generate the documentation for 1.7

author: Enrico Tassi 2018-04-20 10:54:22 +0200
committer: Enrico Tassi 2018-04-20 10:54:22 +0200
commit: ed05182cece6bb3706e09b2ce14af4a41a2e8141 (patch)
tree: e850d7314b6372d0476cf2ffaf7d3830721db7b1 /docs/htmldoc/mathcomp.fingroup.action.html
parent: 3d196f44681fb3b23ff8a79fbd44e12308680531 (diff)
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+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>mathcomp.fingroup.action</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library mathcomp.fingroup.action</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">
+ Group action: orbits, stabilisers, transitivity.                           
+        is_action D to == the function to : T -&gt; aT -&gt; T defines an action  
+                          of D : {set aT} on T.                             
+            action D T == structure for a function defining an action of D. 
+            act_dom to == the domain D of to : action D rT.                 
+    {action: aT &amp;-&gt; T} == structure for a total action.                     
+                       := action [set: aT] T                                
+   TotalAction to1 toM == the constructor for total actions; to1 and toM    
+                          are the proofs of the action identities for 1 and 
+                          a * b, respectively.                              
+   is_groupAction R to == to is a group action on range R: for all a in D,  
+                          the permutation induced by to a is in Aut R. Thus 
+                          the action of D must be trivial outside R.        
+       groupAction D R == the structure for group actions of D on R. This   
+                          is a telescope on action D rT.                    
+         gact_range to == the range R of to : groupAction D R.              
+     GroupAction toAut == constructs a groupAction for action to from       
+                          toAut : actm to @* D \subset Aut R (actm to is    
+                          the morphism to {perm rT} associated to 'to').    
+      orbit to A x == the orbit of x under the action of A via to.          
+ orbit_transversal to A S == a transversal of the partition orbit to A @: S 
+                      of S, provided A acts on S via to.                    
+    amove to A x y == the set of a in A whose action sends x to y.          
+      'C_A[x | to] == the stabiliser of x : rT in A :&amp;: D.                  
+      'C_A(S | to) == the pointwise stabiliser of S : {set rT} in D :&amp;: A.  
+      'N_A(S | to) == the global stabiliser of S : {set rT} in D :&amp;: A.     
+  'Fix(S | to) [a] == the set of fixpoints of a in S.                       
+  'Fix(S | to)(A) == the set of fixpoints of A in S.                       
+ In the first three <i>A can be omitted and defaults to the domain D of to;   
+ in the last two S can be omitted and defaults to [set: T], so 'Fix_to[a]   
+ is the set of all fixpoints of a.                                          
+   The domain restriction ensures that stabilisers have a canonical group   
+ structure, but note that 'Fix sets are generally not groups. Indeed, we    
+ provide alternative definitions when to is a group action on R:            
+      'C</i>(G | to)(A) == the centraliser in R :&amp;: G of the group action of   
+                        D :&amp;: A via to                                      
+      'C(G | to) [a] == the centraliser in R :&amp;: G of a \in D, via to.      
+   These sets are groups when G is; G can be omitted: 'C(|to)(A) is the     
+ centraliser in R of the action of D :&amp;: A via to.                          
+          [acts A, on S | to] == A \subset D acts on the set S via to.      
+          {acts A, on S | to} == A acts on the set S (Prop statement).      
+    {acts A, on group G | to} == [acts A, on S | to] /\ G \subset R, i.e.,  
+                                 A \subset D acts on G \subset R, via       
+                                 to : groupAction D R.                      
+    [transitive A, on S | to] == A acts transitively on S.                  
+      [faithful A, on S | to] == A acts faithfully on S.                    
+      acts_irreducibly to A G == A acts irreducibly via the groupAction to  
+                                 on the nontrivial group G, i.e., A does    
+                                 not act on any nontrivial subgroup of G.   
+ Important caveat: the definitions of orbit, amove, 'Fix(S | to)(A),       
+ transitive and faithful assume that A is a subset of the domain D. As most 
+ of the permutation actions we consider are total this is usually harmless. 
+ (Note that the theory of partial actions is only partially developed.)     
+   In all of the above, to is expected to be the actual action structure,   
+ not merely the function. There is a special scope %act for actions, and    
+ constructions and notations for many classical actions:                    
+       'P == natural action of a permutation group via aperm.               
+       'J == internal group action (conjugation) via conjg (_ ^ _).         
+       'R == regular group action (right translation) via mulg (_ * _).     
+            (However, to limit ambiguity, _ * _ is NOT a canonical action.) 
+     to^* == the action induced by to on {set rT} via to^* (== setact to).  
+      'Js == the internal action on subsets via _ :^ _, equivalent to 'J^*. 
+      'Rs == the regular action on subsets via rcoset, equivalent to 'R^*.  
+      'JG == the conjugation action on {group rT} via (_ :^ _)%G.           
+   to / H == the action induced by to on coset_of H via qact to H, and      
+             restricted to (qact_dom to H) == 'N(rcosets H 'N(H) | to^* ).  
+       'Q == the action induced to cosets by conjugation; the domain is     
+             qact_dom 'J H, which is provably equal to 'N(H).               
+  to %% A == the action of coset_of A via modact to A, with domain D / A    
+             and support restricted to 'C(D :&amp;: A | to).                    
+ to \ sAD == the action of A via ract to sAD == to, if sAD : A \subset D.   
+  [Aut G] == the permutation action restricted to Aut G, via autact G.      
+  &lt; [nRA]&gt; == the action of A on R via actby nRA == to in A and on R, and    
+             the trivial action elsewhere; here nRA : [acts A, on R | to]   
+             or nRA : {acts A, on group R | to}.                            
+     to^? == the action induced by to on sT : @subType rT P, via subact to  
+             with domain subact_dom P to == 'N( [set x | P x] | to).         
+  <tt>phi</tt> == the action of phi : D &gt;-&gt; {perm rT}, via mact phi.             
+  to \o f == the composite action (with domain f @*^-1 D) of the action to  
+             with f : {morphism G &gt;-&gt; aT}, via comp_act to f. Here f must   
+             be the actual morphism object (e.g., coset_morphism H), not    
+             the underlying function (e.g., coset H).                       
+ The explicit application of an action to is usually written (to%act x a),  
+ but %act can be omitted if to is an abstract action or a set action to0^*. 
+ Note that this form will simplify and expose the acting function.          
+   There is a %gact scope for group actions; the notations above are        
+ recognised in %gact when they denote canonical group actions.              
+   Actions can be used to define morphisms:                                 
+ actperm to == the morphism D &gt;-&gt; {perm rT} induced by to.                  
+  actm to a == if a \in D the function on D induced by the action to, else  
+               the identity function. If to is a group action with range R  
+               then actm to a is canonically a morphism on R.               
+ We also define here the restriction operation on permutations (the domain  
+ of this operations is a stabiliser), and local automorphism groups:        
+  restr_perm S p == if p acts on S, the permutation with support in S that  
+                    coincides with p on S; else the identity. Note that     
+                    restr_perm is a permutation group morphism that maps    
+                    Aut G to Aut S when S is a subgroup of G.               
+      Aut_in A G == the local permutation group 'N_A(G | 'P) / 'C_A(G | 'P) 
+                    Usually A is an automorphism group, and then Aut_in A G 
+                    is isomorphic to a subgroup of Aut G, specifically      
+                    restr_perm @* A.                                        
+  Finally, gproduct.v will provide a semi-direct group construction that    
+ maps an external group action to an internal one; the theory of morphisms  
+ between such products makes use of the following definition:               
+  morph_act to to' f fA &lt;=&gt; the action of to' on the images of f and fA is  
+                   the image of the action of to, i.e., for all x and a we  
+                   have f (to x a) = to' (f x) (fA a). Note that there is   
+                   no mention of the domains of to and to'; if needed, this 
+                   predicate should be restricted via the {in ...} notation 
+                   and domain conditions should be added.                    
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ActionDef"><span class="id" title="section">ActionDef</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="ActionDef.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="ActionDef.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="ActionDef.rT"><span class="id" title="variable">rT</span></a> : <span class="id" title="keyword">Type</span>).<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.action.html#ActionDef.aT"><span class="id" title="variable">aT</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#ActionDef.rT"><span class="id" title="variable">rT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="act_morph"><span class="id" title="definition">act_morph</span></a> <span class="id" title="var">to</span> <span class="id" title="var">x</span> := <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="is_action"><span class="id" title="definition">is_action</span></a> <span class="id" title="var">to</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_injective"><span class="id" title="definition">left_injective</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActionDef.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#act_morph"><span class="id" title="definition">act_morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.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#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Record</span> <a name="action"><span class="id" title="record">action</span></a> := <a name="Action"><span class="id" title="constructor">Action</span></a> {<a name="act"><span class="id" title="projection">act</span></a> :&gt; <a class="idref" href="mathcomp.fingroup.action.html#ActionDef.rT"><span class="id" title="variable">rT</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#ActionDef.aT"><span class="id" title="variable">aT</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#ActionDef.rT"><span class="id" title="variable">rT</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#act"><span class="id" title="method">act</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="clone_action"><span class="id" title="definition">clone_action</span></a> <span class="id" title="var">to</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <span class="id" title="var">_</span> <span class="id" title="var">toP</span> := <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">type</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><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#action"><span class="id" title="record">action</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <span class="id" title="var">k</span> ⇒ <a class="idref" href="mathcomp.fingroup.action.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="var">toP</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#ActionDef"><span class="id" title="section">ActionDef</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Need to close the Section here to avoid re-declaring all Argument Scopes  
+</div>
+<div class="code">
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">action_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">act</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="5c606fa2629c1fac8f3ee3f6e9ad2934"><span class="id" title="notation">&quot;</span></a>{ 'action' aT &amp;-&gt; T }" := (<a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">:</span></a> <span class="id" title="var">aT</span><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">]</span></a> <span class="id" title="var">T</span>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "{ 'action'  aT  &amp;-&gt;  T }") : <span class="id" title="var">type_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">&quot;</span></a>[ 'action' 'of' to ]" := (<a class="idref" href="mathcomp.fingroup.action.html#clone_action"><span class="id" title="definition">clone_action</span></a> (@<a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">to</span>))<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'action'  'of'  to ]") : <span class="id" title="var">form_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="act_dom"><span class="id" title="definition">act_dom</span></a> <span class="id" title="var">aT</span> <span class="id" title="var">D</span> <span class="id" title="var">rT</span> <span class="id" title="keyword">of</span> @<a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.fingroup.action.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#rT"><span class="id" title="variable">rT</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#D"><span class="id" title="variable">D</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="TotalAction"><span class="id" title="section">TotalAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="TotalAction.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="TotalAction.rT"><span class="id" title="variable">rT</span></a> : <span class="id" title="keyword">Type</span>) (<a name="TotalAction.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#rT"><span class="id" title="variable">rT</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#aT"><span class="id" title="variable">aT</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#rT"><span class="id" title="variable">rT</span></a>).<br/>
+<span class="id" title="keyword">Hypotheses</span> (<a name="TotalAction.to1"><span class="id" title="variable">to1</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#TotalAction.to"><span class="id" title="variable">to</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> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>) (<a name="TotalAction.toM"><span class="id" title="variable">toM</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.action.html#act_morph"><span class="id" title="definition">act_morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="is_total_action"><span class="id" title="lemma">is_total_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalAction.to"><span class="id" title="variable">to</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="TotalAction"><span class="id" title="definition">TotalAction</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#is_total_action"><span class="id" title="lemma">is_total_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#TotalAction"><span class="id" title="section">TotalAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ActionDefs"><span class="id" title="section">ActionDefs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="ActionDefs.aT"><span class="id" title="variable">aT</span></a> <a name="ActionDefs.aT'"><span class="id" title="variable">aT'</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="ActionDefs.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="ActionDefs.D'"><span class="id" title="variable">D'</span></a> : <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.fingroup.action.html#aT'"><span class="id" title="variable">aT'</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="morph_act"><span class="id" title="definition">morph_act</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">rT'</span> (<span class="id" title="var">to</span> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActionDefs.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#rT"><span class="id" title="variable">rT</span></a>) (<span class="id" title="var">to'</span> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActionDefs.D'"><span class="id" title="variable">D'</span></a> <a class="idref" href="mathcomp.fingroup.action.html#rT'"><span class="id" title="variable">rT'</span></a>) <span class="id" title="var">f</span> <span class="id" title="var">fA</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">a</span>, <a class="idref" href="mathcomp.fingroup.action.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to'"><span class="id" title="variable">to'</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#fA"><span class="id" title="variable">fA</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="ActionDefs.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>. <span class="comment">(*&nbsp;Most&nbsp;definitions&nbsp;require&nbsp;a&nbsp;finType&nbsp;structure&nbsp;on&nbsp;rT&nbsp;*)</span><br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">to</span> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActionDefs.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActionDefs.rT"><span class="id" title="variable">rT</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">A</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.fingroup.action.html#ActionDefs.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">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.fingroup.action.html#ActionDefs.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/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="actm"><span class="id" title="definition">actm</span></a> <span class="id" title="var">to</span> <span class="id" title="var">a</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.action.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#ActionDefs.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</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.fingroup.action.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.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="setact"><span class="id" title="definition">setact</span></a> <span class="id" title="var">to</span> <span class="id" title="var">S</span> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="orbit"><span class="id" title="definition">orbit</span></a> <span class="id" title="var">to</span> <span class="id" title="var">A</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="amove"><span class="id" title="definition">amove</span></a> <span class="id" title="var">to</span> <span class="id" title="var">A</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="afix"><span class="id" title="definition">afix</span></a> <span class="id" title="var">to</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#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.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">]]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="astab"><span class="id" title="definition">astab</span></a> <span class="id" title="var">S</span> <span class="id" title="var">to</span> := <a class="idref" href="mathcomp.fingroup.action.html#ActionDefs.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">]]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="astabs"><span class="id" title="definition">astabs</span></a> <span class="id" title="var">S</span> <span class="id" title="var">to</span> := <a class="idref" href="mathcomp.fingroup.action.html#ActionDefs.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#aa0c936d7feb8f26825e601735c0125f"><span class="id" title="notation">@^-1:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="acts_on"><span class="id" title="definition">acts_on</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</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.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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#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.fingroup.action.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.fingroup.action.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.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="atrans"><span class="id" title="definition">atrans</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span> := <a class="idref" href="mathcomp.fingroup.action.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.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="faithful"><span class="id" title="definition">faithful</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span> := <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#astab"><span class="id" title="definition">astab</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</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#c33afa16525556de4ed568ad52c9389f"><span class="id" title="notation">[1]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#ActionDefs"><span class="id" title="section">ActionDefs</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">&quot;</span></a>to ^*" := (<a class="idref" href="mathcomp.fingroup.action.html#setact"><span class="id" title="definition">setact</span></a> <span class="id" title="var">to</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "to ^*") : <span class="id" title="var">fun_scope</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">&quot;</span></a>''Fix_' to ( A )" := (<a class="idref" href="mathcomp.fingroup.action.html#afix"><span class="id" title="definition">afix</span></a> <span class="id" title="var">to</span> <span class="id" title="var">A</span>)<br/>
+&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">to</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''Fix_' to ( A )") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ camlp4 grammar factoring  
+</div>
+<div class="code">
+<span class="id" title="keyword">Notation</span> <a name="ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">&quot;</span></a>''Fix_' ( to ) ( A )" := <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><span class="id" title="var">A</span><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">&quot;</span></a>''Fix_' ( S | to ) ( A )" := (<span class="id" title="var">S</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><span class="id" title="var">A</span><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''Fix_' ( S  |  to ) ( A )") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">&quot;</span></a>''Fix_' to [ a ]" := (<a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">to</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''Fix_' to [ a ]") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">&quot;</span></a>''Fix_' ( S | to ) [ a ]" := (<span class="id" title="var">S</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">[</span></a><span class="id" title="var">a</span><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">]</span></a>)<br/>
+&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''Fix_' ( S  |  to ) [ a ]") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">&quot;</span></a>''C' ( S | to )" := (<a class="idref" href="mathcomp.fingroup.action.html#astab"><span class="id" title="definition">astab</span></a> <span class="id" title="var">S</span> <span class="id" title="var">to</span>)<br/>
+&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''C' ( S  |  to )") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="652d6cc67746e5361142d90686607781"><span class="id" title="notation">&quot;</span></a>''C_' A ( S | to )" := (<span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">A</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''C_' A ( S  |  to )") : <span class="id" title="var">group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="44caf6cd334a2a1e387cb945c4fdb874"><span class="id" title="notation">&quot;</span></a>''C_' ( A ) ( S | to )" := <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">C_A</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">&quot;</span></a>''C' [ x | to ]" := (<a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''C' [ x  |  to ]") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">&quot;</span></a>''C_' A [ x | to ]" := (<span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">]</span></a>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">A</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''C_' A [ x  |  to ]") : <span class="id" title="var">group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="1fb3703b929cf964cd25eac50eea7151"><span class="id" title="notation">&quot;</span></a>''C_' ( A ) [ x | to ]" := <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_A</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a><br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">&quot;</span></a>''N' ( S | to )" := (<a class="idref" href="mathcomp.fingroup.action.html#astabs"><span class="id" title="definition">astabs</span></a> <span class="id" title="var">S</span> <span class="id" title="var">to</span>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''N' ( S  |  to )") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">&quot;</span></a>''N_' A ( S | to )" := (<span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</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><span class="id" title="var">S</span> <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">A</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''N_' A ( S  |  to )") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">&quot;</span></a>[ 'acts' A , 'on' S | to ]" := (<span class="id" title="var">A</span> <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.ssreflect.finset.html#pred_of_set"><span class="id" title="abbreviation">pred_of_set</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><span class="id" title="var">S</span> <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'acts'  A ,  'on'  S  |  to ]") : <span class="id" title="var">form_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="05449e0e2f87f06fca16a62c49e8b809"><span class="id" title="notation">&quot;</span></a>{ 'acts' A , 'on' S | to }" := (<a class="idref" href="mathcomp.fingroup.action.html#acts_on"><span class="id" title="definition">acts_on</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "{ 'acts'  A ,  'on'  S  |  to }") : <span class="id" title="var">form_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">&quot;</span></a>[ 'transitive' A , 'on' S | to ]" := (<a class="idref" href="mathcomp.fingroup.action.html#atrans"><span class="id" title="definition">atrans</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'transitive'  A ,  'on'  S  |  to ]") : <span class="id" title="var">form_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">&quot;</span></a>[ 'faithful' A , 'on' S | to ]" := (<a class="idref" href="mathcomp.fingroup.action.html#faithful"><span class="id" title="definition">faithful</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'faithful'  A ,  'on'  S  |  to ]") : <span class="id" title="var">form_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="RawAction"><span class="id" title="section">RawAction</span></a>.<br/>
+</div>
+
+<div class="doc">
+ Lemmas that do not require the group structure on the action domain. 
