docs/htmldoc/mathcomp.solvable.jordanholder.html


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<h1 class="libtitle">Library mathcomp.solvable.jordanholder</h1>

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

<br/>
</div>

<div class="doc">
 This files establishes Jordan-Holder theorems for finite groups. These     
 theorems state the uniqueness up to permutation and isomorphism for the    
 series of quotient built from the successive elements of any composition   
 series of the same group. These quotients are also called factors of the   
 composition series. To avoid the heavy use of highly polymorphic lists     
 describing these quotient series, we introduce sections.                   
 This library defines:                                                      
         (G1 / G2)%sec == alias for the pair (G1, G2) of groups in the same 
                          finGroupType, coerced to the actual quotient group
                          group G1 / G2. We call this pseudo-quotient a     
                          section of G1 and G2.                             
    section_isog s1 s2 == s1 and s2 respectively coerce to isomorphic       
                          quotient groups.                                  
        section_repr s == canonical representative of the isomorphism class 
                          of the section s.                                 
         mksrepr G1 G2 == canonical representative of the isomorphism class 
                          of (G1 / G2)%sec.                                 
         mkfactors G s == if s is [:: s1, s2, ..., sn], constructs the list 
                     [:: mksrepr G s1, mksrepr s1 s2, ..., mksrepr sn-1 sn] 
             comps G s == s is a composition series for G i.e. s is a       
                          decreasing sequence of subgroups of G             
                          in which two adjacent elements are maxnormal one  
                          in the other and the last element of s is 1.      
 Given aT and rT two finGroupTypes, (D : {group rT}), (A : {group aT}) and  
 (to : groupAction A D) an external action.                                 
        maxainv to B C == C is a maximal proper normal subgroup of B        
                          invariant by (the external action of A via) to.   
          asimple to B == the maximal proper normal subgroup of B invariant 
                          by the external action to is trivial.             
         acomps to G s == s is a composition series for G invariant by to,  
                          i.e. s is a decreasing sequence of subgroups of G 
                          in which two adjacent elements are maximally      
                          invariant by to one in the other and the          
                          last element of s is 1.                           
 We prove two versions of the result:                                       
<ul class="doclist">
<li> JordanHolderUniqueness establishes the uniqueness up to permutation   
       and isomorphism of the lists of factors in composition series of a   
       given group.                                                         

</li>
<li> StrongJordanHolderUniqueness extends the result to composition series 
       invariant by an external group action.                               

</li>
</ul>
 See also "The Rooster and the Butterflies", proceedings of Calculemus 2013,
 by Assia Mahboubi.                                                          
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/>

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

<br/>
<span class="id" title="keyword">Inductive</span> <a name="section"><span class="id" title="inductive">section</span></a> (<span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) := <a name="GSection"><span class="id" title="constructor">GSection</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">section_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">sec</span>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="mkSec"><span class="id" title="definition">mkSec</span></a> (<span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<span class="id" title="var">G1</span> <span class="id" title="var">G2</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) := <a class="idref" href="mathcomp.solvable.jordanholder.html#GSection"><span class="id" title="constructor">GSection</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#G1"><span class="id" title="variable">G1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G2"><span class="id" title="variable">G2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Infix</span> <a name="173b3ede47946e274e545159d060f62f"><span class="id" title="notation">&quot;</span></a>/" := <a class="idref" href="mathcomp.solvable.jordanholder.html#mkSec"><span class="id" title="definition">mkSec</span></a> : <span class="id" title="var">section_scope</span>.<br/>

<br/>
<span class="id" title="keyword">Coercion</span> <span class="id" title="var">pair_of_section</span> <span class="id" title="var">gT</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gT"><span class="id" title="variable">gT</span></a>) := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.solvable.jordanholder.html#GSection"><span class="id" title="constructor">GSection</span></a> <span class="id" title="var">u</span> := <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">u</span>.<br/>

<br/>
<span class="id" title="keyword">Coercion</span> <span class="id" title="var">quotient_of_section</span> <span class="id" title="var">gT</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gT"><span class="id" title="variable">gT</span></a>) : <a class="idref" href="mathcomp.fingroup.fingroup.html#GroupSet.sort"><span class="id" title="definition">GroupSet.sort</span></a> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a>.<br/>

