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| author | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
|---|---|---|
| committer | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
| commit | 6b59540a2460633df4e3d8347cb4dfe2fb3a3afb (patch) | |
| tree | 1239c1d5553d51a7d73f2f8b465f6a23178ff8a0 /docs/htmldoc/mathcomp.solvable.pgroup.html | |
| parent | dd82aaeae7e9478efc178ce8430986649555b032 (diff) | |
removing everything but index which redirects to the new page
Diffstat (limited to 'docs/htmldoc/mathcomp.solvable.pgroup.html')
| -rw-r--r-- | docs/htmldoc/mathcomp.solvable.pgroup.html | 1026 |
1 files changed, 0 insertions, 1026 deletions
diff --git a/docs/htmldoc/mathcomp.solvable.pgroup.html b/docs/htmldoc/mathcomp.solvable.pgroup.html deleted file mode 100644 index 50ff768..0000000 --- a/docs/htmldoc/mathcomp.solvable.pgroup.html +++ /dev/null @@ -1,1026 +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.pgroup</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.solvable.pgroup</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"> - Standard group notions and constructions based on the prime decomposition - of the order of the group or its elements: - pi.-group G <=> G is a pi-group, i.e., pi.-nat #|G|. -<ul class="doclist"> -<li>> Recall that here and in the sequel pi can be a single prime p. - -</li> -</ul> - pi.-subgroup(H) G <=> H is a pi-subgroup of G. - := (H \subset G) && pi.-group H. -<ul class="doclist"> -<li>> This is provided mostly as a shorhand, with few associated lemmas. - However, we do establish some results on maximal pi-subgroups. - pi.-elt x <=> x is a pi-element. - := pi.-nat # [x] or pi.-group < [x]>. - x.`<i>pi == the pi-constituent of x: the (unique) pi-element - y \in < [x]> such that x * y^-1 is a pi'-element. - pi.-Hall(G) H <=> H is a Hall pi-subgroup of G. - := [&& H \subset G, pi.-group H & pi^'.-nat #|G : H| ]. - -</li> -<li>> This is also eqivalent to H \subset G /\ #|H| = #|G|`<i>pi. - p.-Sylow(G) P <=> P is a Sylow p-subgroup of G. - -</li> -<li>> This is the display and preferred input notation for p.-Hall(G) P. - 'Syl_p(G) == the set of the p-Sylow subgroups of G. - := [set P : {group _} | p.-Sylow(G) P]. - p_group P <=> P is a p-group for some prime p. - Hall G H <=> H is a Hall pi-subgroup of G for some pi. - := coprime #|H| #|G : H| && (H \subset G). - Sylow G P <=> P is a Sylow p-subgroup of G for some p. - := p_group P && Hall G P. - 'O_pi(G) == the pi-core (largest normal pi-subgroup) of G. - -</li> -</ul> - pcore_mod pi G H == the pi-core of G mod H. - := G :&: (coset H @*^-1 'O_pi(G / H)). - 'O</i>{pi2, pi1}(G) == the pi1,pi2-core of G. - := the pi1-core of G mod 'O_pi2(G). -<ul class="doclist"> -<li>> We have 'O{pi2, pi1}(G) / 'O_pi2(G) = 'O_pi1(G / 'O_pi2(G)) - with 'O_pi2(G) <| 'O{pi2, pi1}(G) <| G. - -</li> -</ul> - 'O{pn, ..., p1}(G) == the p1, ..., pn-core of G. - := the p1-core of G mod 'O{pn, ..., p2}(G). - Note that notions are always defined on sets even though their name - indicates "group" properties; the actual definition of the notion never - tests for the group property, since this property will always be provided - by a (canonical) group structure. Similarly, p-group properties assume - without test that p is a prime. -</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="PgroupDefs"><span class="id" title="section">PgroupDefs</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - We defer the definition of the functors ('0_p(G), etc) because they need - to quantify over the finGroupType explicitly. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="PgroupDefs.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">Type</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PgroupDefs.gT"><span class="id" title="variable">gT</span></a>) (<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.pgroup.html#PgroupDefs.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">pi</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</span></a>) (<span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="pgroup"><span class="id" title="definition">pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">nat</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#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="psubgroup"><span class="id" title="definition">psubgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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="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.pgroup.html#pgroup"><span class="id" title="definition">pgroup</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="p_group"><span class="id" title="definition">p_group</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.solvable.pgroup.html#pgroup"><span class="id" title="definition">pgroup</span></a> (<a class="idref" href="mathcomp.ssreflect.prime.html#pdiv"><span class="id" title="definition">pdiv</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#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>) <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="p_elt"><span class="id" title="definition">p_elt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">nat</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="constt"><span class="id" title="definition">constt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">pi</span> := <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.div.html#chinese"><span class="id" title="definition">chinese</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">_pi</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a> 1 0<a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Hall"><span class="id" title="definition">Hall</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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="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.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="pHall"><span class="id" title="definition">pHall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pgroup"><span class="id" title="definition">pgroup</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">nat</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Syl"><span class="id" title="definition">Syl</span></a> <span class="id" title="var">p</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#6ac1c202d9451e0a66d4fc93e5d9b44b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#6ac1c202d9451e0a66d4fc93e5d9b44b"><span class="id" title="notation">set</span></a> <span class="id" title="var">P</span> <a class="idref" href="mathcomp.ssreflect.finset.html#6ac1c202d9451e0a66d4fc93e5d9b44b"><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.pgroup.html#PgroupDefs.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.finset.html#6ac1c202d9451e0a66d4fc93e5d9b44b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pHall"><span class="id" title="definition">pHall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#6ac1c202d9451e0a66d4fc93e5d9b44b"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Sylow"><span class="id" title="definition">Sylow</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> := <a class="idref" href="mathcomp.solvable.pgroup.html#p_group"><span class="id" title="definition">p_group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#PgroupDefs"><span class="id" title="section">PgroupDefs</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">"</span></a>pi .-group" := (<a class="idref" href="mathcomp.solvable.pgroup.html#pgroup"><span class="id" title="definition">pgroup</span></a> <span class="id" title="var">pi</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "pi .-group") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">"</span></a>pi .-subgroup ( A )" := (<a class="idref" href="mathcomp.solvable.pgroup.html#psubgroup"><span class="id" title="definition">psubgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "pi .-subgroup ( A )") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">"</span></a>pi .-elt" := (<a class="idref" href="mathcomp.solvable.pgroup.html#p_elt"><span class="id" title="definition">p_elt</span></a> <span class="id" title="var">pi</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "pi .-elt") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">"</span></a>x .`_ pi" := (<a class="idref" href="mathcomp.solvable.pgroup.html#constt"><span class="id" title="definition">constt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">pi</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 3, <span class="id" title="var">format</span> "x .`_ pi") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">"</span></a>pi .-Hall ( G )" := (<a class="idref" href="mathcomp.solvable.pgroup.html#pHall"><span class="id" title="definition">pHall</span></a> <span class="id" title="var">pi</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">format</span> "pi .-Hall ( G )") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">"</span></a>p .-Sylow ( G )" := <a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred_of_nat"><span class="id" title="definition">nat_pred_of_nat</span></a> <span class="id" title="var">p</span><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">).-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><span class="id" title="var">G</span><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "p .-Sylow ( G )") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="2cfdfd6f05a28cf984c6e60aeed6378f"><span class="id" title="notation">"</span></a>''Syl_' p ( G )" := (<a class="idref" href="mathcomp.solvable.pgroup.html#Syl"><span class="id" title="definition">Syl</span></a> <span class="id" title="var">p</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">p</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''Syl_' p ( G )") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="PgroupProps"><span class="id" title="section">PgroupProps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="PgroupProps.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">pi</span> <span class="id" title="var">rho</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</span></a>) (<span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PgroupProps.gT"><span class="id" title="variable">gT</span></a>) (<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> <span class="id" title="var">D</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.pgroup.html#PgroupProps.