+ Some lemmas like actMin would be actually be valid for arbitrary rT, 
+ e.g., for actions on a function type, but would be difficult to use  
+ as a view due to the confusion between parameters and assumptions.    
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RawAction.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="RawAction.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="RawAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="RawAction.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#rT"><span class="id" title="variable">rT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#RawAction.aT"><span class="id" title="variable">aT</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.action.html#RawAction.rT"><span class="id" title="variable">rT</span></a>) (<span class="id" title="var">A</span> <span class="id" title="var">B</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.fingroup.action.html#RawAction.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">S</span> <span class="id" title="var">T</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.fingroup.action.html#RawAction.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/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="act_inj"><span class="id" title="lemma">act_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_injective"><span class="id" title="definition">left_injective</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actMin"><span class="id" title="lemma">actMin</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#act_morph"><span class="id" title="definition">act_morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.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#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actmEfun"><span class="id" title="lemma">actmEfun</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#RawAction.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#actm"><span class="id" title="definition">actm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actmE"><span class="id" title="lemma">actmE</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#RawAction.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#actm"><span class="id" title="definition">actm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="setactE"><span class="id" title="lemma">setactE</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="mem_setact"><span class="id" title="lemma">mem_setact</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="card_setact"><span class="id" title="lemma">card_setact</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="setact_is_action"><span class="id" title="lemma">setact_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">set_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#setact_is_action"><span class="id" title="lemma">setact_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbitE"><span class="id" title="lemma">orbitE</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbitP"><span class="id" title="lemma">orbitP</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> <span class="id" title="var">y</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.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.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#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="mem_orbit"><span class="id" title="lemma">mem_orbit</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#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="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixP"><span class="id" title="lemma">afixP</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="mathcomp.fingroup.action.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#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="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixS"><span class="id" title="lemma">afixS</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.action.html#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#B"><span class="id" title="variable">B</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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixU"><span class="id" title="lemma">afixU</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix1P"><span class="id" title="lemma">afix1P</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">]</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabIdom"><span class="id" title="lemma">astabIdom</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">C_D</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_dom"><span class="id" title="lemma">astab_dom</span></a> <span class="id" title="var">S</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_act"><span class="id" title="lemma">astab_act</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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#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.fingroup.action.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.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabS"><span class="id" title="lemma">astabS</span></a> <span class="id" title="var">S1</span> <span class="id" title="var">S2</span> : <a class="idref" href="mathcomp.fingroup.action.html#S1"><span class="id" title="variable">S1</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#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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S2"><span class="id" title="variable">S2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabsIdom"><span class="id" title="lemma">astabsIdom</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">N_D</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><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.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.fingroup.action.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.fingroup.action.html#RawAction.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="astabs_dom"><span class="id" title="lemma">astabs_dom</span></a> <span class="id" title="var">S</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.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.fingroup.action.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.fingroup.action.html#RawAction.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> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs_act"><span class="id" title="lemma">astabs_act</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.html#RawAction.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> <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.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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#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.fingroup.action.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.fingroup.action.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>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_sub"><span class="id" title="lemma">astab_sub</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</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.fingroup.action.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.fingroup.action.html#RawAction.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="astabsC"><span class="id" title="lemma">astabsC</span></a> <span class="id" title="var">S</span> : <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.ssreflect.finset.html#08cc1b0d2ac8db12b5c416dfc52232cc"><span class="id" title="notation">~:</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#RawAction.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> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.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.fingroup.action.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.fingroup.action.html#RawAction.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="astabsI"><span class="id" title="lemma">astabsI</span></a> <span class="id" title="var">S</span> <span class="id" title="var">T</span> : <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.fingroup.action.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.fingroup.action.html#RawAction.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> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</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.fingroup.action.html#T"><span class="id" title="variable">T</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#RawAction.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> <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.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#T"><span class="id" title="variable">T</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#RawAction.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="astabs_setact"><span class="id" title="lemma">astabs_setact</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.html#RawAction.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> <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#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1_set"><span class="id" title="lemma">astab1_set</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#set_action"><span class="id" title="definition">set_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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.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.fingroup.action.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.fingroup.action.html#RawAction.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="astabs_set1"><span class="id" title="lemma">astabs_set1</span></a> <span class="id" title="var">x</span> : <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.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</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#RawAction.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> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_dom"><span class="id" title="lemma">acts_dom</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> : <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#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.action.html#S"><span class="id" title="variable">S</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#RawAction.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#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#RawAction.D"><span class="id" title="variable">D</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_act"><span class="id" title="lemma">acts_act</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> : <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#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.action.html#S"><span class="id" title="variable">S</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#RawAction.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#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.fingroup.action.html#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.fingroup.action.html#S"><span class="id" title="variable">S</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#RawAction.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>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabCin"><span class="id" title="lemma">astabCin</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#RawAction.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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">Section</span> <a name="RawAction.ActsSetop"><span class="id" title="section">ActsSetop</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RawAction.ActsSetop.A"><span class="id" title="variable">A</span></a> : <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.fingroup.action.html#RawAction.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="RawAction.ActsSetop.S"><span class="id" title="variable">S</span></a> <a name="RawAction.ActsSetop.T"><span class="id" title="variable">T</span></a> : <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.fingroup.action.html#RawAction.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/>
+<span class="id" title="keyword">Hypotheses</span> (<a name="RawAction.ActsSetop.AactS"><span class="id" title="variable">AactS</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.fingroup.action.html#RawAction.ActsSetop.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.action.html#RawAction.ActsSetop.S"><span class="id" title="variable">S</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#RawAction.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 name="RawAction.ActsSetop.AactT"><span class="id" title="variable">AactT</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.fingroup.action.html#RawAction.ActsSetop.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.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</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#RawAction.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="astabU"><span class="id" title="lemma">astabU</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabsU"><span class="id" title="lemma">astabsU</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.fingroup.action.html#RawAction.ActsSetop.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.fingroup.action.html#RawAction.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> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</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.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</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#RawAction.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> <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.fingroup.action.html#RawAction.ActsSetop.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</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#RawAction.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="astabsD"><span class="id" title="lemma">astabsD</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.fingroup.action.html#RawAction.ActsSetop.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.fingroup.action.html#RawAction.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> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</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.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</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#RawAction.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> <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.fingroup.action.html#RawAction.ActsSetop.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</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#RawAction.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="actsI"><span class="id" title="lemma">actsI</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.fingroup.action.html#RawAction.ActsSetop.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.action.html#RawAction.ActsSetop.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</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#RawAction.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="actsU"><span class="id" title="lemma">actsU</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.fingroup.action.html#RawAction.ActsSetop.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.action.html#RawAction.ActsSetop.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</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#RawAction.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="actsD"><span class="id" title="lemma">actsD</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.fingroup.action.html#RawAction.ActsSetop.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.action.html#RawAction.ActsSetop.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop.T"><span class="id" title="variable">T</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#RawAction.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">End</span> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.ActsSetop"><span class="id" title="section">ActsSetop</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_in_orbit"><span class="id" title="lemma">acts_in_orbit</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<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#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.action.html#S"><span class="id" title="variable">S</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#RawAction.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#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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.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#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#S"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="subset_faithful"><span class="id" title="lemma">subset_faithful</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">faithful</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><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#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">faithful</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a name="RawAction.Reindex"><span class="id" title="section">Reindex</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RawAction.Reindex.vT"><span class="id" title="variable">vT</span></a> : <span class="id" title="keyword">Type</span>) (<a name="RawAction.Reindex.idx"><span class="id" title="variable">idx</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#vT"><span class="id" title="variable">vT</span></a>) (<a name="RawAction.Reindex.op"><span class="id" title="variable">op</span></a> : <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.com_law"><span class="id" title="record">Monoid.com_law</span></a> <a class="idref" href="mathcomp.fingroup.action.html#idx"><span class="id" title="variable">idx</span></a>) (<a name="RawAction.Reindex.S"><span class="id" title="variable">S</span></a> : <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.fingroup.action.html#RawAction.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/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="reindex_astabs"><span class="id" title="lemma">reindex_astabs</span></a> <span class="id" title="var">a</span> <span class="id" title="var">F</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#RawAction.Reindex.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.fingroup.action.html#RawAction.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> <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.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.op"><span class="id" title="variable">op</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.idx"><span class="id" title="variable">idx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.action.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.action.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.op"><span class="id" title="variable">op</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.idx"><span class="id" title="variable">idx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.action.html#F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="reindex_acts"><span class="id" title="lemma">reindex_acts</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">F</span> : <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#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.action.html#RawAction.Reindex.S"><span class="id" title="variable">S</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#RawAction.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#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#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><br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.op"><span class="id" title="variable">op</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.idx"><span class="id" title="variable">idx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.action.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.action.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.op"><span class="id" title="variable">op</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.idx"><span class="id" title="variable">idx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9b4515ceb280b6b5a2638c4e28ba3f31"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.action.html#F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#RawAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#RawAction.Reindex"><span class="id" title="section">Reindex</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#RawAction"><span class="id" title="section">RawAction</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Warning: this directive depends on names of bound variables in the 
+ definition of injective, in ssrfun.v.                               