<br/>
<span class="id" title="keyword">Coercion</span> <span class="id" title="var">section_group</span> <span class="id" title="var">gT</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gT"><span class="id" title="variable">gT</span></a>) : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">)}</span></a> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">]</span></a>.<br/>

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

<br/>
<span class="id" title="keyword">Variables</span> (<a name="Sections.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a>).<br/>

<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">section_subType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#d716e206e5129c6b3a60f0f640eaaeb0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#d716e206e5129c6b3a60f0f640eaaeb0"><span class="id" title="notation">newType</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#d716e206e5129c6b3a60f0f640eaaeb0"><span class="id" title="notation">for</span></a> @<a class="idref" href="mathcomp.solvable.jordanholder.html#pair_of_section"><span class="id" title="definition">pair_of_section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#d716e206e5129c6b3a60f0f640eaaeb0"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="section_eqMixin"><span class="id" title="definition">section_eqMixin</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">eqMixin</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">&lt;:]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">section_eqType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a>) <a class="idref" href="mathcomp.solvable.jordanholder.html#section_eqMixin"><span class="id" title="definition">section_eqMixin</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="section_choiceMixin"><span class="id" title="definition">section_choiceMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">&lt;:]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">section_choiceType</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a>) <a class="idref" href="mathcomp.solvable.jordanholder.html#section_choiceMixin"><span class="id" title="definition">section_choiceMixin</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="section_countMixin"><span class="id" title="definition">section_countMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">&lt;:]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">section_countType</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Exports.CountType"><span class="id" title="abbreviation">CountType</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a>) <a class="idref" href="mathcomp.solvable.jordanholder.html#section_countMixin"><span class="id" title="definition">section_countMixin</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">section_subCountType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">subCountType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="section_finMixin"><span class="id" title="definition">section_finMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">finMixin</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">&lt;:]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">section_finType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.FinType"><span class="id" title="abbreviation">FinType</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a>) <a class="idref" href="mathcomp.solvable.jordanholder.html#section_finMixin"><span class="id" title="definition">section_finMixin</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">section_subFinType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#a701c7b60b4a16f07950761d8bf90924"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#a701c7b60b4a16f07950761d8bf90924"><span class="id" title="notation">subFinType</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#a701c7b60b4a16f07950761d8bf90924"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#a701c7b60b4a16f07950761d8bf90924"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">section_group</span>.<br/>

<br/>
</div>

<div class="doc">
 Isomorphic sections  
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Definition</span> <a name="section_isog"><span class="id" title="definition">section_isog</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">rel</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section"><span class="id" title="inductive">section</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">]</span></a>.<br/>

<br/>
</div>

<div class="doc">
 A witness of the isomorphism class of a section  
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="section_repr"><span class="id" title="definition">section_repr</span></a> <span class="id" title="var">s</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#odflt"><span class="id" title="abbreviation">odflt</span></a> (1 <a class="idref" href="mathcomp.solvable.jordanholder.html#173b3ede47946e274e545159d060f62f"><span class="id" title="notation">/</span></a> 1)%<span class="id" title="var">sec</span> (<a class="idref" href="mathcomp.ssreflect.fintype.html#pick"><span class="id" title="definition">pick</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#section_isog"><span class="id" title="definition">section_isog</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>)).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="mksrepr"><span class="id" title="definition">mksrepr</span></a> <span class="id" title="var">G1</span> <span class="id" title="var">G2</span> := <a class="idref" href="mathcomp.solvable.jordanholder.html#section_repr"><span class="id" title="definition">section_repr</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#mkSec"><span class="id" title="definition">mkSec</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G1"><span class="id" title="variable">G1</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G2"><span class="id" title="variable">G2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="section_reprP"><span class="id" title="lemma">section_reprP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#section_repr"><span class="id" title="definition">section_repr</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="section_repr_isog"><span class="id" title="lemma">section_repr_isog</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section_repr"><span class="id" title="definition">section_repr</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section_repr"><span class="id" title="definition">section_repr</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s2"><span class="id" title="variable">s2</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="mkfactors"><span class="id" title="definition">mkfactors</span></a> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#section_repr"><span class="id" title="definition">section_repr</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> (@<a class="idref" href="mathcomp.solvable.jordanholder.html#mkSec"><span class="id" title="definition">mkSec</span></a> <span class="id" title="var">_</span>) <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.jordanholder.html#Sections"><span class="id" title="section">Sections</span></a>.<br/>