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> <span class="id" title="var">K</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.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.pgroup.html#PgroupProps.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="trivgVpdiv"><span class="id" title="lemma">trivgVpdiv</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.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> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><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.pgroup.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#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="prime_subgroupVti"><span class="id" title="lemma">prime_subgroupVti</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pgroupE"><span class="id" title="lemma">pgroupE</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">nat</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#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">Lemma</span> <a name="sub_pgroup"><span class="id" title="lemma">sub_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_pgroup"><span class="id" title="lemma">eq_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_p'group"><span class="id" title="lemma">eq_p'group</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pgroupNK"><span class="id" title="lemma">pgroupNK</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pi_pgroup"><span class="id" title="lemma">pi_pgroup</span></a> <span class="id" title="var">p</span> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pi_p'group"><span class="id" title="lemma">pi_p'group</span></a> <span class="id" title="var">p</span> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><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.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pi'_p'group"><span class="id" title="lemma">pi'_p'group</span></a> <span class="id" title="var">p</span> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.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.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p'groupEpi"><span class="id" title="lemma">p'groupEpi</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><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="pgroup_pi"><span class="id" title="lemma">pgroup_pi</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="partG_eq1"><span class="id" title="lemma">partG_eq1</span></a> <span class="id" title="var">pi</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.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">_pi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pgroupP"><span class="id" title="lemma">pgroupP</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">p</span>, <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><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.pgroup.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pgroup1"><span class="id" title="lemma">pgroup1</span></a> <span class="id" title="var">pi</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</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.pgroup.html#PgroupProps.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="pgroupS"><span class="id" title="lemma">pgroupS</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="oddSg"><span class="id" title="lemma">oddSg</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</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">Lemma</span> <a name="odd_pgroup_odd"><span class="id" title="lemma">odd_pgroup_odd</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_pgroup"><span class="id" title="lemma">card_pgroup</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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> (<a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="properG_ltn_log"><span class="id" title="lemma">properG_ltn_log</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pgroupM"><span class="id" title="lemma">pgroupM</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pgroupJ"><span class="id" title="lemma">pgroupJ</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pgroup_p"><span class="id" title="lemma">pgroup_p</span></a> <span class="id" title="var">p</span> <span class="id" title="var">P</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p_group"><span class="id" title="definition">p_group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_groupP"><span class="id" title="lemma">p_groupP</span></a> <span class="id" title="var">P</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#p_group"><span class="id" title="definition">p_group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pgroup_pdiv"><span class="id" title="lemma">pgroup_pdiv</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</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><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><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.pgroup.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.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</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">m</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.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="coprime_p'group"><span class="id" title="lemma">coprime_p'group</span></a> <span class="id" title="var">p</span> <span class="id" title="var">K</span> <span class="id" title="var">R</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><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.pgroup.html#R"><span class="id" title="variable">R</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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</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.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_Hall"><span class="id" title="lemma">card_Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</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> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">_pi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHall_sub"><span class="id" title="lemma">pHall_sub</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHall_pgroup"><span class="id" title="lemma">pHall_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHallP"><span class="id" title="lemma">pHallP</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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#ba2b0e492d2b4675a0acf3ea92aabadd"><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.pgroup.html#H"><span class="id" title="variable">H</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> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">_pi</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHallE"><span class="id" title="lemma">pHallE</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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="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.pgroup.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.pgroup.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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">_pi</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="coprime_mulpG_Hall"><span class="id" title="lemma">coprime_mulpG_Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">R</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="coprime_mulGp_Hall"><span class="id" title="lemma">coprime_mulGp_Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">R</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_pHall"><span class="id" title="lemma">eq_in_pHall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_pHall"><span class="id" title="lemma">eq_pHall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_p'Hall"><span class="id" title="lemma">eq_p'Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHallNK"><span class="id" title="lemma">pHallNK</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subHall_Hall"><span class="id" title="lemma">subHall_Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subHall_Sylow"><span class="id" title="lemma">subHall_Sylow</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">p</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">P</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.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.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">Sylow</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">Sylow</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHall_Hall"><span class="id" title="lemma">pHall_Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Hall_pi"><span class="id" title="lemma">Hall_pi</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="HallP"><span class="id" title="lemma">HallP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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="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">pi</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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_Hall"><span class="id" title="lemma">sdprod_Hall</span></a> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="coprime_sdprod_Hall_l"><span class="id" title="lemma">coprime_sdprod_Hall_l</span></a> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><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.pgroup.html#H"><span class="id" title="variable">H</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> <a class="idref" href="mathcomp.solvable.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="coprime_sdprod_Hall_r"><span class="id" title="lemma">coprime_sdprod_Hall_r</span></a> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><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.pgroup.html#H"><span class="id" title="variable">H</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> <a class="idref" href="mathcomp.solvable.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="compl_pHall"><span class="id" title="lemma">compl_pHall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.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.pgroup.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="compl_p'Hall"><span class="id" title="lemma">compl_p'Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.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.pgroup.