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="495b9dab41eed2da7a7d91b1cb8498af"><span class="id" title="notation">&quot;</span></a>to ^*" := (<a class="idref" href="mathcomp.fingroup.action.html#set_action"><span class="id" title="definition">set_action</span></a> <span class="id" title="var">to</span>) : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PartialAction"><span class="id" title="section">PartialAction</span></a>.<br/>
+</div>
+
+<div class="doc">
+ Lemmas that require a (partial) group domain.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="PartialAction.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="PartialAction.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.fingroup.action.html#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="PartialAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+<span class="id" title="keyword">Variable</span> <a name="PartialAction.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.rT"><span class="id" title="variable">rT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.aT"><span class="id" title="variable">aT</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.rT"><span class="id" title="variable">rT</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#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.fingroup.action.html#PartialAction.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">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.fingroup.action.html#PartialAction.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">Implicit</span> <span class="id" title="keyword">Types</span> <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.fingroup.action.html#PartialAction.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/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="act1"><span class="id" title="lemma">act1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</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="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actKin"><span class="id" title="lemma">actKin</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_loop"><span class="id" title="definition">right_loop</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#invg"><span class="id" title="definition">invg</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actKVin"><span class="id" title="lemma">actKVin</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#rev_right_loop"><span class="id" title="definition">rev_right_loop</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#invg"><span class="id" title="definition">invg</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="setactVin"><span class="id" title="lemma">setactVin</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#PartialAction.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#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#aa0c936d7feb8f26825e601735c0125f"><span class="id" title="notation">@^-1:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actXin"><span class="id" title="lemma">actXin</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.fingroup.action.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#PartialAction.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#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#86a04fb564fb97d388cad84a3a204260"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.fingroup.action.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.fingroup.action.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix1"><span class="id" title="lemma">afix1</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixD1"><span class="id" title="lemma">afixD1</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">^#</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_refl"><span class="id" title="lemma">orbit_refl</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="contra_orbit"><span class="id" title="lemma">contra_orbit</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.action.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#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_in_sym"><span class="id" title="lemma">orbit_in_sym</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#symmetric"><span class="id" title="definition">symmetric</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#orbit_rel"><span class="id" title="abbreviation">orbit_rel</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_in_trans"><span class="id" title="lemma">orbit_in_trans</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#orbit_rel"><span class="id" title="abbreviation">orbit_rel</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_in_eqP"><span class="id" title="lemma">orbit_in_eqP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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="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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_in_transl"><span class="id" title="lemma">orbit_in_transl</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_act_in"><span class="id" title="lemma">orbit_act_in</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> <span class="id" title="var">G</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#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#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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_actr_in"><span class="id" title="lemma">orbit_actr_in</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> <span class="id" title="var">G</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#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#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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.action.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_inv_in"><span class="id" title="lemma">orbit_inv_in</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#PartialAction.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_lcoset_in"><span class="id" title="lemma">orbit_lcoset_in</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#PartialAction.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#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#PartialAction.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7a1b86b3dd47c101ea1643c2a591eaf1"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_rcoset_in"><span class="id" title="lemma">orbit_rcoset_in</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#PartialAction.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#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#PartialAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.action.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_conjsg_in"><span class="id" title="lemma">orbit_conjsg_in</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#PartialAction.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#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#PartialAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.action.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://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.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit1P"><span class="id" title="lemma">orbit1P</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a>) (<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="card_orbit1"><span class="id" title="lemma">card_orbit1</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_partition"><span class="id" title="lemma">orbit_partition</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<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#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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#PartialAction.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.ssreflect.finset.html#partition"><span class="id" title="definition">partition</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="orbit_transversal"><span class="id" title="definition">orbit_transversal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#transversal"><span class="id" title="definition">transversal</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_transversalP"><span class="id" title="lemma">orbit_transversalP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> (<span class="id" title="var">P</span> := <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">X</span> := <a class="idref" href="mathcomp.fingroup.action.html#orbit_transversal"><span class="id" title="definition">orbit_transversal</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#PartialAction.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;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#is_transversal"><span class="id" title="definition">is_transversal</span></a> <a class="idref" href="mathcomp.fingroup.action.html#X"><span class="id" title="variable">X</span></a> <a class="idref" href="mathcomp.fingroup.action.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#X"><span class="id" title="variable">X</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#S"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.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#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <span class="id" title="keyword">∀</span> <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.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">&amp;</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.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#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#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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#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#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="group_set_astab"><span class="id" title="lemma">group_set_astab</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">astab_group</span> <span class="id" title="var">S</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#group"><span class="id" title="definition">group</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#group_set_astab"><span class="id" title="lemma">group_set_astab</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix_gen_in"><span class="id" title="lemma">afix_gen_in</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#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#PartialAction.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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&lt;&lt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&gt;&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix_cycle_in"><span class="id" title="lemma">afix_cycle_in</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#PartialAction.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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">]&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixYin"><span class="id" title="lemma">afixYin</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#PartialAction.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#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.fingroup.action.html#PartialAction.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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation">&lt;*&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixMin"><span class="id" title="lemma">afixMin</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#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.fingroup.action.html#PartialAction.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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="sub_astab1_in"><span class="id" title="lemma">sub_astab1_in</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#PartialAction.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#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#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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="group_set_astabs"><span class="id" title="lemma">group_set_astabs</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.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.fingroup.action.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.fingroup.action.html#PartialAction.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">Canonical</span> <span class="id" title="var">astabs_group</span> <span class="id" title="var">S</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#group"><span class="id" title="definition">group</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#group_set_astabs"><span class="id" title="lemma">group_set_astabs</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_norm"><span class="id" title="lemma">astab_norm</span></a> <span class="id" title="var">S</span> : <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.fingroup.action.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.fingroup.action.html#PartialAction.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> <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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_normal"><span class="id" title="lemma">astab_normal</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</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.fingroup.action.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.fingroup.action.html#PartialAction.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="acts_sub_orbit"><span class="id" title="lemma">acts_sub_orbit</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> <span class="id" title="var">x</span> :<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#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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#PartialAction.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</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#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.fingroup.action.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.fingroup.action.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>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_orbit"><span class="id" title="lemma">acts_orbit</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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.fingroup.action.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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</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#PartialAction.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="acts_subnorm_fix"><span class="id" title="lemma">acts_subnorm_fix</span></a> <span class="id" title="var">A</span> : <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.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><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">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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#PartialAction.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="atrans_orbit"><span class="id" title="lemma">atrans_orbit</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PartialAction.OrbitStabilizer"><span class="id" title="section">OrbitStabilizer</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</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.action.html#PartialAction.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="PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.rT"><span class="id" title="variable">rT</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="PartialAction.OrbitStabilizer.sGD"><span class="id" title="variable">sGD</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.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.fingroup.action.html#PartialAction.D"><span class="id" title="variable">D</span></a>.<br/>
+<span class="id" title="keyword">Let</span> <a name="PartialAction.OrbitStabilizer.ssGD"><span class="id" title="variable">ssGD</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#subsetP"><span class="id" title="lemma">subsetP</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.sGD"><span class="id" title="variable">sGD</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="amove_act"><span class="id" title="lemma">amove_act</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#PartialAction.OrbitStabilizer.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.fingroup.action.html#amove"><span class="id" title="definition">amove</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="amove_orbit"><span class="id" title="lemma">amove_orbit</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#amove"><span class="id" title="definition">amove</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="amoveK"><span class="id" title="lemma">amoveK</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#amove"><span class="id" title="definition">amove</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">Ca</span> ⇒ <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Ca"><span class="id" title="variable">Ca</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_stabilizer"><span class="id" title="lemma">orbit_stabilizer</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Ca"><span class="id" title="variable">Ca</span></a>) <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">|</span></a> <span class="id" title="var">Ca</span> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="act_reprK"><span class="id" title="lemma">act_reprK</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#amove"><span class="id" title="definition">amove</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.OrbitStabilizer"><span class="id" title="section">OrbitStabilizer</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="card_orbit_in"><span class="id" title="lemma">card_orbit_in</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="card_orbit_in_stab"><span class="id" title="lemma">card_orbit_in_stab</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_sum_card_orbit"><span class="id" title="lemma">acts_sum_card_orbit</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<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#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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#PartialAction.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.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">(</span></a><span class="id" title="var">T</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#T"><span class="id" title="variable">T</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_setact_in"><span class="id" title="lemma">astab_setact_in</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#PartialAction.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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1_act_in"><span class="id" title="lemma">astab1_act_in</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#PartialAction.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#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a name="Frobenius_Cauchy"><span class="id" title="lemma">Frobenius_Cauchy</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> : <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#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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#PartialAction.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.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">(</span></a><span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atrans_dvd_index_in"><span class="id" title="lemma">atrans_dvd_index_in</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#aa34fd1c61c5cf0a3356b624a5d2afed"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atrans_dvd_in"><span class="id" title="lemma">atrans_dvd_in</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#aa34fd1c61c5cf0a3356b624a5d2afed"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atransPin"><span class="id" title="lemma">atransPin</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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;<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atransP2in"><span class="id" title="lemma">atransP2in</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <span class="id" title="keyword">∀</span> <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.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.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#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#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atrans_acts_in"><span class="id" title="lemma">atrans_acts_in</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#PartialAction.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#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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.fingroup.action.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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#PartialAction.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="subgroup_transitivePin"><span class="id" title="lemma">subgroup_transitivePin</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">S</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.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.fingroup.action.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.fingroup.action.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.fingroup.action.html#PartialAction.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#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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="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#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#PartialAction"><span class="id" title="section">PartialAction</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">&quot;</span></a>''C' ( S | to )" := (<a class="idref" href="mathcomp.fingroup.action.html#astab_group"><span class="id" title="definition">astab_group</span></a> <span class="id" title="var">to</span> <span class="id" title="var">S</span>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="604222e4f2ad6aacb5fe189ffe7e1dce"><span class="id" title="notation">&quot;</span></a>''C_' A ( S | to )" := (<a class="idref" href="mathcomp.fingroup.fingroup.html#setI_group"><span class="id" title="definition">setI_group</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span> <a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="5b118df1249ebe77f4eea4d7d4bb73a3"><span class="id" title="notation">&quot;</span></a>''C_' ( A ) ( S | to )" := (<a class="idref" href="mathcomp.fingroup.fingroup.html#setI_group"><span class="id" title="definition">setI_group</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span> <a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;&nbsp;(<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">&quot;</span></a>''C' [ x | to ]" := (<a class="idref" href="mathcomp.fingroup.action.html#astab_group"><span class="id" title="definition">astab_group</span></a> <span class="id" title="var">to</span> <a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span>%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="619a9997befee6604ab4a6637eedd029"><span class="id" title="notation">&quot;</span></a>''C_' A [ x | to ]" := (<a class="idref" href="mathcomp.fingroup.fingroup.html#setI_group"><span class="id" title="definition">setI_group</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">[</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">]</span></a>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="ecd380e2cecb617207c50384383b0bc7"><span class="id" title="notation">&quot;</span></a>''C_' ( A ) [ x | to ]" := (<a class="idref" href="mathcomp.fingroup.fingroup.html#setI_group"><span class="id" title="definition">setI_group</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">[</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#01e2893fcf35e18b1e93a5c4b5f16e1c"><span class="id" title="notation">]</span></a>)<br/>
+&nbsp;&nbsp;(<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="d8a7e9dd3de10046d82874af3a1cdeb8"><span class="id" title="notation">&quot;</span></a>''N' ( S | to )" := (<a class="idref" href="mathcomp.fingroup.action.html#astabs_group"><span class="id" title="definition">astabs_group</span></a> <span class="id" title="var">to</span> <span class="id" title="var">S</span>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="dd7f1c8bab5f82c80672e99d9005640a"><span class="id" title="notation">&quot;</span></a>''N_' A ( S | to )" := (<a class="idref" href="mathcomp.fingroup.fingroup.html#setI_group"><span class="id" title="definition">setI_group</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.fingroup.action.html#d8a7e9dd3de10046d82874af3a1cdeb8"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d8a7e9dd3de10046d82874af3a1cdeb8"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.action.html#d8a7e9dd3de10046d82874af3a1cdeb8"><span class="id" title="notation">(</span></a><span class="id" title="var">S</span> <a class="idref" href="mathcomp.fingroup.action.html#d8a7e9dd3de10046d82874af3a1cdeb8"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#d8a7e9dd3de10046d82874af3a1cdeb8"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">Group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="TotalActions"><span class="id" title="section">TotalActions</span></a>.<br/>
+</div>
+
+<div class="doc">
+ These lemmas are only established for total actions (domain = [set: rT])  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variable</span> (<a name="TotalActions.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="TotalActions.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="TotalActions.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#5c606fa2629c1fac8f3ee3f6e9ad2934"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.action.html#5c606fa2629c1fac8f3ee3f6e9ad2934"><span class="id" title="notation">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.fingroup.action.html#5c606fa2629c1fac8f3ee3f6e9ad2934"><span class="id" title="notation">&amp;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.action.html#5c606fa2629c1fac8f3ee3f6e9ad2934"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.aT"><span class="id" title="variable">aT</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.rT"><span class="id" title="variable">rT</span></a>) (<span class="id" title="var">A</span> <span class="id" title="var">B</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.fingroup.action.html#TotalActions.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#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.action.html#TotalActions.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">Implicit</span> <span class="id" title="keyword">Type</span> <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.fingroup.action.html#TotalActions.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/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actM"><span class="id" title="lemma">actM</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actK"><span class="id" title="lemma">actK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_loop"><span class="id" title="definition">right_loop</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#invg"><span class="id" title="definition">invg</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actKV"><span class="id" title="lemma">actKV</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#rev_right_loop"><span class="id" title="definition">rev_right_loop</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#invg"><span class="id" title="definition">invg</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actX"><span class="id" title="lemma">actX</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#86a04fb564fb97d388cad84a3a204260"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.fingroup.action.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.fingroup.action.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actCJ"><span class="id" title="lemma">actCJ</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actCJV"><span class="id" title="lemma">actCJV</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a>)) <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_sym"><span class="id" title="lemma">orbit_sym</span></a> <span class="id" title="var">G</span> <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.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_trans"><span class="id" title="lemma">orbit_trans</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#z"><span class="id" title="variable">z</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_eqP"><span class="id" title="lemma">orbit_eqP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> <span class="id" title="var">y</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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_transl"><span class="id" title="lemma">orbit_transl</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_act"><span class="id" title="lemma">orbit_act</span></a> <span class="id" title="var">G</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span>: <a class="idref" href="mathcomp.fingroup.action.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#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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_actr"><span class="id" title="lemma">orbit_actr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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#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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.action.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_eq_mem"><span class="id" title="lemma">orbit_eq_mem</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_inv"><span class="id" title="lemma">orbit_inv</span></a> <span class="id" title="var">A</span> <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.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_lcoset"><span class="id" title="lemma">orbit_lcoset</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7a1b86b3dd47c101ea1643c2a591eaf1"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_rcoset"><span class="id" title="lemma">orbit_rcoset</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.action.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_conjsg"><span class="id" title="lemma">orbit_conjsg</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.action.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://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.fingroup.action.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#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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabP"><span class="id" title="lemma">astabP</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.fingroup.action.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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1P"><span class="id" title="lemma">astab1P</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.fingroup.action.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#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">]</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="sub_astab1"><span class="id" title="lemma">sub_astab1</span></a> <span class="id" title="var">A</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.fingroup.action.html#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#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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="astabC"><span class="id" title="lemma">astabC</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</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.fingroup.action.html#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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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="afix_cycle"><span class="id" title="lemma">afix_cycle</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">]&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#726946a87895fde82e79cb27daba28a9"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix_gen"><span class="id" title="lemma">afix_gen</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&lt;&lt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&gt;&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixM"><span class="id" title="lemma">afixM</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabsP"><span class="id" title="lemma">astabsP</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.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.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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#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.fingroup.action.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.fingroup.action.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.fingroup.action.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.fingroup.action.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.fingroup.action.html#TotalActions.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="card_orbit"><span class="id" title="lemma">card_orbit</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="dvdn_orbit"><span class="id" title="lemma">dvdn_orbit</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#aa34fd1c61c5cf0a3356b624a5d2afed"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="card_orbit_stab"><span class="id" title="lemma">card_orbit_stab</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actsP"><span class="id" title="lemma">actsP</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#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.fingroup.action.html#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.fingroup.action.html#S"><span class="id" title="variable">S</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#TotalActions.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.fingroup.action.html#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.action.html#S"><span class="id" title="variable">S</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#TotalActions.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="setact_orbit"><span class="id" title="lemma">setact_orbit</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_setact"><span class="id" title="lemma">astab_setact</span></a> <span class="id" title="var">S</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1_act"><span class="id" title="lemma">astab1_act</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atransP"><span class="id" title="lemma">atransP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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;<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atransP2"><span class="id" title="lemma">atransP2</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <span class="id" title="keyword">∀</span> <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.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">a</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.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#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#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atrans_acts"><span class="id" title="lemma">atrans_acts</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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.fingroup.action.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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#TotalActions.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="atrans_supgroup"><span class="id" title="lemma">atrans_supgroup</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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.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#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.action.html#S"><span class="id" title="variable">S</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#TotalActions.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="atrans_acts_card"><span class="id" title="lemma">atrans_acts_card</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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/>
+&nbsp;&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#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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#TotalActions.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atrans_dvd"><span class="id" title="lemma">atrans_dvd</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#aa34fd1c61c5cf0a3356b624a5d2afed"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ This is Aschbacher (5.2)  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="acts_fix_norm"><span class="id" title="lemma">acts_fix_norm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.action.html#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.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.fingroup.action.html#B"><span class="id" title="variable">B</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.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#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.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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#TotalActions.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="faithfulP"><span class="id" title="lemma">faithfulP</span></a> <span class="id" title="var">A</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> (<span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="mathcomp.fingroup.action.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#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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">faithful</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ This is the first part of Aschbacher (5.7)  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="astab_trans_gcore"><span class="id" title="lemma">astab_trans_gcore</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> <span class="id" title="var">u</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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#u"><span class="id" title="variable">u</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#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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.fingroup.html#gcore"><span class="id" title="definition">gcore</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ This is Aschbacher (5.20)  
+</div>
+<div class="code">
+<span class="id" title="keyword">Theorem</span> <a name="subgroup_transitiveP"><span class="id" title="lemma">subgroup_transitiveP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">S</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.fingroup.action.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.fingroup.action.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.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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="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#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ This is Aschbacher (5.21)  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="trans_subnorm_fixP"><span class="id" title="lemma">trans_subnorm_fixP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">C</span> := <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d6d8c6555a0082b6cbf651a274d2a4d6"><span class="id" title="notation">]</span></a> <span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><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#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.fingroup.action.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.fingroup.action.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.fingroup.action.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><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.finset.html#2dab1b186f22df1296e4090457f5abb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#6c121d1bcff5b1c0972474f398d18325"><span class="id" title="notation">:^:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#2dab1b186f22df1296e4090457f5abb0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#2dab1b186f22df1296e4090457f5abb0"><span class="id" title="notation">::&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#6c121d1bcff5b1c0972474f398d18325"><span class="id" title="notation">:^:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#C"><span class="id" title="variable">C</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#TotalActions"><span class="id" title="section">TotalActions</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="Restrict"><span class="id" title="section">Restrict</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="Restrict.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Restrict.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="Restrict.rT"><span class="id" title="variable">rT</span></a> : <span class="id" title="keyword">Type</span>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="Restrict.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Restrict.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Restrict.rT"><span class="id" title="variable">rT</span></a>) (<a name="Restrict.A"><span class="id" title="variable">A</span></a> : <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.fingroup.action.html#Restrict.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="ract"><span class="id" title="definition">ract</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.action.html#Restrict.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#Restrict.D"><span class="id" title="variable">D</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#act"><span class="id" title="projection">act</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Restrict.to"><span class="id" title="variable">to</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="Restrict.sAD"><span class="id" title="variable">sAD</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#Restrict.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#Restrict.D"><span class="id" title="variable">D</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="ract_is_action"><span class="id" title="lemma">ract_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Restrict.A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#ract"><span class="id" title="definition">ract</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Restrict.sAD"><span class="id" title="variable">sAD</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">raction</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ract_is_action"><span class="id" title="lemma">ract_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="ractE"><span class="id" title="lemma">ractE</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#raction"><span class="id" title="definition">raction</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Restrict.to"><span class="id" title="variable">to</span></a>.  <br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Other properties of raction need rT : finType; we defer them 
+ until after the definition of actperm.                        