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

<br/>
<span class="id" title="keyword">Variable</span> <a name="CompositionSeries.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#gTg"><span class="id" title="abbreviation">gTg</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gTg"><span class="id" title="abbreviation">gTg</span></a>).<br/>

<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="comps"><span class="id" title="definition">comps</span></a> <span class="id" title="var">G</span> <span class="id" title="var">s</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">G</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#compo"><span class="id" title="abbreviation">compo</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">series</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="compsP"><span class="id" title="lemma">compsP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">s</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">G</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a>  <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">rel</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gTg"><span class="id" title="abbreviation">gTg</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.solvable.jordanholder.html#comps"><span class="id" title="definition">comps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="trivg_comps"><span class="id" title="lemma">trivg_comps</span></a> <span class="id" title="var">G</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#comps"><span class="id" title="definition">comps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#24f47bb7b1a372904563d2bdb8a213a4"><span class="id" title="notation">:==:</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="comps_cons"><span class="id" title="lemma">comps_cons</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#comps"><span class="id" title="definition">comps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#comps"><span class="id" title="definition">comps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="simple_compsP"><span class="id" title="lemma">simple_compsP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#comps"><span class="id" title="definition">comps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">[::</span></a> 1%<span class="id" title="var">G</span><a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">]</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#simple"><span class="id" title="definition">simple</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="exists_comps"><span class="id" title="lemma">exists_comps</span></a> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#gTg"><span class="id" title="abbreviation">gTg</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#comps"><span class="id" title="definition">comps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
</div>

<div class="doc">
 The factors associated to two composition series of the same group are     
 the same up to isomorphism and permutation                                  
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Lemma</span> <a name="JordanHolderUniqueness"><span class="id" title="lemma">JordanHolderUniqueness</span></a> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#gTg"><span class="id" title="abbreviation">gTg</span></a>) (<span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#gTg"><span class="id" title="abbreviation">gTg</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#comps"><span class="id" title="definition">comps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#comps"><span class="id" title="definition">comps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#mkfactors"><span class="id" title="definition">mkfactors</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.solvable.jordanholder.html#mkfactors"><span class="id" title="definition">mkfactors</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s2"><span class="id" title="variable">s2</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.jordanholder.html#CompositionSeries"><span class="id" title="section">CompositionSeries</span></a>.<br/>

<br/>
</div>

<div class="doc">
 Helper lemmas for group actions.                                            
</div>
<div class="code">

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

<br/>
<span class="id" title="keyword">Variables</span> (<a name="MoreGroupAction.aT"><span class="id" title="variable">aT</span></a> <a name="MoreGroupAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/>
<span class="id" title="keyword">Variables</span> (<a name="MoreGroupAction.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="MoreGroupAction.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>
<span class="id" title="keyword">Variable</span> <a name="MoreGroupAction.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.D"><span class="id" title="variable">D</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="gactsP"><span class="id" title="lemma">gactsP</span></a> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> <a class="idref" href="mathcomp.fingroup.action.html#05449e0e2f87f06fca16a62c49e8b809"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.action.html#05449e0e2f87f06fca16a62c49e8b809"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#05449e0e2f87f06fca16a62c49e8b809"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#05449e0e2f87f06fca16a62c49e8b809"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#05449e0e2f87f06fca16a62c49e8b809"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#05449e0e2f87f06fca16a62c49e8b809"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="gactsM"><span class="id" title="lemma">gactsM</span></a> (<span class="id" title="var">N1</span> <span class="id" title="var">N2</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) : <br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="gactsI"><span class="id" title="lemma">gactsI</span></a> (<span class="id" title="var">N1</span> <span class="id" title="var">N2</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) : <br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="gastabsP"><span class="id" title="lemma">gastabsP</span></a> (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">a</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.aT"><span class="id" title="variable">aT</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>) (<a class="idref" href="mathcomp.solvable.jordanholder.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">)</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreGroupAction"><span class="id" title="section">MoreGroupAction</span></a>.<br/>

<br/>
</div>

<div class="doc">
 Helper lemmas for quotient actions.                                         
</div>
<div class="code">