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#745415cc965000e5f864d3e6bd67dced"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_normal_p'HallP"><span class="id" title="lemma">sdprod_normal_p'HallP</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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="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.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_normal_pHallP"><span class="id" title="lemma">sdprod_normal_pHallP</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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="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.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHallJ2"><span class="id" title="lemma">pHallJ2</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHallJnorm"><span class="id" title="lemma">pHallJnorm</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.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.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHallJ"><span class="id" title="lemma">pHallJ</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="HallJ"><span class="id" title="lemma">HallJ</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="psubgroupJ"><span class="id" title="lemma">psubgroupJ</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_groupJ"><span class="id" title="lemma">p_groupJ</span></a> <span class="id" title="var">P</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#p_group"><span class="id" title="definition">p_group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p_group"><span class="id" title="definition">p_group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="SylowJ"><span class="id" title="lemma">SylowJ</span></a> <span class="id" title="var">G</span> <span class="id" title="var">P</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Sylow"><span class="id" title="definition">Sylow</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Sylow"><span class="id" title="definition">Sylow</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_Sylow"><span class="id" title="lemma">p_Sylow</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> <span class="id" title="var">P</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">Sylow</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Sylow"><span class="id" title="definition">Sylow</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHall_subl"><span class="id" title="lemma">pHall_subl</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</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.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Hall1"><span class="id" title="lemma">Hall1</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_group1"><span class="id" title="lemma">p_group1</span></a> : @<a class="idref" href="mathcomp.solvable.pgroup.html#p_group"><span class="id" title="definition">p_group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#PgroupProps.gT"><span class="id" title="variable">gT</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Sylow1"><span class="id" title="lemma">Sylow1</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#Sylow"><span class="id" title="definition">Sylow</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="SylowP"><span class="id" title="lemma">SylowP</span></a> <span class="id" title="var">G</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#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">Sylow</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#Sylow"><span class="id" title="definition">Sylow</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_elt_exp"><span class="id" title="lemma">p_elt_exp</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_p_elt"><span class="id" title="lemma">mem_p_elt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_eltM_norm"><span class="id" title="lemma">p_eltM_norm</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.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.fingroup.fingroup.html#5e5825d099c952c2db2842c142cbde94"><span class="id" title="notation"><[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#5e5825d099c952c2db2842c142cbde94"><span class="id" title="notation">]></span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_eltM"><span class="id" title="lemma">p_eltM</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#commute"><span class="id" title="definition">commute</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_elt1"><span class="id" title="lemma">p_elt1</span></a> <span class="id" title="var">pi</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#PgroupProps.gT"><span class="id" title="variable">gT</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_eltV"><span class="id" title="lemma">p_eltV</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_eltX"><span class="id" title="lemma">p_eltX</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_eltJ"><span class="id" title="lemma">p_eltJ</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_p_elt"><span class="id" title="lemma">sub_p_elt</span></a> <span class="id" title="var">pi1</span> <span class="id" title="var">pi2</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1"><span class="id" title="variable">pi1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2"><span class="id" title="variable">pi2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><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.pgroup.html#pi1"><span class="id" title="variable">pi1</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2"><span class="id" title="variable">pi2</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_p_elt"><span class="id" title="lemma">eq_p_elt</span></a> <span class="id" title="var">pi1</span> <span class="id" title="var">pi2</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi1"><span class="id" title="variable">pi1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2"><span class="id" title="variable">pi2</span></a> <a class="idref" href="http://coq.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.pgroup.html#pi1"><span class="id" title="variable">pi1</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2"><span class="id" title="variable">pi2</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_eltNK"><span class="id" title="lemma">p_eltNK</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_constt"><span class="id" title="lemma">eq_constt</span></a> <span class="id" title="var">pi1</span> <span class="id" title="var">pi2</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi1"><span class="id" title="variable">pi1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2"><span class="id" title="variable">pi2</span></a> <a class="idref" href="http://coq.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.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi1</span></a> <a class="idref" href="http://coq.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.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="consttNK"><span class="id" title="lemma">consttNK</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><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.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cycle_constt"><span class="id" title="lemma">cycle_constt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#5e5825d099c952c2db2842c142cbde94"><span class="id" title="notation"><[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#5e5825d099c952c2db2842c142cbde94"><span class="id" title="notation">]></span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="consttV"><span class="id" title="lemma">consttV</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">).`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="http://coq.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#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">)^-1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt1"><span class="id" title="lemma">constt1</span></a> <span class="id" title="var">pi</span> : 1<a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><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#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#PgroupProps.gT"><span class="id" title="variable">gT</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="consttJ"><span class="id" title="lemma">consttJ</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">).`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="http://coq.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.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_elt_constt"><span class="id" title="lemma">p_elt_constt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="consttC"><span class="id" title="lemma">consttC</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><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.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p'_elt_constt"><span class="id" title="lemma">p'_elt_constt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">)^-1</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="order_constt"><span class="id" title="lemma">order_constt</span></a> <span class="id" title="var">pi</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PgroupProps.gT"><span class="id" title="variable">gT</span></a>) : <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><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#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">`</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#e157af410f295c13bf15c0e174d5741f"><span class="id" title="notation">_pi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="consttM"><span class="id" title="lemma">consttM</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#commute"><span class="id" title="definition">commute</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.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.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">).`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="http://coq.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.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="consttX"><span class="id" title="lemma">consttX</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">).`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="http://coq.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.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt1P"><span class="id" title="lemma">constt1P</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a 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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt_p_elt"><span class="id" title="lemma">constt_p_elt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="http://coq.