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#Restrict"><span class="id" title="section">Restrict</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">&quot;</span></a>to \ sAD" := (<a class="idref" href="mathcomp.fingroup.action.html#raction"><span class="id" title="definition">raction</span></a> <span class="id" title="var">to</span> <span class="id" title="var">sAD</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50) : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ActBy"><span class="id" title="section">ActBy</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="ActBy.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="ActBy.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="ActBy.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="actby_cond"><span class="id" title="definition">actby_cond</span></a> (<span class="id" title="var">A</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.fingroup.action.html#ActBy.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">R</span> (<span class="id" title="var">to</span> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.rT"><span class="id" title="variable">rT</span></a>) : <span class="id" title="keyword">Prop</span> :=<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#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.action.html#R"><span class="id" title="variable">R</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#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">Definition</span> <a name="actby"><span class="id" title="definition">actby</span></a> <span class="id" title="var">A</span> <span class="id" title="var">R</span> <span class="id" title="var">to</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.action.html#actby_cond"><span class="id" title="definition">actby_cond</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> <span class="id" title="var">a</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="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.fingroup.action.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.fingroup.action.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="ActBy.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.fingroup.action.html#ActBy.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="ActBy.R"><span class="id" title="variable">R</span></a> : <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.fingroup.action.html#ActBy.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 name="ActBy.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.rT"><span class="id" title="variable">rT</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="ActBy.nRA"><span class="id" title="variable">nRA</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#actby_cond"><span class="id" title="definition">actby_cond</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.to"><span class="id" title="variable">to</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actby_is_action"><span class="id" title="lemma">actby_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#actby"><span class="id" title="definition">actby</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.nRA"><span class="id" title="variable">nRA</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">action_by</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#actby_is_action"><span class="id" title="lemma">actby_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actbyE"><span class="id" title="lemma">actbyE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#ActBy.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.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#ActBy.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="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">nRA</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">]&gt;</span></a>%<span class="id" title="var">act</span> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix_actby"><span class="id" title="lemma">afix_actby</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">nRA</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">]&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#08cc1b0d2ac8db12b5c416dfc52232cc"><span class="id" title="notation">~:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#ActBy.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_actby"><span class="id" title="lemma">astab_actby</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">nRA</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">]&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">C_A</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#ActBy.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs_actby"><span class="id" title="lemma">astabs_actby</span></a> <span class="id" title="var">S</span> : <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.fingroup.action.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.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">nRA</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">]&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">N_A</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#ActBy.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActBy.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_actby"><span class="id" title="lemma">acts_actby</span></a> (<span class="id" title="var">B</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.fingroup.action.html#ActBy.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">S</span> :<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#B"><span class="id" title="variable">B</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.action.html#S"><span class="id" title="variable">S</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#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">nRA</span></a><a class="idref" href="mathcomp.fingroup.action.html#5bf4be35f60d187d626f827a16d8a344"><span class="id" title="notation">]&gt;</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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#ActBy.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="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#B"><span class="id" title="variable">B</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.action.html#ActBy.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#ActBy.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">End</span> <a class="idref" href="mathcomp.fingroup.action.html#ActBy"><span class="id" title="section">ActBy</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">&quot;</span></a>&lt;[ nRA ] &gt;" := (<a class="idref" href="mathcomp.fingroup.action.html#action_by"><span class="id" title="definition">action_by</span></a> <span class="id" title="var">nRA</span>) : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="SubAction"><span class="id" title="section">SubAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="SubAction.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="SubAction.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.fingroup.action.html#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">Variables</span> (<a name="SubAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="SubAction.sP"><span class="id" title="variable">sP</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.action.html#rT"><span class="id" title="variable">rT</span></a>) (<a name="SubAction.sT"><span class="id" title="variable">sT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#subFinType"><span class="id" title="record">subFinType</span></a> <a class="idref" href="mathcomp.fingroup.action.html#sP"><span class="id" title="variable">sP</span></a>) (<a name="SubAction.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#SubAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#rT"><span class="id" title="variable">rT</span></a>).<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">A</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.fingroup.action.html#SubAction.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.fingroup.action.html#SubAction.sT"><span class="id" title="variable">sT</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <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.fingroup.action.html#SubAction.sT"><span class="id" title="variable">sT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="subact_dom"><span class="id" title="definition">subact_dom</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.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#SubAction.sP"><span class="id" title="variable">sP</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#20dd00d77a881552893c96be95088d1a"><span class="id" title="notation">]</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#SubAction.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/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">subact_dom_group</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#subact_dom"><span class="id" title="definition">subact_dom</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">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">Na</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.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#subact_dom"><span class="id" title="definition">subact_dom</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Lemma</span> <a name="sub_act_proof"><span class="id" title="lemma">sub_act_proof</span></a> <span class="id" title="var">u</span> <span class="id" title="var">Na</span> : <a class="idref" href="mathcomp.fingroup.action.html#SubAction.sP"><span class="id" title="variable">sP</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#SubAction.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.fingroup.action.html#u"><span class="id" title="variable">u</span></a>) (<a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Na"><span class="id" title="variable">Na</span></a>)).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="subact"><span class="id" title="definition">subact</span></a> <span class="id" title="var">u</span> <span class="id" title="var">a</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#insub"><span class="id" title="definition">insub</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">Na</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Sub"><span class="id" title="projection">Sub</span></a> <span class="id" title="var">_</span> (<a class="idref" href="mathcomp.fingroup.action.html#sub_act_proof"><span class="id" title="lemma">sub_act_proof</span></a> <a class="idref" href="mathcomp.fingroup.action.html#u"><span class="id" title="variable">u</span></a> <span class="id" title="var">Na</span>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.fingroup.action.html#u"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="val_subact"><span class="id" title="lemma">val_subact</span></a> <span class="id" title="var">u</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#subact"><span class="id" title="definition">subact</span></a> <a class="idref" href="mathcomp.fingroup.action.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.action.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#subact_dom"><span class="id" title="definition">subact_dom</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.action.html#SubAction.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.fingroup.action.html#u"><span class="id" title="variable">u</span></a>) <a class="idref" href="mathcomp.fingroup.action.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.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.fingroup.action.html#u"><span class="id" title="variable">u</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="subact_is_action"><span class="id" title="lemma">subact_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#subact_dom"><span class="id" title="definition">subact_dom</span></a> <a class="idref" href="mathcomp.fingroup.action.html#subact"><span class="id" title="definition">subact</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">subaction</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#subact_is_action"><span class="id" title="lemma">subact_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_subact"><span class="id" title="lemma">astab_subact</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#subaction"><span class="id" title="definition">subaction</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#subact_dom"><span class="id" title="definition">subact_dom</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#SubAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs_subact"><span class="id" title="lemma">astabs_subact</span></a> <span class="id" title="var">S</span> : <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.fingroup.action.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.fingroup.action.html#subaction"><span class="id" title="definition">subaction</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.fingroup.action.html#subact_dom"><span class="id" title="definition">subact_dom</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</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.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#SubAction.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="afix_subact"><span class="id" title="lemma">afix_subact</span></a> <span class="id" title="var">A</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#subact_dom"><span class="id" title="definition">subact_dom</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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_subaction</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#aa0c936d7feb8f26825e601735c0125f"><span class="id" title="notation">@^-1:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#SubAction"><span class="id" title="section">SubAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="d0c9249f50b96ff03ab4b012119ece66"><span class="id" title="notation">&quot;</span></a>to ^?" := (<a class="idref" href="mathcomp.fingroup.action.html#subaction"><span class="id" title="definition">subaction</span></a> <span class="id" title="var">_</span> <span class="id" title="var">to</span>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "to ^?") : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="QuotientAction"><span class="id" title="section">QuotientAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="QuotientAction.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="QuotientAction.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.fingroup.action.html#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="QuotientAction.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="QuotientAction.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.rT"><span class="id" title="variable">rT</span></a>) (<a name="QuotientAction.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><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.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">Definition</span> <a name="qact_dom"><span class="id" title="definition">qact_dom</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.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</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.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</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#QuotientAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#495b9dab41eed2da7a7d91b1cb8498af"><span class="id" title="notation">^*</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">)</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">qact_dom_group</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</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">Fact</span> <a name="qact_subdomE"><span class="id" title="lemma">qact_subdomE</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#subdom"><span class="id" title="abbreviation">subdom</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</span></a>.<br/>
+ <span class="id" title="keyword">Lemma</span> <a name="qact_proof"><span class="id" title="lemma">qact_proof</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.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#subdom"><span class="id" title="abbreviation">subdom</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="qact"><span class="id" title="definition">qact</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.fingroup.action.html#QuotientAction.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.fingroup.action.html#QuotientAction.aT"><span class="id" title="variable">aT</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.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#act"><span class="id" title="projection">act</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#495b9dab41eed2da7a7d91b1cb8498af"><span class="id" title="notation">^*</span></a><a class="idref" href="mathcomp.fingroup.action.html#d0c9249f50b96ff03ab4b012119ece66"><span class="id" title="notation">^?</span></a> <a class="idref" href="mathcomp.fingroup.action.html#8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">\</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact_proof"><span class="id" title="lemma">qact_proof</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">quotient_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact"><span class="id" title="definition">qact</span></a><a class="idref" href="mathcomp.fingroup.action.html#4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_qact_dom"><span class="id" title="lemma">acts_qact_dom</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.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</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.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.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><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#QuotientAction.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="qactEcond"><span class="id" title="lemma">qactEcond</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.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.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</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><br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#quotient_action"><span class="id" title="definition">quotient_action</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.action.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#qact_dom"><span class="id" title="definition">qact_dom</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="qactE"><span class="id" title="lemma">qactE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.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.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</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.fingroup.action.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#qact_dom"><span class="id" title="definition">qact_dom</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#quotient_action"><span class="id" title="definition">quotient_action</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_quotient"><span class="id" title="lemma">acts_quotient</span></a> (<span class="id" title="var">A</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.fingroup.action.html#QuotientAction.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">B</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.fingroup.action.html#QuotientAction.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;<a class="idref" href="mathcomp.fingroup.action.html#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#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">N_qact_dom</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><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.fingroup.action.html#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.action.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.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#quotient_action"><span class="id" title="definition">quotient_action</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="astabs_quotient"><span class="id" title="lemma">astabs_quotient</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.fingroup.action.html#QuotientAction.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.fingroup.action.html#QuotientAction.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.fingroup.action.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.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.fingroup.action.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.fingroup.action.html#QuotientAction.H"><span class="id" title="variable">H</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#quotient_action"><span class="id" title="definition">quotient_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">N_qact_dom</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#QuotientAction"><span class="id" title="section">QuotientAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="2e4ccc4d97341df4136dd243a5fe1a5c"><span class="id" title="notation">&quot;</span></a>to / H" := (<a class="idref" href="mathcomp.fingroup.action.html#quotient_action"><span class="id" title="definition">quotient_action</span></a> <span class="id" title="var">to</span> <span class="id" title="var">H</span>) : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ModAction"><span class="id" title="section">ModAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="ModAction.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="ModAction.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.fingroup.action.html#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="ModAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+<span class="id" title="keyword">Variable</span> <a name="ModAction.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.rT"><span class="id" title="variable">rT</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.fingroup.action.html#ModAction.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>) (<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.fingroup.action.html#ModAction.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/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ModAction.GenericMod"><span class="id" title="section">GenericMod</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="ModAction.GenericMod.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><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.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/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a name="ModAction.GenericMod.acts_dom"><span class="id" title="variable">acts_dom</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.fingroup.action.html#dom"><span class="id" title="abbreviation">dom</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.fingroup.action.html#range"><span class="id" title="abbreviation">range</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#ModAction.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#acts_act"><span class="id" title="lemma">acts_act</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#acts_subnorm_fix"><span class="id" title="lemma">acts_subnorm_fix</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="modact"><span class="id" title="definition">modact</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">Ha</span> : <a class="idref" href="mathcomp.fingroup.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#range"><span class="id" title="abbreviation">range</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#ModAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Ha"><span class="id" title="variable">Ha</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="modactEcond"><span class="id" title="lemma">modactEcond</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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#dom"><span class="id" title="abbreviation">dom</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#modact"><span class="id" title="definition">modact</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#range"><span class="id" title="abbreviation">range</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="modactE"><span class="id" title="lemma">modactE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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#ModAction.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#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.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.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</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.fingroup.action.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.fingroup.action.html#range"><span class="id" title="abbreviation">range</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#modact"><span class="id" title="definition">modact</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="modact_is_action"><span class="id" title="lemma">modact_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#ModAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#modact"><span class="id" title="definition">modact</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">mod_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#modact_is_action"><span class="id" title="lemma">modact_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ModAction.GenericMod.Stabilizers"><span class="id" title="section">Stabilizers</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="ModAction.GenericMod.Stabilizers.S"><span class="id" title="variable">S</span></a> : <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.fingroup.action.html#ModAction.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/>
+<span class="id" title="keyword">Hypothesis</span> <a name="ModAction.GenericMod.Stabilizers.cSH"><span class="id" title="variable">cSH</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.Stabilizers.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Let</span> <a name="ModAction.GenericMod.Stabilizers.fixSH"><span class="id" title="variable">fixSH</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.Stabilizers.S"><span class="id" title="variable">S</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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#ModAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs_mod"><span class="id" title="lemma">astabs_mod</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.fingroup.action.html#ModAction.GenericMod.Stabilizers.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.fingroup.action.html#mod_action"><span class="id" title="definition">mod_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.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.fingroup.action.html#ModAction.GenericMod.Stabilizers.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.fingroup.action.html#ModAction.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> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_mod"><span class="id" title="lemma">astab_mod</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.Stabilizers.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#mod_action"><span class="id" title="definition">mod_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.Stabilizers.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.Stabilizers"><span class="id" title="section">Stabilizers</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix_mod"><span class="id" title="lemma">afix_mod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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#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.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><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#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#mod_action"><span class="id" title="definition">mod_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#ModAction.GenericMod.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><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.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.GenericMod"><span class="id" title="section">GenericMod</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="modact_faithful"><span class="id" title="lemma">modact_faithful</span></a> <span class="id" title="var">G</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">faithful</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#mod_action"><span class="id" title="definition">mod_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#604222e4f2ad6aacb5fe189ffe7e1dce"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#604222e4f2ad6aacb5fe189ffe7e1dce"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#604222e4f2ad6aacb5fe189ffe7e1dce"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#604222e4f2ad6aacb5fe189ffe7e1dce"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ModAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#604222e4f2ad6aacb5fe189ffe7e1dce"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#ModAction"><span class="id" title="section">ModAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="2763ae1d41d6f4b536ed07b20d50de88"><span class="id" title="notation">&quot;</span></a>to %% H" := (<a class="idref" href="mathcomp.fingroup.action.html#mod_action"><span class="id" title="definition">mod_action</span></a> <span class="id" title="var">to</span> <span class="id" title="var">H</span>) : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ActPerm"><span class="id" title="section">ActPerm</span></a>.<br/>
+</div>
+
+<div class="doc">
