilpotent</span></a> :=<br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">∀</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">(</span></a><span class="id" title="var">G</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#PropertiesDefs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#PropertiesDefs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#PropertiesDefs.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f3be25edeb0349b0a76405eded9d0b98"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b91223a7636398c530555b2312d1e79b"><span class="id" title="notation">:==:</span></a> 1<a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="nil_class"><span class="id" title="definition">nil_class</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> 1 (<a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">n</span> ⇒ <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#PropertiesDefs.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#PropertiesDefs.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="solvable"><span class="id" title="definition">solvable</span></a> :=<br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">∀</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">(</span></a><span class="id" title="var">G</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#PropertiesDefs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#PropertiesDefs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f3be25edeb0349b0a76405eded9d0b98"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b91223a7636398c530555b2312d1e79b"><span class="id" title="notation">:==:</span></a> 1<a class="idref" href="mathcomp.ssreflect.fintype.html#b14c43cd248537980c3a1a815ab087df"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#PropertiesDefs"><span class="id" title="section">PropertiesDefs</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="NilpotentProps"><span class="id" title="section">NilpotentProps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="NilpotentProps.gT"><span class="id" title="variable">gT</span></a>: <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#NilpotentProps.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#NilpotentProps.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilpotent1"><span class="id" title="lemma">nilpotent1</span></a> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80a826bb5c5b3ef58870b90cd3030216"><span class="id" title="notation">[1</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#NilpotentProps.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#80a826bb5c5b3ef58870b90cd3030216"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilpotentS"><span class="id" title="lemma">nilpotentS</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nil_comm_properl"><span class="id" title="lemma">nil_comm_properl</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">A</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">N_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nil_comm_properr"><span class="id" title="lemma">nil_comm_properr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">N_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="centrals_nil"><span class="id" title="lemma">centrals_nil</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#NilpotentProps.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.gseries.html#cc81b5d388c2f555c988b815d6ad5f42"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#cc81b5d388c2f555c988b815d6ad5f42"><span class="id" title="notation">central</span></a><a class="idref" href="mathcomp.solvable.gseries.html#d20e6db07424254201d52f8314a47040"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#d20e6db07424254201d52f8314a47040"><span class="id" title="notation">series</span></a> 1%<span class="id" title="var">G</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> 1%<span class="id" title="var">G</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#NilpotentProps"><span class="id" title="section">NilpotentProps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="LowerCentral"><span class="id" title="section">LowerCentral</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="LowerCentral.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#LowerCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#LowerCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn0"><span class="id" title="lemma">lcn0</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_0</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn1"><span class="id" title="lemma">lcn1</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="lcnSn"><span class="id" title="lemma">lcnSn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bc63483175fac06bcd5541a5e9093b18"><span class="id" title="notation">.+2</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="lcnSnS"><span class="id" title="lemma">lcnSnS</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="lcnE"><span class="id" title="lemma">lcnE</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#lower_central_at_rec"><span class="id" title="definition">lower_central_at_rec</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="lcn2"><span class="id" title="lemma">lcn2</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_2</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_group_set"><span class="id" title="lemma">lcn_group_set</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">lower_central_at_group</span> <span class="id" title="var">n</span> <span class="id" title="var">G</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#Group"><span class="id" title="constructor">Group</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#lcn_group_set"><span class="id" title="lemma">lcn_group_set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_char"><span class="id" title="lemma">lcn_char</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_normal"><span class="id" title="lemma">lcn_normal</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_sub"><span class="id" title="lemma">lcn_sub</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_norm"><span class="id" title="lemma">lcn_norm</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_subS"><span class="id" title="lemma">lcn_subS</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_normalS"><span class="id" title="lemma">lcn_normalS</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_central"><span class="id" title="lemma">lcn_central</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_sub_leq"><span class="id" title="lemma">lcn_sub_leq</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_m</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcnS"><span class="id" title="lemma">lcnS</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_cprod"><span class="id" title="lemma">lcn_cprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_dprod"><span class="id" title="lemma">lcn_dprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="der_cprod"><span