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.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_in_constt"><span class="id" title="lemma">sub_in_constt</span></a> <span class="id" title="var">pi1</span> <span class="id" title="var">pi2</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1"><span class="id" title="variable">pi1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2"><span class="id" title="variable">pi2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi2</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi1</span></a> <a class="idref" href="http://coq.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.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="prod_constt"><span class="id" title="lemma">prod_constt</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#c7ac55b0cccd799c4aff327c5e84fff2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#c7ac55b0cccd799c4aff327c5e84fff2"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#c7ac55b0cccd799c4aff327c5e84fff2"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.fingroup.fingroup.html#c7ac55b0cccd799c4aff327c5e84fff2"><span class="id" title="notation">≤</span></a> <span class="id" title="var">p</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#c7ac55b0cccd799c4aff327c5e84fff2"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#c7ac55b0cccd799c4aff327c5e84fff2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="max_pgroupJ"><span class="id" title="lemma">max_pgroupJ</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">M</span> <span class="id" title="var">G</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">max</span></a> <span class="id" title="var">M</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><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#9dc7aaba1a150ad5c926725e5128fb48"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#9dc7aaba1a150ad5c926725e5128fb48"><span class="id" title="notation">max</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#9dc7aaba1a150ad5c926725e5128fb48"><span class="id" title="notation">of</span></a> <span class="id" title="var">M</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#9dc7aaba1a150ad5c926725e5128fb48"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#9dc7aaba1a150ad5c926725e5128fb48"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="comm_sub_max_pgroup"><span class="id" title="lemma">comm_sub_max_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">M</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">max</span></a> <span class="id" title="var">M</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.fingroup.fingroup.html#commute"><span class="id" title="definition">commute</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#M"><span class="id" title="variable">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="normal_sub_max_pgroup"><span class="id" title="lemma">normal_sub_max_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">M</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">max</span></a> <span class="id" title="var">M</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.pgroup.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.pgroup.html#M"><span class="id" title="variable">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="norm_sub_max_pgroup"><span class="id" title="lemma">norm_sub_max_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">M</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">max</span></a> <span class="id" title="var">M</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.solvable.pgroup.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#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.pgroup.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><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.pgroup.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.pgroup.html#M"><span class="id" title="variable">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_pHall"><span class="id" title="lemma">sub_pHall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</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.pgroup.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.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f0bbce9238fab3dd03626439080979a9"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Hall_max"><span class="id" title="lemma">Hall_max</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">max</span></a> <span class="id" title="var">H</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pHall_id"><span class="id" title="lemma">pHall_id</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="psubgroup1"><span class="id" title="lemma">psubgroup1</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Cauchy"><span class="id" title="lemma">Cauchy</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><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.pgroup.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><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.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - These lemmas actually hold for maximal pi-groups, but below we'll - derive from the Cauchy lemma that a normal max pi-group is Hall. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_normal_Hall"><span class="id" title="lemma">sub_normal_Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.pgroup.html#K"><span class="id" title="variable">K</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.pgroup.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.pgroup.html#K"><span class="id" title="variable">K</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.pgroup.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="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_normal_Hall"><span class="id" title="lemma">mem_normal_Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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="http://coq.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniq_normal_Hall"><span class="id" title="lemma">uniq_normal_Hall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">max</span></a> <span class="id" title="var">K</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><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.pgroup.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f0bbce9238fab3dd03626439080979a9"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#PgroupProps"><span class="id" title="section">PgroupProps</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="NormalHall"><span class="id" title="section">NormalHall</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="NormalHall.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="NormalHall.pi"><span class="id" title="variable">pi</span></a> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</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.pgroup.html#NormalHall.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="normal_max_pgroup_Hall"><span class="id" title="lemma">normal_max_pgroup_Hall</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">max</span></a> <span class="id" title="var">H</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#NormalHall.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><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.pgroup.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.pgroup.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.pgroup.html#NormalHall.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="setI_normal_Hall"><span class="id" title="lemma">setI_normal_Hall</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#NormalHall.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#K"><span class="id" title="variable">K</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.pgroup.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.pgroup.html#NormalHall.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#K"><span class="id" title="variable">K</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#NormalHall"><span class="id" title="section">NormalHall</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Morphim"><span class="id" title="section">Morphim</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Morphim.aT"><span class="id" title="variable">aT</span></a> <a name="Morphim.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="Morphim.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.pgroup.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="Morphim.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.pgroup.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.pgroup.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">Types</span> (<span class="id" title="var">pi</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">P</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.pgroup.html#Morphim.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_pgroup"><span class="id" title="lemma">morphim_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_odd"><span class="id" title="lemma">morphim_odd</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pmorphim_pgroup"><span class="id" title="lemma">pmorphim_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#7ef99623452370540bbc44fd30b0bc94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#7ef99623452370540bbc44fd30b0bc94"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.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.pgroup.html#Morphim.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_p_index"><span class="id" title="lemma">morphim_p_index</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Morphim.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">nat</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#31aa0776b2fdb88da5b5ba70544862a1"><span class="id" title="notation">nat</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_pHall"><span class="id" title="lemma">morphim_pHall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Morphim.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pmorphim_pHall"><span class="id" title="lemma">pmorphim_pHall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Morphim.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.pgroup.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.pgroup.html#Morphim.