+ Morphism to permutations induced by an action.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="ActPerm.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="ActPerm.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="ActPerm.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+<span class="id" title="keyword">Variable</span> <a name="ActPerm.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActPerm.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActPerm.rT"><span class="id" title="variable">rT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="actperm"><span class="id" title="definition">actperm</span></a> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.fingroup.perm.html#perm"><span class="id" title="abbreviation">perm</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#act_inj"><span class="id" title="lemma">act_inj</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActPerm.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actpermM"><span class="id" title="lemma">actpermM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActPerm.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">actperm_morphism</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#actpermM"><span class="id" title="lemma">actpermM</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actpermE"><span class="id" title="lemma">actpermE</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActPerm.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actpermK"><span class="id" title="lemma">actpermK</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.perm.html#aperm"><span class="id" title="definition">aperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActPerm.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="ker_actperm"><span class="id" title="lemma">ker_actperm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActPerm.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#ActPerm"><span class="id" title="section">ActPerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="RestrictActionTheory"><span class="id" title="section">RestrictActionTheory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RestrictActionTheory.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="RestrictActionTheory.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="RestrictActionTheory.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RestrictActionTheory.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.rT"><span class="id" title="variable">rT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="faithful_isom"><span class="id" title="lemma">faithful_isom</span></a> (<span class="id" title="var">A</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.action.html#RestrictActionTheory.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>) <span class="id" title="var">S</span> (<span class="id" title="var">nSA</span> : <a class="idref" href="mathcomp.fingroup.action.html#actby_cond"><span class="id" title="definition">actby_cond</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">faithful</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><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.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#nSA"><span class="id" title="variable">nSA</span></a><a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">]&gt;</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#nSA"><span class="id" title="variable">nSA</span></a><a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">]&gt;</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RestrictActionTheory.A"><span class="id" title="variable">A</span></a> : <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.fingroup.action.html#RestrictActionTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="RestrictActionTheory.sAD"><span class="id" title="variable">sAD</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#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#RestrictActionTheory.D"><span class="id" title="variable">D</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="ractpermE"><span class="id" title="lemma">ractpermE</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">\</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.sAD"><span class="id" title="variable">sAD</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix_ract"><span class="id" title="lemma">afix_ract</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">\</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.sAD"><span class="id" title="variable">sAD</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_ract"><span class="id" title="lemma">astab_ract</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">\</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.sAD"><span class="id" title="variable">sAD</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">C_A</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs_ract"><span class="id" title="lemma">astabs_ract</span></a> <span class="id" title="var">S</span> : <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.fingroup.action.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.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">\</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.sAD"><span class="id" title="variable">sAD</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">N_A</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_ract"><span class="id" title="lemma">acts_ract</span></a> (<span class="id" title="var">B</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.fingroup.action.html#RestrictActionTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">S</span> :<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#B"><span class="id" title="variable">B</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.action.html#S"><span class="id" title="variable">S</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#RestrictActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">\</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory.sAD"><span class="id" title="variable">sAD</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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#RestrictActionTheory.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="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#B"><span class="id" title="variable">B</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.action.html#S"><span class="id" title="variable">S</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#RestrictActionTheory.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">End</span> <a class="idref" href="mathcomp.fingroup.action.html#RestrictActionTheory"><span class="id" title="section">RestrictActionTheory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MorphAct"><span class="id" title="section">MorphAct</span></a>.<br/>
+</div>
+
+<div class="doc">
+ Action induced by a morphism to permutations.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphAct.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="MorphAct.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.fingroup.action.html#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="MorphAct.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+<span class="id" title="keyword">Variable</span> <a name="MorphAct.phi"><span class="id" title="variable">phi</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAct.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">perm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAct.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">}</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="mact"><span class="id" title="definition">mact</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.fingroup.action.html#MorphAct.phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="mact_is_action"><span class="id" title="lemma">mact_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAct.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#mact"><span class="id" title="definition">mact</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">morph_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#mact_is_action"><span class="id" title="lemma">mact_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="mactE"><span class="id" title="lemma">mactE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#morph_action"><span class="id" title="definition">morph_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAct.phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="injm_faithful"><span class="id" title="lemma">injm_faithful</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAct.phi"><span class="id" title="variable">phi</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#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">faithful</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAct.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_action"><span class="id" title="definition">morph_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="perm_mact"><span class="id" title="lemma">perm_mact</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_action"><span class="id" title="definition">morph_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAct.phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#MorphAct"><span class="id" title="section">MorphAct</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="157b1d4873eeace33ab3d1b9523d7613"><span class="id" title="notation">&quot;</span></a>&lt;&lt; phi &gt;&gt;" := (<a class="idref" href="mathcomp.fingroup.action.html#morph_action"><span class="id" title="definition">morph_action</span></a> <span class="id" title="var">phi</span>) : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="CompAct"><span class="id" title="section">CompAct</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="CompAct.gT"><span class="id" title="variable">gT</span></a> <a name="CompAct.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="CompAct.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="CompAct.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#CompAct.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="CompAct.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#CompAct.rT"><span class="id" title="variable">rT</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="CompAct.B"><span class="id" title="variable">B</span></a> : <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.fingroup.action.html#CompAct.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="CompAct.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#CompAct.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="comp_act"><span class="id" title="definition">comp_act</span></a> <span class="id" title="var">x</span> <span class="id" title="var">e</span> := <a class="idref" href="mathcomp.fingroup.action.html#CompAct.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#CompAct.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e"><span class="id" title="variable">e</span></a>).<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_is_action"><span class="id" title="lemma">comp_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#CompAct.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#CompAct.D"><span class="id" title="variable">D</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#comp_act"><span class="id" title="definition">comp_act</span></a>.<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comp_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#comp_is_action"><span class="id" title="lemma">comp_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_actE"><span class="id" title="lemma">comp_actE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">e</span> : <a class="idref" href="mathcomp.fingroup.action.html#comp_action"><span class="id" title="definition">comp_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#CompAct.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#CompAct.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e"><span class="id" title="variable">e</span></a>).  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afix_comp"><span class="id" title="lemma">afix_comp</span></a> (<span class="id" title="var">A</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.fingroup.action.html#CompAct.gT"><span class="id" title="variable">gT</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#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#CompAct.B"><span class="id" title="variable">B</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#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_comp_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><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.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">Fix_to</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#CompAct.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#169c8da9095de246cb2051165c6767e3"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_comp"><span class="id" title="lemma">astab_comp</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#comp_action"><span class="id" title="definition">comp_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#CompAct.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#CompAct.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs_comp"><span class="id" title="lemma">astabs_comp</span></a> <span class="id" title="var">S</span> : <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.fingroup.action.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.fingroup.action.html#comp_action"><span class="id" title="definition">comp_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.fingroup.action.html#CompAct.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</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.fingroup.action.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.fingroup.action.html#CompAct.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.fingroup.action.html#CompAct"><span class="id" title="section">CompAct</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="835827ac44615432d269abaf103548c9"><span class="id" title="notation">&quot;</span></a>to \o f" := (<a class="idref" href="mathcomp.fingroup.action.html#comp_action"><span class="id" title="definition">comp_action</span></a> <span class="id" title="var">to</span> <span class="id" title="var">f</span>) : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="PermAction"><span class="id" title="section">PermAction</span></a>.<br/>
+</div>
+
+<div class="doc">
+  Natural action of permutation groups.                         
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="PermAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">c</span> : <a class="idref" href="mathcomp.fingroup.action.html#gT"><span class="id" title="abbreviation">gT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="aperm_is_action"><span class="id" title="lemma">aperm_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> (@<a class="idref" href="mathcomp.fingroup.perm.html#aperm"><span class="id" title="definition">aperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#PermAction.rT"><span class="id" title="variable">rT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">perm_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#aperm_is_action"><span class="id" title="lemma">aperm_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="pcycleE"><span class="id" title="lemma">pcycleE</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.perm.html#pcycle"><span class="id" title="definition">pcycle</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#perm_action"><span class="id" title="definition">perm_action</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">]&gt;</span></a>%<span class="id" title="var">g</span>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="perm_act1P"><span class="id" title="lemma">perm_act1P</span></a> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.perm.html#aperm"><span class="id" title="definition">aperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="perm_faithful"><span class="id" title="lemma">perm_faithful</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">faithful</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#perm_action"><span class="id" title="definition">perm_action</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actperm_id"><span class="id" title="lemma">actperm_id</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#perm_action"><span class="id" title="definition">perm_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#p"><span class="id" title="variable">p</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#PermAction"><span class="id" title="section">PermAction</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">&quot;</span></a>'P" := (<a class="idref" href="mathcomp.fingroup.action.html#perm_action"><span class="id" title="definition">perm_action</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">action_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="ActpermOrbits"><span class="id" title="section">ActpermOrbits</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="ActpermOrbits.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="ActpermOrbits.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.fingroup.action.html#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="ActpermOrbits.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+<span class="id" title="keyword">Variable</span> <a name="ActpermOrbits.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActpermOrbits.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActpermOrbits.rT"><span class="id" title="variable">rT</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbit_morphim_actperm"><span class="id" title="lemma">orbit_morphim_actperm</span></a> (<span class="id" title="var">A</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.fingroup.action.html#ActpermOrbits.aT"><span class="id" title="variable">aT</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#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#ActpermOrbits.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#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActpermOrbits.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActpermOrbits.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="pcycle_actperm"><span class="id" title="lemma">pcycle_actperm</span></a> (<span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#ActpermOrbits.aT"><span class="id" title="variable">aT</span></a>) :<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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#ActpermOrbits.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.perm.html#pcycle"><span class="id" title="definition">pcycle</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActpermOrbits.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ActpermOrbits.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">]&gt;</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#ActpermOrbits"><span class="id" title="section">ActpermOrbits</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="RestrictPerm"><span class="id" title="section">RestrictPerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RestrictPerm.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="RestrictPerm.S"><span class="id" title="variable">S</span></a> : <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.fingroup.action.html#T"><span class="id" title="variable">T</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="restr_perm"><span class="id" title="definition">restr_perm</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#subxx"><span class="id" title="lemma">subxx</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.fingroup.action.html#RestrictPerm.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.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</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#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">]&gt;</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">restr_perm_morphism</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#restr_perm"><span class="id" title="definition">restr_perm</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="restr_perm_on"><span class="id" title="lemma">restr_perm_on</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.fingroup.perm.html#perm_on"><span class="id" title="definition">perm_on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictPerm.S"><span class="id" title="variable">S</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#restr_perm"><span class="id" title="definition">restr_perm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#p"><span class="id" title="variable">p</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="triv_restr_perm"><span class="id" title="lemma">triv_restr_perm</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.fingroup.action.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</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.fingroup.action.html#RestrictPerm.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.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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#restr_perm"><span class="id" title="definition">restr_perm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="restr_permE"><span class="id" title="lemma">restr_permE</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.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.fingroup.action.html#RestrictPerm.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.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictPerm.S"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">p</span>, <a class="idref" href="mathcomp.fingroup.action.html#restr_perm"><span class="id" title="definition">restr_perm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="ker_restr_perm"><span class="id" title="lemma">ker_restr_perm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.fingroup.action.html#restr_perm"><span class="id" title="definition">restr_perm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RestrictPerm.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="im_restr_perm"><span class="id" title="lemma">im_restr_perm</span></a> <span class="id" title="var">p</span> : <a class="idref" href="mathcomp.fingroup.action.html#restr_perm"><span class="id" title="definition">restr_perm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RestrictPerm.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.fingroup.action.html#RestrictPerm.S"><span class="id" title="variable">S</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#RestrictPerm"><span class="id" title="section">RestrictPerm</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="AutIn"><span class="id" title="section">AutIn</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="AutIn.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/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="Aut_in"><span class="id" title="definition">Aut_in</span></a> <span class="id" title="var">A</span> (<span class="id" title="var">B</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.fingroup.action.html#AutIn.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) := <a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">N_A</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.fingroup.action.html#d17cdb562dee90224eb90588920818d9"><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.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">C_A</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> <a name="AutIn.G"><span class="id" title="variable">G</span></a> <a name="AutIn.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><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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/>
+<span class="id" title="keyword">Hypothesis</span> <a name="AutIn.sHG"><span class="id" title="variable">sHG</span></a>: <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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.fingroup.action.html#AutIn.G"><span class="id" title="variable">G</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="Aut_restr_perm"><span class="id" title="lemma">Aut_restr_perm</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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.fingroup.action.html#restr_perm"><span class="id" title="definition">restr_perm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.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.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.H"><span class="id" title="variable">H</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="restr_perm_Aut"><span class="id" title="lemma">restr_perm_Aut</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#restr_perm"><span class="id" title="definition">restr_perm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.H"><span class="id" title="variable">H</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="Aut_in_isog"><span class="id" title="lemma">Aut_in_isog</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#Aut_in"><span class="id" title="definition">Aut_in</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#AutIn.H"><span class="id" title="variable">H</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.fingroup.action.html#restr_perm"><span class="id" title="definition">restr_perm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.G"><span class="id" title="variable">G</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="Aut_sub_fullP"><span class="id" title="lemma">Aut_sub_fullP</span></a> :<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> (<span class="id" title="keyword">∀</span> <span class="id" title="var">h</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>, <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.action.html#AutIn.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">g</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</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><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#g"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.action.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">]</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.fingroup.action.html#Aut_in"><span class="id" title="definition">Aut_in</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#AutIn.H"><span class="id" title="variable">H</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.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutIn.H"><span class="id" title="variable">H</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#AutIn"><span class="id" title="section">AutIn</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="InjmAutIn"><span class="id" title="section">InjmAutIn</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="InjmAutIn.gT"><span class="id" title="variable">gT</span></a> <a name="InjmAutIn.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="InjmAutIn.D"><span class="id" title="variable">D</span></a> <a name="InjmAutIn.G"><span class="id" title="variable">G</span></a> <a name="InjmAutIn.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><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.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 name="InjmAutIn.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/>
+<span class="id" title="keyword">Hypotheses</span> (<a name="InjmAutIn.injf"><span class="id" title="variable">injf</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.f"><span class="id" title="variable">f</span></a>) (<a name="InjmAutIn.sGD"><span class="id" title="variable">sGD</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.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.fingroup.action.html#InjmAutIn.D"><span class="id" title="variable">D</span></a>) (<a name="InjmAutIn.sHG"><span class="id" title="variable">sHG</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.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.fingroup.action.html#InjmAutIn.G"><span class="id" title="variable">G</span></a>).<br/>
+<span class="id" title="keyword">Let</span> <a name="InjmAutIn.sHD"><span class="id" title="variable">sHD</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#subset_trans"><span class="id" title="lemma">subset_trans</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.sHG"><span class="id" title="variable">sHG</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.sGD"><span class="id" title="variable">sGD</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs_Aut_isom"><span class="id" title="lemma">astabs_Aut_isom</span></a> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#fGisom"><span class="id" title="abbreviation">fGisom</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#InjmAutIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.H"><span class="id" title="variable">H</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#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#InjmAutIn.H"><span class="id" title="variable">H</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#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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="isom_restr_perm"><span class="id" title="lemma">isom_restr_perm</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.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.fingroup.action.html#fHisom"><span class="id" title="abbreviation">fHisom</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#inH"><span class="id" title="abbreviation">inH</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#infH"><span class="id" title="abbreviation">infH</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#fGisom"><span class="id" title="abbreviation">fGisom</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="restr_perm_isom"><span class="id" title="lemma">restr_perm_isom</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#inH"><span class="id" title="abbreviation">inH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#infH"><span class="id" title="abbreviation">infH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.G"><span class="id" title="variable">G</span></a>)) <a class="idref" href="mathcomp.fingroup.action.html#fHisom"><span class="id" title="abbreviation">fHisom</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="injm_Aut_sub"><span class="id" title="lemma">injm_Aut_sub</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#Aut_in"><span class="id" title="definition">Aut_in</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.G"><span class="id" title="variable">G</span></a>)) (<a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.H"><span class="id" title="variable">H</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.fingroup.action.html#Aut_in"><span class="id" title="definition">Aut_in</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.H"><span class="id" title="variable">H</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="injm_Aut_full"><span class="id" title="lemma">injm_Aut_full</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#Aut_in"><span class="id" title="definition">Aut_in</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.G"><span class="id" title="variable">G</span></a>)) (<a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.H"><span class="id" title="variable">H</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.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#Aut_in"><span class="id" title="definition">Aut_in</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.H"><span class="id" title="variable">H</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.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#InjmAutIn"><span class="id" title="section">InjmAutIn</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="GroupAction"><span class="id" title="section">GroupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="GroupAction.aT"><span class="id" title="variable">aT</span></a> <a name="GroupAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="GroupAction.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="GroupAction.R"><span class="id" title="variable">R</span></a> : <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.fingroup.action.html#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/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="is_groupAction"><span class="id" title="definition">is_groupAction</span></a> (<span class="id" title="var">to</span> : <a class="idref" href="mathcomp.fingroup.action.html#actT"><span class="id" title="abbreviation">actT</span></a>) :=<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupAction.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.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.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupAction.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Structure</span> <a name="groupAction"><span class="id" title="record">groupAction</span></a> := <a name="GroupAction"><span class="id" title="constructor">GroupAction</span></a> {<a name="gact"><span class="id" title="projection">gact</span></a> :&gt; <a class="idref" href="mathcomp.fingroup.action.html#actT"><span class="id" title="abbreviation">actT</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#gact"><span class="id" title="method">gact</span></a>}.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="clone_groupAction"><span class="id" title="definition">clone_groupAction</span></a> <span class="id" title="var">to</span> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <span class="id" title="var">_</span> <span class="id" title="var">toA</span> := <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">type</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="var">_</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <span class="id" title="var">k</span> ⇒ <a class="idref" href="mathcomp.fingroup.action.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="var">toA</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="section">GroupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">groupAction_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">gact</span>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="9c48f0d46104b2a59cfe6fb489695b9b"><span class="id" title="notation">&quot;</span></a>[ 'groupAction' 'of' to ]" :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.fingroup.action.html#clone_groupAction"><span class="id" title="definition">clone_groupAction</span></a> (@<a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">to</span>))<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'groupAction'  'of'  to ]") : <span class="id" title="var">form_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="GroupActionDefs"><span class="id" title="section">GroupActionDefs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="GroupActionDefs.aT"><span class="id" title="variable">aT</span></a> <a name="GroupActionDefs.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="GroupActionDefs.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="GroupActionDefs.R"><span class="id" title="variable">R</span></a> : <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.fingroup.action.html#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/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">A</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.fingroup.action.html#GroupActionDefs.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">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.fingroup.action.html#GroupActionDefs.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/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">to</span> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionDefs.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionDefs.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="gact_range"><span class="id" title="definition">gact_range</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionDefs.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionDefs.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#GroupActionDefs.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="gacent"><span class="id" title="definition">gacent</span></a> <span class="id" title="var">to</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionDefs.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionDefs.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="acts_on_group"><span class="id" title="definition">acts_on_group</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span> := <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#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.action.html#S"><span class="id" title="variable">S</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#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#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#GroupActionDefs.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Coercion</span> <span class="id" title="var">actby_cond_group</span> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span> : <a class="idref" href="mathcomp.fingroup.action.html#acts_on_group"><span class="id" title="definition">acts_on_group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</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#actby_cond"><span class="id" title="definition">actby_cond</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#to"><span class="id" title="variable">to</span></a> :=<br/>