class="id" title="lemma">der_cprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="der_dprod"><span class="id" title="lemma">der_dprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_bigcprod"><span class="id" title="lemma">lcn_bigcprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">I</span> <span class="id" title="var">r</span> <span class="id" title="var">P</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#LowerCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_bigdprod"><span class="id" title="lemma">lcn_bigdprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">I</span> <span class="id" title="var">r</span> <span class="id" title="var">P</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#LowerCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="der_bigcprod"><span class="id" title="lemma">der_bigcprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">I</span> <span class="id" title="var">r</span> <span class="id" title="var">P</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#LowerCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="der_bigdprod"><span class="id" title="lemma">der_bigdprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">I</span> <span class="id" title="var">r</span> <span class="id" title="var">P</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#LowerCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilpotent_class"><span class="id" title="lemma">nilpotent_class</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_nil_classP"><span class="id" title="lemma">lcn_nil_classP</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1) (<a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcnP"><span class="id" title="lemma">lcnP</span></a> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1) (<a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="abelian_nil"><span class="id" title="lemma">abelian_nil</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nil_class0"><span class="id" title="lemma">nil_class0</span></a> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b91223a7636398c530555b2312d1e79b"><span class="id" title="notation">:==:</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nil_class1"><span class="id" title="lemma">nil_class1</span></a> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cprod_nil"><span class="id" title="lemma">cprod_nil</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mulg_nil"><span class="id" title="lemma">mulg_nil</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_nil"><span class="id" title="lemma">dprod_nil</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="bigdprod_nil"><span class="id" title="lemma">bigdprod_nil</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">A_</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#LowerCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#A_"><span class="id" title="variable">A_</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#LowerCentral"><span class="id" title="section">LowerCentral</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="b8882f631c26b24161bbbe8eed0d3c8b"><span class="id" title="notation">"</span></a>''L_' n ( G )" := (<a class="idref" href="mathcomp.solvable.nilpotent.html#lower_central_at_group"><span class="id" title="definition">lower_central_at_group</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span>) : <span class="id" title="var">Group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lcn_cont"><span class="id" title="lemma">lcn_cont</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous"><span class="id" title="definition">GFunctor.continuous</span></a> (@<a class="idref" href="mathcomp.solvable.nilpotent.html#lower_central_at"><span class="id" title="definition">lower_central_at</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">lcn_igFun</span> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">igFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#lcn_sub"><span class="id" title="lemma">lcn_sub</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#lcn_cont"><span class="id" title="lemma">lcn_cont</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">lcn_gFun</span> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">gFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#lcn_cont"><span class="id" title="lemma">lcn_cont</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">lcn_mgFun</span> <span class="id" title="var">n</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#f3da3221c5171e732a65fec8cc2ba4fa"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f3da3221c5171e732a65fec8cc2ba4fa"><span class="id" title="notation">mgFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#f3da3221c5171e732a65fec8cc2ba4fa"><span class="id" title="notation">by</span></a> <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> ⇒ @<a class="idref" href="mathcomp.solvable.nilpotent.html#lcnS"><span class="id" title="lemma">lcnS</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f3da3221c5171e732a65fec8cc2ba4fa"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="UpperCentralFunctor"><span class="id" title="section">UpperCentralFunctor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="UpperCentralFunctor.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_pmap"><span class="id" title="lemma">ucn_pmap</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">hZ</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pmap"><span class="id" title="record">GFunctor.pmap</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> @<a class="idref" href="mathcomp.solvable.nilpotent.html#upper_central_at"><span class="id" title="definition">upper_central_at</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor.n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#hZ"><span class="id" title="variable">hZ</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Now extract all the intermediate facts of the last proof. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_group_set"><span class="id" title="lemma">ucn_group_set</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</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.