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#7ef99623452370540bbc44fd30b0bc94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#7ef99623452370540bbc44fd30b0bc94"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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><br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_Hall"><span class="id" title="lemma">morphim_Hall</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Morphim.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.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Hall"><span class="id" title="definition">Hall</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_pSylow"><span class="id" title="lemma">morphim_pSylow</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> <span class="id" title="var">P</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</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.pgroup.html#Morphim.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.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">Sylow</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">Sylow</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#081d3e80d093e95dd63e6bafc24fef78"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#P"><span class="id" title="variable">P</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_p_group"><span class="id" title="lemma">morphim_p_group</span></a> <span class="id" title="var">P</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#p_group"><span class="id" title="definition">p_group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#p_group"><span class="id" title="definition">p_group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#P"><span class="id" title="variable">P</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_Sylow"><span class="id" title="lemma">morphim_Sylow</span></a> <span class="id" title="var">G</span> <span class="id" title="var">P</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</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.pgroup.html#Morphim.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.pgroup.html#Sylow"><span class="id" title="definition">Sylow</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Sylow"><span class="id" title="definition">Sylow</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#P"><span class="id" title="variable">P</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morph_p_elt"><span class="id" title="lemma">morph_p_elt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morph_constt"><span class="id" title="lemma">morph_constt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.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.pgroup.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">.`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a> <a class="idref" href="http://coq.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.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">).`</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a2b55bf62bd0599fa8bbde6701c3511d"><span class="id" title="notation">_pi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#Morphim"><span class="id" title="section">Morphim</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pquotient"><span class="id" title="section">Pquotient</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Pquotient.pi"><span class="id" title="variable">pi</span></a> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</span></a>) (<a name="Pquotient.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="Pquotient.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Pquotient.G"><span class="id" title="variable">G</span></a> <a name="Pquotient.H"><span class="id" title="variable">H</span></a> <a name="Pquotient.K"><span class="id" title="variable">K</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.pgroup.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/> -<span class="id" title="keyword">Hypothesis</span> <a name="Pquotient.piK"><span class="id" title="variable">piK</span></a> : <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_pgroup"><span class="id" title="lemma">quotient_pgroup</span></a> : <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.H"><span class="id" title="variable">H</span></a>). <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_pHall"><span class="id" title="lemma">quotient_pHall</span></a> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</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.pgroup.html#Pquotient.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><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.pgroup.html#Pquotient.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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.pgroup.html#Pquotient.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_odd"><span class="id" title="lemma">quotient_odd</span></a> : <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.H"><span class="id" title="variable">H</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">Lemma</span> <a name="pquotient_pgroup"><span class="id" title="lemma">pquotient_pgroup</span></a> : <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><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.pgroup.html#Pquotient.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pquotient_pHall"><span class="id" title="lemma">pquotient_pHall</span></a> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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.pgroup.html#Pquotient.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.K"><span class="id" title="variable">K</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltn_log_quotient"><span class="id" title="lemma">ltn_log_quotient</span></a> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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.pgroup.html#Pquotient.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.pgroup.html#Pquotient.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.pgroup.html#Pquotient.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.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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.pgroup.html#Pquotient.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient.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>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#Pquotient"><span class="id" title="section">Pquotient</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Application of card_Aut_cyclic to internal faithful action on cyclic - p-subgroups. -</div> -<div class="code"> -<span class="id" title="keyword">Section</span> <a name="InnerAutCyclicPgroup"><span class="id" title="section">InnerAutCyclicPgroup</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="InnerAutCyclicPgroup.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="InnerAutCyclicPgroup.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="InnerAutCyclicPgroup.G"><span class="id" title="variable">G</span></a> <a name="InnerAutCyclicPgroup.C"><span class="id" title="variable">C</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.pgroup.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/> -<span class="id" title="keyword">Hypothesis</span> <a name="InnerAutCyclicPgroup.nCG"><span class="id" title="variable">nCG</span></a> : <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.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.pgroup.html#InnerAutCyclicPgroup.C"><span class="id" title="variable">C</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="logn_quotient_cent_cyclic_pgroup"><span class="id" title="lemma">logn_quotient_cent_cyclic_pgroup</span></a> : <br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.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.pgroup.html#InnerAutCyclicPgroup.C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.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.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.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.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><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.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</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="p'group_quotient_cent_prime"><span class="id" title="lemma">p'group_quotient_cent_prime</span></a> :<br/> - <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.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.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup.C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#InnerAutCyclicPgroup"><span class="id" title="section">InnerAutCyclicPgroup</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="PcoreDef"><span class="id" title="section">PcoreDef</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - A functor needs to quantify over the finGroupType just beore the set. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="PcoreDef.pi"><span class="id" title="variable">pi</span></a> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</span></a>) (<a name="PcoreDef.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="PcoreDef.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.pgroup.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="pcore"><span class="id" title="definition">pcore</span></a> := <a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">(</span></a><span class="id" title="var">G</span> <a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">max</span></a> <span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#PcoreDef.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#PcoreDef.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#e6278be94ff3547a5d055628cea199c4"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pcore_group</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.pgroup.html#PcoreDef.