+&nbsp;&nbsp;@<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#proj1"><span class="id" title="lemma">proj1</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="acts_irreducibly"><span class="id" title="definition">acts_irreducibly</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> <span class="id" title="var">to</span> :=<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">min</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">of</span></a> <span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#36625695d37b6869c156bfcdf13834f7"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><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.fingroup.action.html#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.action.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.fingroup.action.html#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="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionDefs"><span class="id" title="section">GroupActionDefs</span></a>.<br/>
+
+<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">&quot;</span></a>''C_' ( | to ) ( A )" := (<a class="idref" href="mathcomp.fingroup.action.html#gacent"><span class="id" title="definition">gacent</span></a> <span class="id" title="var">to</span> <span class="id" title="var">A</span>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''C_' ( | to ) ( A )") : <span class="id" title="var">group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">&quot;</span></a>''C_' ( G | to ) ( A )" := (<span class="id" title="var">G</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><span class="id" title="var">A</span><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''C_' ( G  |  to ) ( A )") : <span class="id" title="var">group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">&quot;</span></a>''C_' ( | to ) [ a ]" := <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''C_' ( | to ) [ a ]") : <span class="id" title="var">group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="f003dba8fcf02dcc50c37ba6ada8b94d"><span class="id" title="notation">&quot;</span></a>''C_' ( G | to ) [ a ]" := <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">(</span></a><span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">|</span></a> <span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''C_' ( G  |  to ) [ a ]") : <span class="id" title="var">group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">&quot;</span></a>{ 'acts' A , 'on' 'group' G | to }" := (<a class="idref" href="mathcomp.fingroup.action.html#acts_on_group"><span class="id" title="definition">acts_on_group</span></a> <span class="id" title="var">A</span> <span class="id" title="var">G</span> <span class="id" title="var">to</span>)<br/>
+&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "{ 'acts'  A ,  'on'  'group'  G  |  to }") : <span class="id" title="var">form_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="RawGroupAction"><span class="id" title="section">RawGroupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RawGroupAction.aT"><span class="id" title="variable">aT</span></a> <a name="RawGroupAction.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="RawGroupAction.D"><span class="id" title="variable">D</span></a> : <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.fingroup.action.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="RawGroupAction.R"><span class="id" title="variable">R</span></a> : <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.fingroup.action.html#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/>
+<span class="id" title="keyword">Variable</span> <a name="RawGroupAction.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.fingroup.action.html#RawGroupAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actperm_Aut"><span class="id" title="lemma">actperm_Aut</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="im_actperm_Aut"><span class="id" title="lemma">im_actperm_Aut</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.D"><span class="id" title="variable">D</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.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gact_out"><span class="id" title="lemma">gact_out</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#RawGroupAction.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#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#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gactM"><span class="id" title="lemma">gactM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actmM"><span class="id" title="lemma">actmM</span></a> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#actm"><span class="id" title="definition">actm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">act_morphism</span> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#actmM"><span class="id" title="lemma">actmM</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="morphim_actm"><span class="id" title="lemma">morphim_actm</span></a> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> (<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.fingroup.action.html#RawGroupAction.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.fingroup.action.html#S"><span class="id" title="variable">S</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#RawGroupAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#actm"><span class="id" title="definition">actm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#54d9268ab0ed9ca3b605c1679d80e7b8"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="RawGroupAction.a"><span class="id" title="variable">a</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.aT"><span class="id" title="variable">aT</span></a>) (<a name="RawGroupAction.A"><span class="id" title="variable">A</span></a> <a name="RawGroupAction.B"><span class="id" title="variable">B</span></a> : <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.fingroup.action.html#RawGroupAction.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="RawGroupAction.S"><span class="id" title="variable">S</span></a> : <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.fingroup.action.html#RawGroupAction.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/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentIdom"><span class="id" title="lemma">gacentIdom</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentIim"><span class="id" title="lemma">gacentIim</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentS"><span class="id" title="lemma">gacentS</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.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#RawGroupAction.B"><span class="id" title="variable">B</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#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</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#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentU"><span class="id" title="lemma">gacentU</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Hypotheses</span> (<a name="RawGroupAction.Da"><span class="id" title="variable">Da</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.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#RawGroupAction.D"><span class="id" title="variable">D</span></a>) (<a name="RawGroupAction.sAD"><span class="id" title="variable">sAD</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.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#RawGroupAction.D"><span class="id" title="variable">D</span></a>) (<a name="RawGroupAction.sSR"><span class="id" title="variable">sSR</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.S"><span class="id" title="variable">S</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#RawGroupAction.R"><span class="id" title="variable">R</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentE"><span class="id" title="lemma">gacentE</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacent1E"><span class="id" title="lemma">gacent1E</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><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.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="subgacentE"><span class="id" title="lemma">subgacentE</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><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.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="subgacent1E"><span class="id" title="lemma">subgacent1E</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#f003dba8fcf02dcc50c37ba6ada8b94d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f003dba8fcf02dcc50c37ba6ada8b94d"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#f003dba8fcf02dcc50c37ba6ada8b94d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f003dba8fcf02dcc50c37ba6ada8b94d"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f003dba8fcf02dcc50c37ba6ada8b94d"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#f003dba8fcf02dcc50c37ba6ada8b94d"><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.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction.a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#d332864708f1d0e9a3a13805c0663964"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#RawGroupAction"><span class="id" title="section">RawGroupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="GroupActionTheory"><span class="id" title="section">GroupActionTheory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> <a name="GroupActionTheory.aT"><span class="id" title="variable">aT</span></a> <a name="GroupActionTheory.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="GroupActionTheory.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.fingroup.action.html#GroupActionTheory.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="GroupActionTheory.R"><span class="id" title="variable">R</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.action.html#GroupActionTheory.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="GroupActionTheory.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.fingroup.action.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.action.html#R"><span class="id" title="variable">R</span></a>).<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">A</span> <span class="id" title="var">B</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.fingroup.action.html#GroupActionTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">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.fingroup.action.html#GroupActionTheory.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">Implicit</span> <span class="id" title="keyword">Type</span> <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.fingroup.action.html#GroupActionTheory.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/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">M</span> <span class="id" title="var">N</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.action.html#GroupActionTheory.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="gact1"><span class="id" title="lemma">gact1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> 1 <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gactV"><span class="id" title="lemma">gactV</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gactX"><span class="id" title="lemma">gactX</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">n</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#86a04fb564fb97d388cad84a3a204260"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.fingroup.action.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gactJ"><span class="id" title="lemma">gactJ</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.fingroup.action.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.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gactR"><span class="id" title="lemma">gactR</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</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.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d5d566f861753940ef0e9a18d348c2b8"><span class="id" title="notation">[~</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d5d566f861753940ef0e9a18d348c2b8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d5d566f861753940ef0e9a18d348c2b8"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gact_stable"><span class="id" title="lemma">gact_stable</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.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</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.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</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#GroupActionTheory.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>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="group_set_gacent"><span class="id" title="lemma">group_set_gacent</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gacent_group</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#Group"><span class="id" title="constructor">Group</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#group_set_gacent"><span class="id" title="lemma">group_set_gacent</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacent1"><span class="id" title="lemma">gacent1</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a>1<a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacent_gen"><span class="id" title="lemma">gacent_gen</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#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#GroupActionTheory.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#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&lt;&lt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&gt;&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentD1"><span class="id" title="lemma">gacentD1</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">^#</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacent_cycle"><span class="id" title="lemma">gacent_cycle</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.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#GroupActionTheory.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#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">]&gt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#61d1bba23e27ff161a7de70cf7f24d12"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentY"><span class="id" title="lemma">gacentY</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#GroupActionTheory.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#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.fingroup.action.html#GroupActionTheory.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#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation">&lt;*&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentM"><span class="id" title="lemma">gacentM</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#GroupActionTheory.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#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.fingroup.action.html#GroupActionTheory.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#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1"><span class="id" title="lemma">astab1</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab_range"><span class="id" title="lemma">astab_range</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentC"><span class="id" title="lemma">gacentC</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#GroupActionTheory.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#S"><span class="id" title="variable">S</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#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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="astab_gen"><span class="id" title="lemma">astab_gen</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&lt;&lt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&gt;&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabM"><span class="id" title="lemma">astabM</span></a> <span class="id" title="var">M</span> <span class="id" title="var">N</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</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#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#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.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.action.html#N"><span class="id" title="variable">N</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#N"><span class="id" title="variable">N</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs1"><span class="id" title="lemma">astabs1</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>1 <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.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> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabs_range"><span class="id" title="lemma">astabs_range</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.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</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#GroupActionTheory.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> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.D"><span class="id" title="variable">D</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabsD1"><span class="id" title="lemma">astabsD1</span></a> <span class="id" title="var">S</span> : <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.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">^#</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#GroupActionTheory.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> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.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.fingroup.action.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.fingroup.action.html#GroupActionTheory.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="gacts_range"><span class="id" title="lemma">gacts_range</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#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#GroupActionTheory.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#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_subnorm_gacent"><span class="id" title="lemma">acts_subnorm_gacent</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#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#GroupActionTheory.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.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><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">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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#GroupActionTheory.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="acts_subnorm_subgacent"><span class="id" title="lemma">acts_subnorm_subgacent</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#GroupActionTheory.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.fingroup.action.html#B"><span class="id" title="variable">B</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.action.html#S"><span class="id" title="variable">S</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#GroupActionTheory.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.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><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">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><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#GroupActionTheory.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="acts_gen"><span class="id" title="lemma">acts_gen</span></a> <span class="id" title="var">A</span> <span class="id" title="var">S</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#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.fingroup.action.html#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.action.html#S"><span class="id" title="variable">S</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#GroupActionTheory.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.fingroup.action.html#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.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&lt;&lt;</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&gt;&gt;</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#GroupActionTheory.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="acts_joing"><span class="id" title="lemma">acts_joing</span></a> <span class="id" title="var">A</span> <span class="id" title="var">M</span> <span class="id" title="var">N</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</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#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#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.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#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.fingroup.action.html#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.action.html#M"><span class="id" title="variable">M</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#GroupActionTheory.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.fingroup.action.html#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.action.html#N"><span class="id" title="variable">N</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#GroupActionTheory.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#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.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation">&lt;*&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#N"><span class="id" title="variable">N</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#GroupActionTheory.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="injm_actm"><span class="id" title="lemma">injm_actm</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#actm"><span class="id" title="definition">actm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="im_actm"><span class="id" title="lemma">im_actm</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#actm"><span class="id" title="definition">actm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_char"><span class="id" title="lemma">acts_char</span></a> <span class="id" title="var">G</span> <span class="id" title="var">M</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#GroupActionTheory.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#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#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.fingroup.action.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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</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#GroupActionTheory.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="gacts_char"><span class="id" title="lemma">gacts_char</span></a> <span class="id" title="var">G</span> <span class="id" title="var">M</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#GroupActionTheory.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#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">}</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Section</span> <a name="GroupActionTheory.Restrict"><span class="id" title="section">Restrict</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="GroupActionTheory.Restrict.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.fingroup.action.html#GroupActionTheory.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="GroupActionTheory.Restrict.sAD"><span class="id" title="variable">sAD</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#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#GroupActionTheory.D"><span class="id" title="variable">D</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="ract_is_groupAction"><span class="id" title="lemma">ract_is_groupAction</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">\</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Restrict.sAD"><span class="id" title="variable">sAD</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ract_groupAction</span> := <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ract_is_groupAction"><span class="id" title="lemma">ract_is_groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacent_ract"><span class="id" title="lemma">gacent_ract</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#ract_groupAction"><span class="id" title="definition">ract_groupAction</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Restrict.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Restrict"><span class="id" title="section">Restrict</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="GroupActionTheory.ActBy"><span class="id" title="section">ActBy</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="GroupActionTheory.ActBy.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.fingroup.action.html#GroupActionTheory.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="GroupActionTheory.ActBy.G"><span class="id" title="variable">G</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.action.html#GroupActionTheory.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="GroupActionTheory.ActBy.nGAg"><span class="id" title="variable">nGAg</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#942d78c074baecb495cbc019394dd5b3"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actby_is_groupAction"><span class="id" title="lemma">actby_is_groupAction</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.ActBy.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.ActBy.nGAg"><span class="id" title="variable">nGAg</span></a><a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">]&gt;</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">actby_groupAction</span> := <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#actby_is_groupAction"><span class="id" title="lemma">actby_is_groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacent_actby"><span class="id" title="lemma">gacent_actby</span></a> <span class="id" title="var">B</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#actby_groupAction"><span class="id" title="definition">actby_groupAction</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.ActBy.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.ActBy.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.ActBy"><span class="id" title="section">ActBy</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="GroupActionTheory.Quotient"><span class="id" title="section">Quotient</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="GroupActionTheory.Quotient.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><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.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="acts_qact_dom_norm"><span class="id" title="lemma">acts_qact_dom_norm</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.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Quotient.H"><span class="id" title="variable">H</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.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.fingroup.action.html#GroupActionTheory.Quotient.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</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#GroupActionTheory.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>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="qact_is_groupAction"><span class="id" title="lemma">qact_is_groupAction</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Quotient.H"><span class="id" title="variable">H</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.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.fingroup.action.html#GroupActionTheory.Quotient.H"><span class="id" title="variable">H</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">quotient_groupAction</span> := <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#qact_is_groupAction"><span class="id" title="lemma">qact_is_groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="qact_domE"><span class="id" title="lemma">qact_domE</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Quotient.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.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#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.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Quotient.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</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.fingroup.action.html#GroupActionTheory.Quotient.H"><span class="id" title="variable">H</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#GroupActionTheory.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.fingroup.action.html#GroupActionTheory.Quotient"><span class="id" title="section">Quotient</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="GroupActionTheory.Mod"><span class="id" title="section">Mod</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="GroupActionTheory.Mod.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><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.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/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="modact_is_groupAction"><span class="id" title="lemma">modact_is_groupAction</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2763ae1d41d6f4b536ed07b20d50de88"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">mod_groupAction</span> := <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#modact_is_groupAction"><span class="id" title="lemma">modact_is_groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="modgactE"><span class="id" title="lemma">modgactE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod.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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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#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.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><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#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2763ae1d41d6f4b536ed07b20d50de88"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">act</span> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacent_mod"><span class="id" title="lemma">gacent_mod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">M</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod.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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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#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.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.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</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><br/>