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">upper_central_at_group</span> <span class="id" title="var">gT</span> <span class="id" title="var">G</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#Group"><span class="id" title="constructor">Group</span></a> (@<a class="idref" href="mathcomp.solvable.nilpotent.html#ucn_group_set"><span class="id" title="lemma">ucn_group_set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_sub"><span class="id" title="lemma">ucn_sub</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_ucn"><span class="id" title="lemma">morphim_ucn</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous"><span class="id" title="definition">GFunctor.pcontinuous</span></a> (@<a class="idref" href="mathcomp.solvable.nilpotent.html#upper_central_at"><span class="id" title="definition">upper_central_at</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor.n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ucn_igFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">igFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#ucn_sub"><span class="id" title="lemma">ucn_sub</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#morphim_ucn"><span class="id" title="lemma">morphim_ucn</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ucn_gFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">gFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#morphim_ucn"><span class="id" title="lemma">morphim_ucn</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ucn_pgFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#f4a0ec7c18bd128b271d4428328fd43b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f4a0ec7c18bd128b271d4428328fd43b"><span class="id" title="notation">pgFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#f4a0ec7c18bd128b271d4428328fd43b"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#morphim_ucn"><span class="id" title="lemma">morphim_ucn</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f4a0ec7c18bd128b271d4428328fd43b"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="UpperCentralFunctor.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="UpperCentralFunctor.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_char"><span class="id" title="lemma">ucn_char</span></a> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor.G"><span class="id" title="variable">G</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_norm"><span class="id" title="lemma">ucn_norm</span></a> : <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">)</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_normal"><span class="id" title="lemma">ucn_normal</span></a> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor.G"><span class="id" title="variable">G</span></a>. <br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentralFunctor"><span class="id" title="section">UpperCentralFunctor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="2b049c2c133c0b952cef84279630b623"><span class="id" title="notation">"</span></a>''Z_' n ( G )" := (<a class="idref" href="mathcomp.solvable.nilpotent.html#upper_central_at_group"><span class="id" title="definition">upper_central_at_group</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span>) : <span class="id" title="var">Group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="UpperCentral"><span class="id" title="section">UpperCentral</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="UpperCentral.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn0"><span class="id" title="lemma">ucn0</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_0</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucnSn"><span class="id" title="lemma">ucnSn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#320f70d30c9a649ec82642b364681418"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucnE"><span class="id" title="lemma">ucnE</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#upper_central_at_rec"><span class="id" title="definition">upper_central_at_rec</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_subS"><span class="id" title="lemma">ucn_subS</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_sub_geq"><span class="id" title="lemma">ucn_sub_geq</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#73030c22bc0b1fa771c65aa5414c65f9"><span class="id" title="notation">≥</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_m</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_central"><span class="id" title="lemma">ucn_central</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_normalS"><span class="id" title="lemma">ucn_normalS</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_comm"><span class="id" title="lemma">ucn_comm</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn1"><span class="id" title="lemma">ucn1</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucnSnR"><span class="id" title="lemma">ucnSnR</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f5e197738ea7c8e266850f6045418c3f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#f5e197738ea7c8e266850f6045418c3f"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#f5e197738ea7c8e266850f6045418c3f"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_cprod"><span class="id" title="lemma">ucn_cprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_dprod"><span class="id" title="lemma">ucn_dprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_bigcprod"><span class="id" title="lemma">ucn_bigcprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">I</span> <span class="id" title="var">r</span> <span class="id" title="var">P</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_bigdprod"><span class="id" title="lemma">ucn_bigdprod</span></a> <span class="id" title="var">n</span> <span class="id" title="var">I</span> <span class="id" title="var">r</span> <span class="id" title="var">P</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentral.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_lcnP"><span class="id" title="lemma">ucn_lcnP</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucnP"><span class="id" title="lemma">ucnP</span></a> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_nil_classP"><span class="id" title="lemma">ucn_nil_classP</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_id"><span class="id" title="lemma">ucn_id</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">('</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ucn_nilpotent"><span class="id" title="lemma">ucn_nilpotent</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nil_class_ucn"><span class="id" title="lemma">nil_class_ucn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#UpperCentral"><span class="id" title="section">UpperCentral</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MorphNil"><span class="id" title="section">MorphNil</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MorphNil.aT"><span class="id" title="variable">aT</span></a> <a name="MorphNil.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="MorphNil.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="MorphNil.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_lcn"><span class="id" title="lemma">morphim_lcn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">L_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#e51b1b2b9737232f812351f799d286f3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_ucn"><span class="id" title="lemma">injm_ucn</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_nil"><span class="id" title="lemma">morphim_nil</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_nil"><span class="id" title="lemma">injm_nil</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nil_class_morphim"><span class="id" title="lemma">nil_class_morphim</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nil_class_injm"><span class="id" title="lemma">nil_class_injm</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphNil"><span class="id" title="section">MorphNil</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="QuotientNil"><span class="id" title="section">QuotientNil</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="QuotientNil.