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="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pcore"><span class="id" title="definition">pcore</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#PcoreDef"><span class="id" title="section">PcoreDef</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">"</span></a>''O_' pi ( G )" := (<a class="idref" href="mathcomp.solvable.pgroup.html#pcore"><span class="id" title="definition">pcore</span></a> <span class="id" title="var">pi</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">pi</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''O_' pi ( G )") : <span class="id" title="var">group_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="41d567f4cadf395a6bc9313f2e91ac15"><span class="id" title="notation">"</span></a>''O_' pi ( G )" := (<a class="idref" href="mathcomp.solvable.pgroup.html#pcore_group"><span class="id" title="definition">pcore_group</span></a> <span class="id" title="var">pi</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="PseriesDefs"><span class="id" title="section">PseriesDefs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="PseriesDefs.pis"><span class="id" title="variable">pis</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</span></a>) (<a name="PseriesDefs.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="PseriesDefs.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.pgroup.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="pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <span class="id" title="var">pi</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.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#PseriesDefs.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.pgroup.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pcore_mod_group</span> <span class="id" title="var">pi</span> <span class="id" title="var">B</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.pgroup.html#PseriesDefs.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/> - <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="pseries"><span class="id" title="definition">pseries</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> 1 (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#PseriesDefs.pis"><span class="id" title="variable">pis</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_group_set"><span class="id" title="lemma">pseries_group_set</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pseries_group</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.pgroup.html#PseriesDefs.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"><span class="id" title="definition">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pseries_group_set"><span class="id" title="lemma">pseries_group_set</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#PseriesDefs"><span class="id" title="section">PseriesDefs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">"</span></a>''O_{' p1 , .. , pn } ( A )" :=<br/> - (<a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#ConsPred"><span class="id" title="abbreviation">ConsPred</span></a> <span class="id" title="var">p1</span> .. (<a class="idref" href="mathcomp.solvable.pgroup.html#ConsPred"><span class="id" title="abbreviation">ConsPred</span></a> <span class="id" title="var">pn</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>) ..) <span class="id" title="var">A</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''O_{' p1 , .. , pn } ( A )") : <span class="id" title="var">group_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="79b9d7f8350df9b24b8430dc1a350b11"><span class="id" title="notation">"</span></a>''O_{' p1 , .. , pn } ( A )" :=<br/> - (<a class="idref" href="mathcomp.solvable.pgroup.html#pseries_group"><span class="id" title="definition">pseries_group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#ConsPred"><span class="id" title="abbreviation">ConsPred</span></a> <span class="id" title="var">p1</span> .. (<a class="idref" href="mathcomp.solvable.pgroup.html#ConsPred"><span class="id" title="abbreviation">ConsPred</span></a> <span class="id" title="var">pn</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>) ..) <span class="id" title="var">A</span>) : <span class="id" title="var">Group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="PCoreProps"><span class="id" title="section">PCoreProps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="PCoreProps.pi"><span class="id" title="variable">pi</span></a> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</span></a>) (<a name="PCoreProps.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> : <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.pgroup.html#PCoreProps.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> <span class="id" title="var">M</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.pgroup.html#PCoreProps.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="pcore_psubgroup"><span class="id" title="lemma">pcore_psubgroup</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">subgroup</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#40070e138310190e4cd3ac9d3a86440d"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_pgroup"><span class="id" title="lemma">pcore_pgroup</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_sub"><span class="id" title="lemma">pcore_sub</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_sub_Hall"><span class="id" title="lemma">pcore_sub_Hall</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_max"><span class="id" title="lemma">pcore_max</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_pgroup_id"><span class="id" title="lemma">pcore_pgroup_id</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_normal"><span class="id" title="lemma">pcore_normal</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="normal_Hall_pcore"><span class="id" title="lemma">normal_Hall_pcore</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_Hall_pcore"><span class="id" title="lemma">eq_Hall_pcore</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_Hall_pcore"><span class="id" title="lemma">sub_Hall_pcore</span></a> <span class="id" title="var">G</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#K"><span class="id" title="variable">K</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.pgroup.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.pgroup.html#K"><span class="id" title="variable">K</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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#K"><span class="id" title="variable">K</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_Hall_pcore"><span class="id" title="lemma">mem_Hall_pcore</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_Hall_pcoreP"><span class="id" title="lemma">sdprod_Hall_pcoreP</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_pcore_HallP"><span class="id" title="lemma">sdprod_pcore_HallP</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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="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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps.pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcoreJ"><span class="id" title="lemma">pcoreJ</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#PCoreProps"><span class="id" title="section">PCoreProps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MorphPcore"><span class="id" title="section">MorphPcore</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">pi</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</span></a>) (<span class="id" title="var">gT</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>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_pcore"><span class="id" title="lemma">morphim_pcore</span></a> <span class="id" title="var">pi</span> : <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.pgroup.html#pcore"><span class="id" title="definition">pcore</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcoreS"><span class="id" title="lemma">pcoreS</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">gT</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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pcore_igFun</span> <span class="id" title="var">pi</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.pgroup.html#pcore_sub"><span class="id" title="lemma">pcore_sub</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#morphim_pcore"><span class="id" title="lemma">morphim_pcore</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</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">pcore_gFun</span> <span class="id" title="var">pi</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.pgroup.html#morphim_pcore"><span class="id" title="lemma">morphim_pcore</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</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">pcore_pgFun</span> <span class="id" title="var">pi</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.pgroup.html#morphim_pcore"><span class="id" title="lemma">morphim_pcore</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</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">Lemma</span> <a name="pcore_char"><span class="id" title="lemma">pcore_char</span></a> <span class="id" title="var">pi</span> <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.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MorphPcore.PcoreMod"><span class="id" title="section">PcoreMod</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="MorphPcore.PcoreMod.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pmap"><span class="id" title="record">GFunctor.pmap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_mod_sub"><span class="id" title="lemma">pcore_mod_sub</span></a> <span class="id" title="var">pi</span> <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.pgroup.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.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#MorphPcore.PcoreMod.F"><span class="id" title="variable">F</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_pcore_mod"><span class="id" title="lemma">quotient_pcore_mod</span></a> <span class="id" title="var">pi</span> <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.pgroup.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>) (<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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_pcore_mod"><span class="id" title="lemma">morphim_pcore_mod</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">D</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.pgroup.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>) (<span class="id" title="var">f</span> : <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.pgroup.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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#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.