+&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#mod_groupAction"><span class="id" title="definition">mod_groupAction</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><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.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#80a58c159fbe61cbb7996107e4e1da9b"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_irr_mod"><span class="id" title="lemma">acts_irr_mod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">M</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod.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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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#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.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.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</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.fingroup.action.html#acts_irreducibly"><span class="id" title="definition">acts_irreducibly</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</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#acts_irreducibly"><span class="id" title="definition">acts_irreducibly</span></a> (<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#GroupActionTheory.Mod.H"><span class="id" title="variable">H</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#mod_groupAction"><span class="id" title="definition">mod_groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.Mod"><span class="id" title="section">Mod</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="modact_coset_astab"><span class="id" title="lemma">modact_coset_astab</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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#GroupActionTheory.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#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2763ae1d41d6f4b536ed07b20d50de88"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#241fd4471d2bcb8850364ee8be828b4a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">act</span> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="acts_irr_mod_astab"><span class="id" title="lemma">acts_irr_mod_astab</span></a> <span class="id" title="var">G</span> <span class="id" title="var">M</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#acts_irreducibly"><span class="id" title="definition">acts_irreducibly</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</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#acts_irreducibly"><span class="id" title="definition">acts_irreducibly</span></a> (<a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#652d6cc67746e5361142d90686607781"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#mod_groupAction"><span class="id" title="definition">mod_groupAction</span></a> <span class="id" title="var">_</span>).<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="GroupActionTheory.CompAct"><span class="id" title="section">CompAct</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="GroupActionTheory.CompAct.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="GroupActionTheory.CompAct.G"><span class="id" title="variable">G</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.action.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 name="GroupActionTheory.CompAct.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="comp_is_groupAction"><span class="id" title="lemma">comp_is_groupAction</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.R"><span class="id" title="variable">R</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#comp_action"><span class="id" title="definition">comp_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.CompAct.f"><span class="id" title="variable">f</span></a>).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">comp_groupAction</span> := <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#comp_is_groupAction"><span class="id" title="lemma">comp_is_groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacent_comp"><span class="id" title="lemma">gacent_comp</span></a> <span class="id" title="var">U</span> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#comp_groupAction"><span class="id" title="definition">comp_groupAction</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#U"><span class="id" title="variable">U</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.CompAct.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#U"><span class="id" title="variable">U</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory.CompAct"><span class="id" title="section">CompAct</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#GroupActionTheory"><span class="id" title="section">GroupActionTheory</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="172d6d57d3711fc7c98e224a1ba08aa5"><span class="id" title="notation">&quot;</span></a>''C_' ( | to ) ( A )" := (<a class="idref" href="mathcomp.fingroup.action.html#gacent_group"><span class="id" title="definition">gacent_group</span></a> <span class="id" title="var">to</span> <span class="id" title="var">A</span>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="21d18710edc732547be12d2aad92fc55"><span class="id" title="notation">&quot;</span></a>''C_' ( G | to ) ( A )" :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="mathcomp.fingroup.fingroup.html#setI_group"><span class="id" title="definition">setI_group</span></a> <span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.action.html#172d6d57d3711fc7c98e224a1ba08aa5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#172d6d57d3711fc7c98e224a1ba08aa5"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#172d6d57d3711fc7c98e224a1ba08aa5"><span class="id" title="notation">(|</span></a><span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#172d6d57d3711fc7c98e224a1ba08aa5"><span class="id" title="notation">)(</span></a><span class="id" title="var">A</span><a class="idref" href="mathcomp.fingroup.action.html#172d6d57d3711fc7c98e224a1ba08aa5"><span class="id" title="notation">)</span></a>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="4356e7bfd850205078f909c5323b38d5"><span class="id" title="notation">&quot;</span></a>''C_' ( | to ) [ a ]" := (<a class="idref" href="mathcomp.fingroup.action.html#gacent_group"><span class="id" title="definition">gacent_group</span></a> <span class="id" title="var">to</span> <a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span>%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a>) : <span class="id" title="var">Group_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="cd7ed9e932fd7248ad04885bdad6bf8e"><span class="id" title="notation">&quot;</span></a>''C_' ( G | to ) [ a ]" :=<br/>
+&nbsp;&nbsp;(<a class="idref" href="mathcomp.fingroup.fingroup.html#setI_group"><span class="id" title="definition">setI_group</span></a> <span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.action.html#4356e7bfd850205078f909c5323b38d5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#4356e7bfd850205078f909c5323b38d5"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#4356e7bfd850205078f909c5323b38d5"><span class="id" title="notation">(|</span></a><span class="id" title="var">to</span><a class="idref" href="mathcomp.fingroup.action.html#4356e7bfd850205078f909c5323b38d5"><span class="id" title="notation">)[</span></a><span class="id" title="var">a</span><a class="idref" href="mathcomp.fingroup.action.html#4356e7bfd850205078f909c5323b38d5"><span class="id" title="notation">]</span></a>) : <span class="id" title="var">Group_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="9590c91d152ef8d2e38e09f16e80a814"><span class="id" title="notation">&quot;</span></a>to \ sAD" := (<a class="idref" href="mathcomp.fingroup.action.html#ract_groupAction"><span class="id" title="definition">ract_groupAction</span></a> <span class="id" title="var">to</span> <span class="id" title="var">sAD</span>) : <span class="id" title="var">groupAction_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="6ecf966a3859b7d1751aa41c1213e466"><span class="id" title="notation">&quot;</span></a>&lt;[ nGA ] &gt;" := (<a class="idref" href="mathcomp.fingroup.action.html#actby_groupAction"><span class="id" title="definition">actby_groupAction</span></a> <span class="id" title="var">nGA</span>) : <span class="id" title="var">groupAction_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="2756d213ba2c3f60f8b9fc530f7c4cdc"><span class="id" title="notation">&quot;</span></a>to / H" := (<a class="idref" href="mathcomp.fingroup.action.html#quotient_groupAction"><span class="id" title="definition">quotient_groupAction</span></a> <span class="id" title="var">to</span> <span class="id" title="var">H</span>) : <span class="id" title="var">groupAction_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="1f68cf81bb7cfeedd6167d6fc0c539cc"><span class="id" title="notation">&quot;</span></a>to %% H" := (<a class="idref" href="mathcomp.fingroup.action.html#mod_groupAction"><span class="id" title="definition">mod_groupAction</span></a> <span class="id" title="var">to</span> <span class="id" title="var">H</span>) : <span class="id" title="var">groupAction_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="66f424a4bc70f81c17ca6c6cb9254216"><span class="id" title="notation">&quot;</span></a>to \o f" := (<a class="idref" href="mathcomp.fingroup.action.html#comp_groupAction"><span class="id" title="definition">comp_groupAction</span></a> <span class="id" title="var">to</span> <span class="id" title="var">f</span>) : <span class="id" title="var">groupAction_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Operator group isomorphism.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Section</span> <a name="MorphAction"><span class="id" title="section">MorphAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphAction.aT1"><span class="id" title="variable">aT1</span></a> <a name="MorphAction.aT2"><span class="id" title="variable">aT2</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="MorphAction.rT1"><span class="id" title="variable">rT1</span></a> <a name="MorphAction.rT2"><span class="id" title="variable">rT2</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphAction.D1"><span class="id" title="variable">D1</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.action.html#MorphAction.aT1"><span class="id" title="variable">aT1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="MorphAction.D2"><span class="id" title="variable">D2</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.action.html#MorphAction.aT2"><span class="id" title="variable">aT2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphAction.to1"><span class="id" title="variable">to1</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.D1"><span class="id" title="variable">D1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.rT1"><span class="id" title="variable">rT1</span></a>) (<a name="MorphAction.to2"><span class="id" title="variable">to2</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.D2"><span class="id" title="variable">D2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.rT2"><span class="id" title="variable">rT2</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphAction.A"><span class="id" title="variable">A</span></a> : <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.fingroup.action.html#MorphAction.aT1"><span class="id" title="variable">aT1</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="MorphAction.R"><span class="id" title="variable">R</span></a> <a name="MorphAction.S"><span class="id" title="variable">S</span></a> : <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.fingroup.action.html#MorphAction.rT1"><span class="id" title="variable">rT1</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphAction.h"><span class="id" title="variable">h</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.rT1"><span class="id" title="variable">rT1</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#MorphAction.rT2"><span class="id" title="variable">rT2</span></a>) (<a name="MorphAction.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.D1"><span class="id" title="variable">D1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.aT2"><span class="id" title="variable">aT2</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/>
+<span class="id" title="keyword">Hypotheses</span> (<a name="MorphAction.actsDR"><span class="id" title="variable">actsDR</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.fingroup.action.html#MorphAction.D1"><span class="id" title="variable">D1</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.fingroup.action.html#MorphAction.R"><span class="id" title="variable">R</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#MorphAction.to1"><span class="id" title="variable">to1</span></a><a class="idref" href="mathcomp.fingroup.action.html#05449e0e2f87f06fca16a62c49e8b809"><span class="id" title="notation">}</span></a>) (<a name="MorphAction.injh"><span class="id" title="variable">injh</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&amp;,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.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#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="MorphAction.defD2"><span class="id" title="variable">defD2</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.D1"><span class="id" title="variable">D1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.D2"><span class="id" title="variable">D2</span></a>.<br/>
+<span class="id" title="keyword">Hypotheses</span> (<a name="MorphAction.sSR"><span class="id" title="variable">sSR</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.S"><span class="id" title="variable">S</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#MorphAction.R"><span class="id" title="variable">R</span></a>) (<a name="MorphAction.sAD1"><span class="id" title="variable">sAD1</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.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#MorphAction.D1"><span class="id" title="variable">D1</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="MorphAction.hfJ"><span class="id" title="variable">hfJ</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.D1"><span class="id" title="variable">D1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_act"><span class="id" title="definition">morph_act</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.to1"><span class="id" title="variable">to1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.to2"><span class="id" title="variable">to2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="morph_astabs"><span class="id" title="lemma">morph_astabs</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</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.fingroup.action.html#MorphAction.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.fingroup.action.html#MorphAction.to1"><span class="id" title="variable">to1</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.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.fingroup.action.html#MorphAction.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.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.fingroup.action.html#MorphAction.to2"><span class="id" title="variable">to2</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="morph_astab"><span class="id" title="lemma">morph_astab</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.to1"><span class="id" title="variable">to1</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphAction.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.to2"><span class="id" title="variable">to2</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="morph_afix"><span class="id" title="lemma">morph_afix</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.to1"><span class="id" title="variable">to1</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><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.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphAction.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.to2"><span class="id" title="variable">to2</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphAction.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#MorphAction"><span class="id" title="section">MorphAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="MorphGroupAction"><span class="id" title="section">MorphGroupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphGroupAction.aT1"><span class="id" title="variable">aT1</span></a> <a name="MorphGroupAction.aT2"><span class="id" title="variable">aT2</span></a> <a name="MorphGroupAction.rT1"><span class="id" title="variable">rT1</span></a> <a name="MorphGroupAction.rT2"><span class="id" title="variable">rT2</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="MorphGroupAction.D1"><span class="id" title="variable">D1</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.action.html#MorphGroupAction.aT1"><span class="id" title="variable">aT1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="MorphGroupAction.D2"><span class="id" title="variable">D2</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.action.html#MorphGroupAction.aT2"><span class="id" title="variable">aT2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphGroupAction.R1"><span class="id" title="variable">R1</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.action.html#MorphGroupAction.rT1"><span class="id" title="variable">rT1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="MorphGroupAction.R2"><span class="id" title="variable">R2</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.action.html#MorphGroupAction.rT2"><span class="id" title="variable">rT2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphGroupAction.to1"><span class="id" title="variable">to1</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.D1"><span class="id" title="variable">D1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.R1"><span class="id" title="variable">R1</span></a>) (<a name="MorphGroupAction.to2"><span class="id" title="variable">to2</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.D2"><span class="id" title="variable">D2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.R2"><span class="id" title="variable">R2</span></a>).<br/>
+<span class="id" title="keyword">Variables</span> (<a name="MorphGroupAction.h"><span class="id" title="variable">h</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.rT2"><span class="id" title="variable">rT2</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) (<a name="MorphGroupAction.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.D1"><span class="id" title="variable">D1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.aT2"><span class="id" title="variable">aT2</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/>
+<span class="id" title="keyword">Hypotheses</span> (<a name="MorphGroupAction.iso_h"><span class="id" title="variable">iso_h</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.h"><span class="id" title="variable">h</span></a>) (<a name="MorphGroupAction.iso_f"><span class="id" title="variable">iso_f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.D1"><span class="id" title="variable">D1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.D2"><span class="id" title="variable">D2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.f"><span class="id" title="variable">f</span></a>).<br/>
+<span class="id" title="keyword">Hypothesis</span> <a name="MorphGroupAction.hfJ"><span class="id" title="variable">hfJ</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.D1"><span class="id" title="variable">D1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_act"><span class="id" title="definition">morph_act</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.to1"><span class="id" title="variable">to1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.to2"><span class="id" title="variable">to2</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#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.fingroup.action.html#MorphGroupAction.aT1"><span class="id" title="variable">aT1</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</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.fingroup.action.html#MorphGroupAction.rT1"><span class="id" title="variable">rT1</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">M</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.action.html#MorphGroupAction.rT1"><span class="id" title="variable">rT1</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="morph_gastabs"><span class="id" title="lemma">morph_gastabs</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#MorphGroupAction.R1"><span class="id" title="variable">R1</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#MorphGroupAction.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</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.fingroup.action.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.fingroup.action.html#MorphGroupAction.to1"><span class="id" title="variable">to1</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.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.fingroup.action.html#MorphGroupAction.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.action.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.fingroup.action.html#MorphGroupAction.to2"><span class="id" title="variable">to2</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="morph_gastab"><span class="id" title="lemma">morph_gastab</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</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#MorphGroupAction.R1"><span class="id" title="variable">R1</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#MorphGroupAction.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.to1"><span class="id" title="variable">to1</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.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.action.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.to2"><span class="id" title="variable">to2</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="morph_gacent"><span class="id" title="lemma">morph_gacent</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#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#MorphGroupAction.D1"><span class="id" title="variable">D1</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#MorphGroupAction.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.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.to1"><span class="id" title="variable">to1</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.to2"><span class="id" title="variable">to2</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="morph_gact_irr"><span class="id" title="lemma">morph_gact_irr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">M</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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#MorphGroupAction.D1"><span class="id" title="variable">D1</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#M"><span class="id" title="variable">M</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#MorphGroupAction.R1"><span class="id" title="variable">R1</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#acts_irreducibly"><span class="id" title="definition">acts_irreducibly</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.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.action.html#M"><span class="id" title="variable">M</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.to2"><span class="id" title="variable">to2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#acts_irreducibly"><span class="id" title="definition">acts_irreducibly</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction.to1"><span class="id" title="variable">to1</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#MorphGroupAction"><span class="id" title="section">MorphGroupAction</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+ Conjugation and right translation actions.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Section</span> <a name="InternalActionDefs"><span class="id" title="section">InternalActionDefs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="InternalActionDefs.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">A</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.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#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.action.html#InternalActionDefs.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/>
+</div>
+
+<div class="doc">
+ This is not a Canonical action because it is seldom used, and it would     
+ cause too many spurious matches (any group product would be viewed as an   
+ action!).                                                                   
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a name="mulgr_action"><span class="id" title="definition">mulgr_action</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#TotalAction"><span class="id" title="definition">TotalAction</span></a> (@<a class="idref" href="mathcomp.fingroup.fingroup.html#mulg1"><span class="id" title="lemma">mulg1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a>) (@<a class="idref" href="mathcomp.fingroup.fingroup.html#mulgA"><span class="id" title="lemma">mulgA</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">conjg_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#TotalAction"><span class="id" title="definition">TotalAction</span></a> (@<a class="idref" href="mathcomp.fingroup.fingroup.html#conjg1"><span class="id" title="lemma">conjg1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a>) (@<a class="idref" href="mathcomp.fingroup.fingroup.html#conjgM"><span class="id" title="lemma">conjgM</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="conjg_is_groupAction"><span class="id" title="lemma">conjg_is_groupAction</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <a class="idref" href="mathcomp.fingroup.action.html#conjg_action"><span class="id" title="definition">conjg_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">conjg_groupAction</span> := <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#conjg_is_groupAction"><span class="id" title="lemma">conjg_is_groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="rcoset_is_action"><span class="id" title="lemma">rcoset_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> (@<a class="idref" href="mathcomp.fingroup.fingroup.html#rcoset"><span class="id" title="definition">rcoset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">rcoset_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#rcoset_is_action"><span class="id" title="lemma">rcoset_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">conjsg_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#TotalAction"><span class="id" title="definition">TotalAction</span></a> (@<a class="idref" href="mathcomp.fingroup.fingroup.html#conjsg1"><span class="id" title="lemma">conjsg1</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a>) (@<a class="idref" href="mathcomp.fingroup.fingroup.html#conjsgM"><span class="id" title="lemma">conjsgM</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="conjG_is_action"><span class="id" title="lemma">conjG_is_action</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_action"><span class="id" title="definition">is_action</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> (@<a class="idref" href="mathcomp.fingroup.fingroup.html#conjG_group"><span class="id" title="definition">conjG_group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs.gT"><span class="id" title="variable">gT</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="conjG_action"><span class="id" title="definition">conjG_action</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#Action"><span class="id" title="constructor">Action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#conjG_is_action"><span class="id" title="lemma">conjG_is_action</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#InternalActionDefs"><span class="id" title="section">InternalActionDefs</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">&quot;</span></a>'R" := (@<a class="idref" href="mathcomp.fingroup.action.html#mulgr_action"><span class="id" title="definition">mulgr_action</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">action_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">&quot;</span></a>'Rs" := (@<a class="idref" href="mathcomp.fingroup.action.html#rcoset_action"><span class="id" title="definition">rcoset_action</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">action_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">&quot;</span></a>'J" := (@<a class="idref" href="mathcomp.fingroup.action.html#conjg_action"><span class="id" title="definition">conjg_action</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">action_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="4ff57707e8e91cdc242380a7a9082e82"><span class="id" title="notation">&quot;</span></a>'J" := (@<a class="idref" href="mathcomp.fingroup.action.html#conjg_groupAction"><span class="id" title="definition">conjg_groupAction</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">groupAction_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="f05bcd008f250563fa67537a776ea961"><span class="id" title="notation">&quot;</span></a>'Js" := (@<a class="idref" href="mathcomp.fingroup.action.html#conjsg_action"><span class="id" title="definition">conjsg_action</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">action_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="cdd023ccc2efe92dd3cc3bafec9e78af"><span class="id" title="notation">&quot;</span></a>'JG" := (@<a class="idref" href="mathcomp.fingroup.action.html#conjG_action"><span class="id" title="definition">conjG_action</span></a> <span class="id" title="var">_</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">action_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">&quot;</span></a>'Q" := (<a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2e4ccc4d97341df4136dd243a5fe1a5c"><span class="id" title="notation">/</span></a> <span class="id" title="var">_</span>)%<span class="id" title="var">act</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">action_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="a18f700a73b74a8543514202563c13c4"><span class="id" title="notation">&quot;</span></a>'Q" := (<a class="idref" href="mathcomp.fingroup.action.html#4ff57707e8e91cdc242380a7a9082e82"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#4ff57707e8e91cdc242380a7a9082e82"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2756d213ba2c3f60f8b9fc530f7c4cdc"><span class="id" title="notation">/</span></a> <span class="id" title="var">_</span>)%<span class="id" title="var">gact</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8) : <span class="id" title="var">groupAction_scope</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="InternalGroupAction"><span class="id" title="section">InternalGroupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="InternalGroupAction.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#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.fingroup.action.html#InternalGroupAction.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">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.fingroup.action.html#InternalGroupAction.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/>
+<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.gT"><span class="id" title="variable">gT</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+  Various identities for actions on groups.  