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#QuotientNil.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_ucn_add"><span class="id" title="lemma">quotient_ucn_add</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_m</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">Z_n</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#c1fbc0daf497c5c35b43512087ca0b96"><span class="id" title="notation">))</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_nil"><span class="id" title="lemma">isog_nil</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">G</span> (<span class="id" title="var">L</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#L"><span class="id" title="variable">L</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_nil_class"><span class="id" title="lemma">isog_nil_class</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">G</span> (<span class="id" title="var">L</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#L"><span class="id" title="variable">L</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_nil"><span class="id" title="lemma">quotient_nil</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_center_nil"><span class="id" title="lemma">quotient_center_nil</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nil_class_quotient_center"><span class="id" title="lemma">nil_class_quotient_center</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">).-1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilpotent_sub_norm"><span class="id" title="lemma">nilpotent_sub_norm</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">N_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f0bbce9238fab3dd03626439080979a9"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilpotent_proper_norm"><span class="id" title="lemma">nilpotent_proper_norm</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">N_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilpotent_subnormal"><span class="id" title="lemma">nilpotent_subnormal</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#52cdd2d549fa2e18a6c3687e75ce9c7c"><span class="id" title="notation"><|<|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="TI_center_nil"><span class="id" title="lemma">TI_center_nil</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f0bbce9238fab3dd03626439080979a9"><span class="id" title="notation">:=:</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="meet_center_nil"><span class="id" title="lemma">meet_center_nil</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="center_nil_eq1"><span class="id" title="lemma">center_nil_eq1</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b91223a7636398c530555b2312d1e79b"><span class="id" title="notation">:==:</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cyclic_nilpotent_quo_der1_cyclic"><span class="id" title="lemma">cyclic_nilpotent_quo_der1_cyclic</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#QuotientNil"><span class="id" title="section">QuotientNil</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Solvable"><span class="id" title="section">Solvable</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Solvable.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#Solvable.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilpotent_sol"><span class="id" title="lemma">nilpotent_sol</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="abelian_sol"><span class="id" title="lemma">abelian_sol</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="solvable1"><span class="id" title="lemma">solvable1</span></a> : <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80a826bb5c5b3ef58870b90cd3030216"><span class="id" title="notation">[1</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#Solvable.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#80a826bb5c5b3ef58870b90cd3030216"><span class="id" title="notation">]</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="solvableS"><span class="id" title="lemma">solvableS</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sol_der1_proper"><span class="id" title="lemma">sol_der1_proper</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="derivedP"><span class="id" title="lemma">derivedP</span></a> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a><a class="idref" href="mathcomp.solvable.nilpotent.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1) (<a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#Solvable"><span class="id" title="section">Solvable</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MorphSol"><span class="id" title="section">MorphSol</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MorphSol.gT"><span class="id" title="variable">gT</span></a> <a name="MorphSol.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="MorphSol.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="MorphSol.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Variable</span> <a name="MorphSol.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_sol"><span class="id" title="lemma">morphim_sol</span></a> : <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_sol"><span class="id" title="lemma">injm_sol</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#MorphSol"><span class="id" title="section">MorphSol</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="QuotientSol"><span class="id" title="section">QuotientSol</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="QuotientSol.gT"><span class="id" title="variable">gT</span></a> <a name="QuotientSol.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#QuotientSol.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_sol"><span class="id" title="lemma">isog_sol</span></a> <span class="id" title="var">G</span> (<span class="id" title="var">L</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#QuotientSol.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#L"><span class="id" title="variable">L</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_sol"><span class="id" title="lemma">quotient_sol</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="series_sol"><span class="id" title="lemma">series_sol</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> (<a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="metacyclic_sol"><span class="id" title="lemma">metacyclic_sol</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.cyclic.html#metacyclic"><span class="id" title="definition">metacyclic</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.nilpotent.html#QuotientSol"><span class="id" title="section">QuotientSol</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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