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#MorphPcore.PcoreMod.F"><span class="id" title="variable">F</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#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.pgroup.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#MorphPcore.PcoreMod.F"><span class="id" title="variable">F</span></a> <span class="id" title="var">_</span> (<a class="idref" href="mathcomp.solvable.pgroup.html#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.pgroup.html#G"><span class="id" title="variable">G</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_mod_res"><span class="id" title="lemma">pcore_mod_res</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">D</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.pgroup.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>) (<span class="id" title="var">f</span> : <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.pgroup.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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#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.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#MorphPcore.PcoreMod.F"><span class="id" title="variable">F</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.solvable.pgroup.html#D"><span class="id" title="variable">D</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.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#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.pgroup.html#D"><span class="id" title="variable">D</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#MorphPcore.PcoreMod.F"><span class="id" title="variable">F</span></a> <span class="id" title="var">_</span> (<a class="idref" href="mathcomp.solvable.pgroup.html#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.pgroup.html#D"><span class="id" title="variable">D</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_mod1"><span class="id" title="lemma">pcore_mod1</span></a> <span class="id" title="var">pi</span> <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.pgroup.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.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#MorphPcore.PcoreMod"><span class="id" title="section">PcoreMod</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_rcons"><span class="id" title="lemma">pseries_rcons</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">pis</span> <span class="id" title="var">gT</span> (<span class="id" title="var">A</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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_subfun"><span class="id" title="lemma">pseries_subfun</span></a> <span class="id" title="var">pis</span> :<br/> - <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.closed"><span class="id" title="definition">GFunctor.closed</span></a> (@<a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous"><span class="id" title="definition">GFunctor.pcontinuous</span></a> (@<a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_sub"><span class="id" title="lemma">pseries_sub</span></a> <span class="id" title="var">pis</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.closed"><span class="id" title="definition">GFunctor.closed</span></a> (@<a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_pseries"><span class="id" title="lemma">morphim_pseries</span></a> <span class="id" title="var">pis</span> : <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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseriesS"><span class="id" title="lemma">pseriesS</span></a> <span class="id" title="var">pis</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.hereditary"><span class="id" title="definition">GFunctor.hereditary</span></a> (@<a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pseries_igFun</span> <span class="id" title="var">pis</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.pgroup.html#pseries_sub"><span class="id" title="lemma">pseries_sub</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#morphim_pseries"><span class="id" title="lemma">morphim_pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</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">pseries_gFun</span> <span class="id" title="var">pis</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.pgroup.html#morphim_pseries"><span class="id" title="lemma">morphim_pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</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">pseries_pgFun</span> <span class="id" title="var">pis</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.pgroup.html#morphim_pseries"><span class="id" title="lemma">morphim_pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</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">Lemma</span> <a name="pseries_char"><span class="id" title="lemma">pseries_char</span></a> <span class="id" title="var">pis</span> <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.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</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.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_normal"><span class="id" title="lemma">pseries_normal</span></a> <span class="id" title="var">pis</span> <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.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseriesJ"><span class="id" title="lemma">pseriesJ</span></a> <span class="id" title="var">pis</span> <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.pgroup.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>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries1"><span class="id" title="lemma">pseries1</span></a> <span class="id" title="var">pi</span> <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.pgroup.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.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">O_</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">}(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_pop"><span class="id" title="lemma">pseries_pop</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">pis</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_pop2"><span class="id" title="lemma">pseries_pop2</span></a> <span class="id" title="var">pi1</span> <span class="id" title="var">pi2</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi1</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">O_</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#pi1"><span class="id" title="variable">pi1</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2"><span class="id" title="variable">pi2</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">}(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi2</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_sub_catl"><span class="id" title="lemma">pseries_sub_catl</span></a> <span class="id" title="var">pi1s</span> <span class="id" title="var">pi2s</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_pseries"><span class="id" title="lemma">quotient_pseries</span></a> <span class="id" title="var">pis</span> <span class="id" title="var">pi</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_norm2"><span class="id" title="lemma">pseries_norm2</span></a> <span class="id" title="var">pi1s</span> <span class="id" title="var">pi2s</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</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="pseries_sub_catr"><span class="id" title="lemma">pseries_sub_catr</span></a> <span class="id" title="var">pi1s</span> <span class="id" title="var">pi2s</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_pseries2"><span class="id" title="lemma">quotient_pseries2</span></a> <span class="id" title="var">pi1</span> <span class="id" title="var">pi2</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">O_</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#pi1"><span class="id" title="variable">pi1</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2"><span class="id" title="variable">pi2</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><span class="id" title="notation">}(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#bc707905e1798e6e2fa9aec727f7b1fb"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi1</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi2</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi1</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">))</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_pseries_cat"><span class="id" title="lemma">quotient_pseries_cat</span></a> <span class="id" title="var">pi1s</span> <span class="id" title="var">pi2s</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_catl_id"><span class="id" title="lemma">pseries_catl_id</span></a> <span class="id" title="var">pi1s</span> <span class="id" title="var">pi2s</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_char_catl"><span class="id" title="lemma">pseries_char_catl</span></a> <span class="id" title="var">pi1s</span> <span class="id" title="var">pi2s</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_catr_id"><span class="id" title="lemma">pseries_catr_id</span></a> <span class="id" title="var">pi1s</span> <span class="id" title="var">pi2s</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pseries_char_catr"><span class="id" title="lemma">pseries_char_catr</span></a> <span class="id" title="var">pi1s</span> <span class="id" title="var">pi2s</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#pi1s"><span class="id" title="variable">pi1s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi2s"><span class="id" title="variable">pi2s</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_modp"><span class="id" title="lemma">pcore_modp</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">gT</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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pcore_mod"><span class="id" title="definition">pcore_mod</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pquotient_pcore"><span class="id" title="lemma">pquotient_pcore</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">gT</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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="trivg_pcore_quotient"><span class="id" title="lemma">trivg_pcore_quotient</span></a> <span class="id" title="var">pi</span> <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.