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbitR"><span class="id" title="lemma">orbitR</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7a1b86b3dd47c101ea1643c2a591eaf1"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1R"><span class="id" title="lemma">astab1R</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabR"><span class="id" title="lemma">astabR</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabsR"><span class="id" title="lemma">astabsR</span></a> <span class="id" title="var">G</span> : <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.fingroup.action.html#G"><span class="id" title="variable">G</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#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="atransR"><span class="id" title="lemma">atransR</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="faithfulR"><span class="id" title="lemma">faithfulR</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">faithful</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#78176c90a6fd5acd7c52e3779488c085"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e1bf5287bfc6397badc2a71c227e8d0"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="Cayley_repr"><span class="id" title="definition">Cayley_repr</span></a> <span class="id" title="var">G</span> := <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">&lt;[</span></a><a class="idref" href="mathcomp.fingroup.action.html#atrans_acts"><span class="id" title="lemma">atrans_acts</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#atransR"><span class="id" title="lemma">atransR</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>)<a class="idref" href="mathcomp.fingroup.action.html#43026b5788d0e3ac602219a41766cc2f"><span class="id" title="notation">]&gt;</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Theorem</span> <a name="Cayley_isom"><span class="id" title="lemma">Cayley_isom</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#Cayley_repr"><span class="id" title="definition">Cayley_repr</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.fingroup.action.html#Cayley_repr"><span class="id" title="definition">Cayley_repr</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>).<br/>
+ 
+<br/>
+<span class="id" title="keyword">Theorem</span> <a name="Cayley_isog"><span class="id" title="lemma">Cayley_isog</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</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.fingroup.action.html#Cayley_repr"><span class="id" title="definition">Cayley_repr</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbitJ"><span class="id" title="lemma">orbitJ</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#30988ee242f08216f4b40cf90b42b816"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixJ"><span class="id" title="lemma">afixJ</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><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.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabJ"><span class="id" title="lemma">astabJ</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1J"><span class="id" title="lemma">astab1J</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabsJ"><span class="id" title="lemma">astabsJ</span></a> <span class="id" title="var">A</span> : <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.fingroup.action.html#A"><span class="id" title="variable">A</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#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.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.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="setactJ"><span class="id" title="lemma">setactJ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a><a class="idref" href="mathcomp.fingroup.action.html#495b9dab41eed2da7a7d91b1cb8498af"><span class="id" title="notation">^*</span></a>%<span class="id" title="var">act</span> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentJ"><span class="id" title="lemma">gacentJ</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#4ff57707e8e91cdc242380a7a9082e82"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#4ff57707e8e91cdc242380a7a9082e82"><span class="id" title="notation">J</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbitRs"><span class="id" title="lemma">orbitRs</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">Rs</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="sub_afixRs_norms"><span class="id" title="lemma">sub_afixRs_norms</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> <span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">Rs</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#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#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="sub_afixRs_norm"><span class="id" title="lemma">sub_afixRs_norm</span></a> <span class="id" title="var">G</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.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">Rs</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.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.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.fingroup.action.html#G"><span class="id" title="variable">G</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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixRs_rcosets"><span class="id" title="lemma">afixRs_rcosets</span></a> <span class="id" title="var">A</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">Rs</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#35fe9120fe28e031e47d4f6cccd1bd3b"><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.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1Rs"><span class="id" title="lemma">astab1Rs</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <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.fingroup.action.html#InternalGroupAction.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">Rs</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actsRs_rcosets"><span class="id" title="lemma">actsRs_rcosets</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <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#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.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">Rs</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="transRs_rcosets"><span class="id" title="lemma">transRs_rcosets</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">transitive</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">Rs</span></a><a class="idref" href="mathcomp.fingroup.action.html#ad5f1da050fedbae022d48bb21530fba"><span class="id" title="notation">]</span></a>.<br/>
+ 
+<br/>
+</div>
+
+<div class="doc">
+ This is the second part of Aschbacher (5.7)  
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a name="astabRs_rcosets"><span class="id" title="lemma">astabRs_rcosets</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#rcosets"><span class="id" title="definition">rcosets</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#45b5ed5ec521d3d65513a2c70294928a"><span class="id" title="notation">Rs</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.fingroup.html#gcore"><span class="id" title="definition">gcore</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="orbitJs"><span class="id" title="lemma">orbitJs</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#orbit"><span class="id" title="definition">orbit</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f05bcd008f250563fa67537a776ea961"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f05bcd008f250563fa67537a776ea961"><span class="id" title="notation">Js</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#6c121d1bcff5b1c0972474f398d18325"><span class="id" title="notation">:^:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a>.  <br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1Js"><span class="id" title="lemma">astab1Js</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f05bcd008f250563fa67537a776ea961"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f05bcd008f250563fa67537a776ea961"><span class="id" title="notation">Js</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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.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.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="card_conjugates"><span class="id" title="lemma">card_conjugates</span></a> <span class="id" title="var">A</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#6c121d1bcff5b1c0972474f398d18325"><span class="id" title="notation">:^:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="afixJG"><span class="id" title="lemma">afixJG</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.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#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#cdd023ccc2efe92dd3cc3bafec9e78af"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#cdd023ccc2efe92dd3cc3bafec9e78af"><span class="id" title="notation">JG</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#ff32a55d89a82185d94af5720a5e1f63"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#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.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.fingroup.action.html#G"><span class="id" title="variable">G</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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astab1JG"><span class="id" title="lemma">astab1JG</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#cdd023ccc2efe92dd3cc3bafec9e78af"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#cdd023ccc2efe92dd3cc3bafec9e78af"><span class="id" title="notation">JG</span></a><a class="idref" href="mathcomp.fingroup.action.html#f5d78ca47c9779b162180a14b237bdf4"><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.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.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="dom_qactJ"><span class="id" title="lemma">dom_qactJ</span></a> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.action.html#qact_dom"><span class="id" title="definition">qact_dom</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</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.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="qactJ"><span class="id" title="lemma">qactJ</span></a> <span class="id" title="var">H</span> (<span class="id" title="var">Hy</span> : <a class="idref" href="mathcomp.fingroup.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a>) <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a>%<span class="id" title="var">act</span> <a class="idref" href="mathcomp.fingroup.action.html#Hy"><span class="id" title="variable">Hy</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.action.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.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.fingroup.action.html#H"><span class="id" title="variable">H</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.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Hy"><span class="id" title="variable">Hy</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><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.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.fingroup.action.html#Hy"><span class="id" title="variable">Hy</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="actsQ"><span class="id" title="lemma">actsQ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">H</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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.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.fingroup.action.html#H"><span class="id" title="variable">H</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.fingroup.action.html#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.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.fingroup.action.html#B"><span class="id" title="variable">B</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.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#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.action.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.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#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</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="astabsQ"><span class="id" title="lemma">astabsQ</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.action.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.fingroup.action.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.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.fingroup.action.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.fingroup.action.html#H"><span class="id" title="variable">H</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#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><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.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.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</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.fingroup.action.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabQ"><span class="id" title="lemma">astabQ</span></a> <span class="id" title="var">H</span> <span class="id" title="var">Abar</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#Abar"><span class="id" title="variable">Abar</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#Abar"><span class="id" title="variable">Abar</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="sub_astabQ"><span class="id" title="lemma">sub_astabQ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">H</span> <span class="id" title="var">Bbar</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#Bbar"><span class="id" title="variable">Bbar</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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="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.fingroup.action.html#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.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.fingroup.action.html#H"><span class="id" title="variable">H</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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.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.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#Bbar"><span class="id" title="variable">Bbar</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="sub_astabQR"><span class="id" title="lemma">sub_astabQR</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">H</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#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.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.fingroup.action.html#H"><span class="id" title="variable">H</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.fingroup.action.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.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.fingroup.action.html#H"><span class="id" title="variable">H</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><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#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#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">]</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="astabQR"><span class="id" title="lemma">astabQR</span></a> <span class="id" title="var">A</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.action.html#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.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.fingroup.action.html#H"><span class="id" title="variable">H</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><br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">in</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.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">]</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#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#11a9aebd9632a5968df4f5811663355a"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="quotient_astabQ"><span class="id" title="lemma">quotient_astabQ</span></a> <span class="id" title="var">H</span> <span class="id" title="var">Abar</span> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#Abar"><span class="id" title="variable">Abar</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><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.fingroup.action.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#Abar"><span class="id" title="variable">Abar</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="conj_astabQ"><span class="id" title="lemma">conj_astabQ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> :<br/>
+&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.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.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.fingroup.action.html#H"><span class="id" title="variable">H</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.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#b45e3930d72b4fd3a2651cb590e50b50"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="InternalGroupAction.CardClass"><span class="id" title="section">CardClass</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> <a name="InternalGroupAction.CardClass.G"><span class="id" title="variable">G</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.action.html#InternalGroupAction.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">Lemma</span> <a name="index_cent1"><span class="id" title="lemma">index_cent1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#30988ee242f08216f4b40cf90b42b816"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="classes_partition"><span class="id" title="lemma">classes_partition</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#partition"><span class="id" title="definition">partition</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="sum_card_class"><span class="id" title="lemma">sum_card_class</span></a> : <a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">(</span></a><span class="id" title="var">C</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="class_formula"><span class="id" title="lemma">class_formula</span></a> : <a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">(</span></a><span class="id" title="var">C</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#537ae7c18e428d4642a9dfb8520f03ee"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> <a class="idref" href="mathcomp.fingroup.action.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="abelian_classP"><span class="id" title="lemma">abelian_classP</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#30988ee242f08216f4b40cf90b42b816"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.action.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="card_classes_abelian"><span class="id" title="lemma">card_classes_abelian</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction.CardClass"><span class="id" title="section">CardClass</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#InternalGroupAction"><span class="id" title="section">InternalGroupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="gacentQ"><span class="id" title="lemma">gacentQ</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">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.fingroup.action.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>) (<span class="id" title="var">A</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.fingroup.action.html#gT"><span class="id" title="variable">gT</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#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.action.html#a18f700a73b74a8543514202563c13c4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#a18f700a73b74a8543514202563c13c4"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><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.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.action.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.action.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Section</span> <a name="AutAct"><span class="id" title="section">AutAct</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Variable</span> (<a name="AutAct.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="AutAct.G"><span class="id" title="variable">G</span></a> : <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.fingroup.action.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a name="autact"><span class="id" title="definition">autact</span></a> := <a class="idref" href="mathcomp.fingroup.action.html#act"><span class="id" title="projection">act</span></a> (<a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6e357cd64401db5e610bc455a18fd25f"><span class="id" title="notation">P</span></a> <a class="idref" href="mathcomp.fingroup.action.html#8328eb29c76b92b5b9da5590d036317b"><span class="id" title="notation">\</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#subsetT"><span class="id" title="lemma">subsetT</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutAct.G"><span class="id" title="variable">G</span></a>)).<br/>
+<span class="id" title="keyword">Canonical</span> <span class="id" title="var">aut_action</span> := <a class="idref" href="mathcomp.fingroup.action.html#4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.action.html#autact"><span class="id" title="definition">autact</span></a><a class="idref" href="mathcomp.fingroup.action.html#4f9e3ea32a2c10a2e344c72a4fd57c91"><span class="id" title="notation">]</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="autactK"><span class="id" title="lemma">autactK</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.action.html#actperm"><span class="id" title="definition">actperm</span></a> <a class="idref" href="mathcomp.fingroup.action.html#aut_action"><span class="id" title="definition">aut_action</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.action.html#a"><span class="id" title="variable">a</span></a>.<br/>
+ 
+<br/>
+<span class="id" title="keyword">Lemma</span> <a name="autact_is_groupAction"><span class="id" title="lemma">autact_is_groupAction</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#is_groupAction"><span class="id" title="definition">is_groupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#AutAct.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#aut_action"><span class="id" title="definition">aut_action</span></a>.<br/>
+ <span class="id" title="keyword">Canonical</span> <span class="id" title="var">aut_groupAction</span> := <a class="idref" href="mathcomp.fingroup.action.html#GroupAction"><span class="id" title="constructor">GroupAction</span></a> <a class="idref" href="mathcomp.fingroup.action.html#autact_is_groupAction"><span class="id" title="lemma">autact_is_groupAction</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.action.html#AutAct"><span class="id" title="section">AutAct</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Notation</span> <a name="33eddbe1b49846aa9c1a4e3a43f0fe2f"><span class="id" title="notation">&quot;</span></a>[ 'Aut' G ]" := (<a class="idref" href="mathcomp.fingroup.action.html#aut_action"><span class="id" title="definition">aut_action</span></a> <span class="id" title="var">G</span>) : <span class="id" title="var">action_scope</span>.<br/>
+<span class="id" title="keyword">Notation</span> <a name="bbbd7c93723feb1df75c7fc60aa27b88"><span class="id" title="notation">&quot;</span></a>[ 'Aut' G ]" := (<a class="idref" href="mathcomp.fingroup.action.html#aut_groupAction"><span class="id" title="definition">aut_groupAction</span></a> <span class="id" title="var">G</span>) : <span class="id" title="var">groupAction_scope</span>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
+\ No newline at end of file
author	Enrico Tassi	2018-04-20 10:54:22 +0200
committer	Enrico Tassi	2018-04-20 10:54:22 +0200
commit	ed05182cece6bb3706e09b2ce14af4a41a2e8141 (patch)
tree	e850d7314b6372d0476cf2ffaf7d3830721db7b1 /docs/htmldoc/mathcomp.fingroup.action.html
parent	3d196f44681fb3b23ff8a79fbd44e12308680531 (diff)