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

<br/>
<span class="id" title="keyword">Variables</span> (<a name="MoreQuotientAction.aT"><span class="id" title="variable">aT</span></a> <a name="MoreQuotientAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/>
<span class="id" title="keyword">Variables</span> (<a name="MoreQuotientAction.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>)(<a name="MoreQuotientAction.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>
<span class="id" title="keyword">Variable</span> <a name="MoreQuotientAction.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.D"><span class="id" title="variable">D</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="qact_dom_doms"><span class="id" title="lemma">qact_dom_doms</span></a> (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.A"><span class="id" title="variable">A</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="acts_qact_doms"><span class="id" title="lemma">acts_qact_doms</span></a> (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.A"><span class="id" title="variable">A</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="qacts_cosetpre"><span class="id" title="lemma">qacts_cosetpre</span></a> (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">K'</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K'"><span class="id" title="variable">K'</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2e4ccc4d97341df4136dd243a5fe1a5c"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K'"><span class="id" title="variable">K'</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="qacts_coset"><span class="id" title="lemma">qacts_coset</span></a> (<span class="id" title="var">H</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2e4ccc4d97341df4136dd243a5fe1a5c"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.jordanholder.html#MoreQuotientAction"><span class="id" title="section">MoreQuotientAction</span></a>.<br/>

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

<br/>
<span class="id" title="keyword">Variables</span> (<a name="StableCompositionSeries.aT"><span class="id" title="variable">aT</span></a> <a name="StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/>
<span class="id" title="keyword">Variables</span> (<a name="StableCompositionSeries.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>)(<a name="StableCompositionSeries.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>
<span class="id" title="keyword">Variable</span> <a name="StableCompositionSeries.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.D"><span class="id" title="variable">D</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="maxainv"><span class="id" title="definition">maxainv</span></a> (<span class="id" title="var">B</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">max</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">of</span></a> <span class="id" title="var">H</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">|</span></a> <br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">]</span></a>.<br/>

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

<br/>
<span class="id" title="keyword">Variables</span> <a name="StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a> <a name="StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxainv_norm"><span class="id" title="lemma">maxainv_norm</span></a> : <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxainv_proper"><span class="id" title="lemma">maxainv_proper</span></a> : <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxainv_sub"><span class="id" title="lemma">maxainv_sub</span></a> : <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxainv_ainvar"><span class="id" title="lemma">maxainv_ainvar</span></a> : <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxainvS"><span class="id" title="lemma">maxainvS</span></a> : <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.N"><span class="id" title="variable">N</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxainv_exists"><span class="id" title="lemma">maxainv_exists</span></a> : <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#36625695d37b6869c156bfcdf13834f7"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">{</span></a><span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N"><span class="id" title="variable">N</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.MaxAinvProps"><span class="id" title="section">MaxAinvProps</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxainvM"><span class="id" title="lemma">maxainvM</span></a> (<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#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#3ead863f625b94b2207b999cfcc823ba"><span class="id" title="notation">:&lt;&gt;:</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="asimple"><span class="id" title="definition">asimple</span></a> (<span class="id" title="var">K</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) := <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> 1.<br/>

<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">H</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="asimpleP"><span class="id" title="lemma">asimpleP</span></a> <span class="id" title="var">K</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4d70beb460e783df2b7b2ad387c00194"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#36625695d37b6869c156bfcdf13834f7"><span class="id" title="notation">:!=:</span></a> 1<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4d70beb460e783df2b7b2ad387c00194"><span class="id" title="notation">&amp;</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">H</span>, <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4d70beb460e783df2b7b2ad387c00194"><span class="id" title="notation">]</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.solvable.jordanholder.html#asimple"><span class="id" title="definition">asimple</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="acomps"><span class="id" title="definition">acomps</span></a> <span class="id" title="var">K</span> <span class="id" title="var">s</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">G</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">rel</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="acompsP"><span class="id" title="lemma">acompsP</span></a> <span class="id" title="var">K</span> <span class="id" title="var">s</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">G</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a>  <a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">rel</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.solvable.jordanholder.html#acomps"><span class="id" title="definition">acomps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="trivg_acomps"><span class="id" title="lemma">trivg_acomps</span></a> <span class="id" title="var">K</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#acomps"><span class="id" title="definition">acomps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#24f47bb7b1a372904563d2bdb8a213a4"><span class="id" title="notation">:==:</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="acomps_cons"><span class="id" title="lemma">acomps_cons</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#acomps"><span class="id" title="definition">acomps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#acomps"><span class="id" title="definition">acomps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="asimple_acompsP"><span class="id" title="lemma">asimple_acompsP</span></a> <span class="id" title="var">K</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.solvable.jordanholder.html#acomps"><span class="id" title="definition">acomps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">[::</span></a> 1%<span class="id" title="var">G</span><a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">]</span></a>) (<a class="idref" href="mathcomp.solvable.jordanholder.html#asimple"><span class="id" title="definition">asimple</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="exists_acomps"><span class="id" title="lemma">exists_acomps</span></a> <span class="id" title="var">K</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#acomps"><span class="id" title="definition">acomps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.jordanholder.html#StableCompositionSeries"><span class="id" title="section">StableCompositionSeries</span></a>.<br/>