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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="pseries_rcons_id"><span class="id" title="lemma">pseries_rcons_id</span></a> <span class="id" title="var">pis</span> <span class="id" title="var">pi</span> <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.pgroup.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/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a>) <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#MorphPcore"><span class="id" title="section">MorphPcore</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqPcore"><span class="id" title="section">EqPcore</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="EqPcore.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">pi</span> <span class="id" title="var">rho</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#nat_pred"><span class="id" title="definition">nat_pred</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.pgroup.html#EqPcore.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="sub_in_pcore"><span class="id" title="lemma">sub_in_pcore</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_pcore"><span class="id" title="lemma">sub_pcore</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_pcore"><span class="id" title="lemma">eq_in_pcore</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_pcore"><span class="id" title="lemma">eq_pcore</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a> <a class="idref" href="http://coq.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcoreNK"><span class="id" title="lemma">pcoreNK</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_p'core"><span class="id" title="lemma">eq_p'core</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a> <a class="idref" href="http://coq.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_rho</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_Hall_p'coreP"><span class="id" title="lemma">sdprod_Hall_p'coreP</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_p'core_HallP"><span class="id" title="lemma">sdprod_p'core_HallP</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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="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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcoreI"><span class="id" title="lemma">pcoreI</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">rho</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#9505898acdd70a74fca20676bf8d8084"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#9505898acdd70a74fca20676bf8d8084"><span class="id" title="notation">predI</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#9505898acdd70a74fca20676bf8d8084"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#rho"><span class="id" title="variable">rho</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#9505898acdd70a74fca20676bf8d8084"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">('</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_rho</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">))</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="bigcap_p'core"><span class="id" title="lemma">bigcap_p'core</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#27fb056d0ab27107566bb9722b69ede4"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#27fb056d0ab27107566bb9722b69ede4"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#27fb056d0ab27107566bb9722b69ede4"><span class="id" title="notation">(</span></a><span class="id" title="var">p</span> <a class="idref" href="mathcomp.ssreflect.finset.html#27fb056d0ab27107566bb9722b69ede4"><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.pgroup.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="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#27fb056d0ab27107566bb9722b69ede4"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#27fb056d0ab27107566bb9722b69ede4"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_p</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="coprime_pcoreC"><span class="id" title="lemma">coprime_pcoreC</span></a> (<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">pi</span> <span class="id" title="var">G</span> (<span class="id" title="var">R</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.pgroup.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.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</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">Lemma</span> <a name="TI_pcoreC"><span class="id" title="lemma">TI_pcoreC</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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="pcore_setI_normal"><span class="id" title="lemma">pcore_setI_normal</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#EqPcore"><span class="id" title="section">EqPcore</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Injm"><span class="id" title="section">Injm</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Injm.aT"><span class="id" title="variable">aT</span></a> <a name="Injm.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="Injm.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.pgroup.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="Injm.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.pgroup.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.pgroup.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">Hypothesis</span> <a name="Injm.injf"><span class="id" title="variable">injf</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.pgroup.html#Injm.f"><span class="id" title="variable">f</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">A</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.pgroup.html#Injm.aT"><span class="id" title="variable">aT</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.pgroup.html#Injm.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="injm_pgroup"><span class="id" title="lemma">injm_pgroup</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Injm.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Injm.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.pgroup.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_pelt"><span class="id" title="lemma">injm_pelt</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Injm.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Injm.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#02ede184c88cd665ffb04f209168388a"><span class="id" title="notation">elt</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_pHall"><span class="id" title="lemma">injm_pHall</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Injm.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.pgroup.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.pgroup.html#Injm.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Injm.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.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Injm.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.pgroup.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">Hall</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#a4de5afc30e4046e35829de6f2bc75f3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_pcore"><span class="id" title="lemma">injm_pcore</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Injm.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.pgroup.html#Injm.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Injm.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.pgroup.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_pseries"><span class="id" title="lemma">injm_pseries</span></a> <span class="id" title="var">pis</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#Injm.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.pgroup.html#Injm.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> (<a class="idref" href="mathcomp.solvable.pgroup.html#Injm.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.pgroup.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.pgroup.html#Injm"><span class="id" title="section">Injm</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Isog"><span class="id" title="section">Isog</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Isog.aT"><span class="id" title="variable">aT</span></a> <a name="Isog.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="Isog.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.pgroup.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="Isog.H"><span class="id" title="variable">H</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.pgroup.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/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_pgroup"><span class="id" title="lemma">isog_pgroup</span></a> <span class="id" title="var">pi</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#Isog.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.pgroup.html#Isog.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Isog.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.pgroup.html#pi"><span class="id" title="variable">pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Isog.H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_pcore"><span class="id" title="lemma">isog_pcore</span></a> <span class="id" title="var">pi</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#Isog.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.pgroup.html#Isog.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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Isog.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</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.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#Isog.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_pseries"><span class="id" title="lemma">isog_pseries</span></a> <span class="id" title="var">pis</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#Isog.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.pgroup.html#Isog.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Isog.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.pgroup.html#pseries"><span class="id" title="definition">pseries</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#pis"><span class="id" title="variable">pis</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#Isog.H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.pgroup.html#Isog"><span class="id" title="section">Isog</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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