<br/>

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

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

<br/>
<span class="id" title="keyword">Variables</span> <a name="StrongJordanHolder.AuxiliaryLemmas.aT"><span class="id" title="variable">aT</span></a> <a name="StrongJordanHolder.AuxiliaryLemmas.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<span class="id" title="keyword">Variables</span> (<a name="StrongJordanHolder.AuxiliaryLemmas.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="StrongJordanHolder.AuxiliaryLemmas.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#D"><span class="id" title="variable">D</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxainv_asimple_quo"><span class="id" title="lemma">maxainv_asimple_quo</span></a> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#asimple"><span class="id" title="definition">asimple</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2756d213ba2c3f60f8b9fc530f7c4cdc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a>) (<a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="asimple_quo_maxainv"><span class="id" title="lemma">asimple_quo_maxainv</span></a> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#asimple"><span class="id" title="definition">asimple</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2756d213ba2c3f60f8b9fc530f7c4cdc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a>) (<a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#maxainv"><span class="id" title="definition">maxainv</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#H"><span class="id" title="variable">H</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="asimpleI"><span class="id" title="lemma">asimpleI</span></a> (<span class="id" title="var">N1</span> <span class="id" title="var">N2</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#asimple"><span class="id" title="definition">asimple</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2756d213ba2c3f60f8b9fc530f7c4cdc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a>) (<a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#asimple"><span class="id" title="definition">asimple</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2756d213ba2c3f60f8b9fc530f7c4cdc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2756d213ba2c3f60f8b9fc530f7c4cdc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#afecdac1581fada66d09c1ab40cfc23e"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a><a class="idref" href="mathcomp.fingroup.action.html#2756d213ba2c3f60f8b9fc530f7c4cdc"><span class="id" title="notation">)</span></a>) (<a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.jordanholder.html#N2"><span class="id" title="variable">N2</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#N1"><span class="id" title="variable">N1</span></a><a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">)</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.AuxiliaryLemmas"><span class="id" title="section">AuxiliaryLemmas</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="StrongJordanHolder.aT"><span class="id" title="variable">aT</span></a> <a name="StrongJordanHolder.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/>
<span class="id" title="keyword">Variables</span> (<a name="StrongJordanHolder.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="StrongJordanHolder.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="StrongJordanHolder.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#D"><span class="id" title="variable">D</span></a>).<br/>

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</div>

<div class="doc">
 The factors associated to two A-stable composition series of the same      
 group are the same up to isomorphism and permutation                        
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<div class="code">

<br/>
<span class="id" title="keyword">Lemma</span> <a name="StrongJordanHolderUniqueness"><span class="id" title="lemma">StrongJordanHolderUniqueness</span></a> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#acomps"><span class="id" title="definition">acomps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#acomps"><span class="id" title="definition">acomps</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.solvable.jordanholder.html#mkfactors"><span class="id" title="definition">mkfactors</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.solvable.jordanholder.html#mkfactors"><span class="id" title="definition">mkfactors</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.jordanholder.html#s2"><span class="id" title="variable">s2</span></a>).<br/>

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<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.jordanholder.html#StrongJordanHolder"><span class="id" title="section">StrongJordanHolder</span></a>.<br/>

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