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<h1 class="libtitle">Library mathcomp.solvable.gfunctor</h1>
<div class="code">
<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/>
Distributed under the terms of CeCILL-B. *)</span><br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/>
<br/>
</div>
<div class="doc">
This file provides basic interfaces for the notion of "generic"
characteristic subgroups; these amount to subfunctors of the identity
functor in some category of groups.
See "Generic Proof Tools And Finite Group Theory",
Francois Garillot, PhD, 2011, Chapter 3.
The implementation proposed here is fairly basic, relying on first order
function matching and on structure telescopes, both of which are somewhat
limited and fragile. It should switch in the future to more general and
more robust quotation matching.
The definitions in this file (types, properties and structures) are all
packaged under the GFunctor submodule, i.e., client code should refer to
GFunctor.continuous, GFunctor.map, etc. Notations, Coercions and Lemmas
are exported and thus directly available, however.
We provide the following:
object_map == the type of the (polymorphic) object map of a group
functor; the %gF scope is bound to object_map.
:= forall gT : finGroupType, {set gT} -> {set gT}.
We define two operations on object_map (with notations in the %gF scope):
F1 \o F2 == the composite map; (F1 \o F2) G expands to F1 (F2 G).
F1 %% F2 == F1 computed modulo F2; we have
(F1 %% F2) G / F2 G = F1 (G / F2 G)
We define the following (type-polymorphic) properties of an object_map F:
group_valued F <-> F G is a group when G is a group
closed F <-> F G is a subgroup o fG when G is a group
continuous F <-> F is continuous with respect to morphism image:
for any f : {morphism G >-> ..}, f @* (F G) is a
a subgroup of F (f @* G); equivalently, F is
functorial in the category Grp of groups.
Most common "characteristic subgroup" are produced
continuous object maps.
iso_continuous F <-> F is continuous with respect to isomorphism image;
equivalently, F is functorial in the Grp groupoid.
The Puig and the Thompson J subgroups are examples
of iso_continuous maps that are not continuous.
pcontinuous F <-> F is continuous with respect to partial morphism
image, i.e., functorial in the category of groups
and partial morphisms. The center and p-core are
examples of pcontinuous maps.
hereditary F <-> inclusion in the image of F is hereditary, i.e.,
for any subgroup H of G, the intersection of H with
F G is included in H. Note that F is pcontinuous
iff it is continuous and hereditary; indeed proofs
of pcontinuous F coerce to proofs of hereditary F
and continuous F.
monotonic F <-> F is monotonic with respect to inclusion: for any
subgroup H of G, F H is a subgroup of F G. The
derived and lower central series are examples of
monotonic maps.
Four structures provide interfaces to these properties:
GFunctor.iso_map == structure for object maps that are group_valued,
closed, and iso_continuous.
[igFun by Fsub & !Fcont] == the iso_map structure for an object map F
such that F G is canonically a group when G is, and
given Fsub : closed F and Fcont : iso_continuous F.
[igFun by Fsub & Fcont] == as above, but expecting Fcont : continuous F.
[igFun of F] == clone an existing GFunctor.iso_map structure for F.
GFunctor.map == structure for continuous object maps, inheriting
from the GFunctor.iso_map structure.
[gFun by Fcont] == the map structure for an F with a canonical iso_map
structure, given Fcont : continuous F.
[gFun of F] == clone an existing GFunctor.map structure for F.
GFunctor.pmap == structure for pcontinuous object maps, inheriting
from the GFunctor.map structure.
[pgFun by Fher] == the pmap structure for an F with a canonical map
structure, given Fher : hereditary F.
[pgFun of F] == clone an existing GFunctor.pmap structure for F.
GFunctor.mono_map == structure for monotonic, continuous object maps
inheriting from the GFunctor.map structure.
[mgFun by Fmon] == the mono_map structure for an F with a canonical
map structure, given Fmon : monotonic F.
[mgFun of F] == clone an existing GFunctor.mono_map structure for F
Lemmas for these group functors use either a 'gF' prefix or an 'F' suffix.
The (F1 \o F2) and (F1 %% F2) operations have canonical GFunctor.map
structures when F1 is monotonic or hereditary, respectively.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/>
<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
<br/>
<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">gFun_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">gF</span>.<br/>
<br/>
<span class="id" title="keyword">Module</span> <a name="GFunctor"><span class="id" title="module">GFunctor</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="GFunctor.object_map"><span class="id" title="definition">object_map</span></a> := <span class="id" title="keyword">∀</span> <span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>, <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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> <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#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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/>
<br/>
<span class="id" title="keyword">Section</span> <a name="GFunctor.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> <span class="id" title="var">hT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a>.<br/>
<br/>
</div>
<div class="doc">
Group closure.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="GFunctor.group_valued"><span class="id" title="definition">group_valued</span></a> := <span class="id" title="keyword">∀</span> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>), <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>).<br/>
<br/>
</div>
<div class="doc">
Subgroup closure.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="GFunctor.closed"><span class="id" title="definition">closed</span></a> := <span class="id" title="keyword">∀</span> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>), <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>.<br/>
<br/>
</div>
<div class="doc">
General functoriality, i.e., continuity of the object map
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="GFunctor.continuous"><span class="id" title="definition">continuous</span></a> :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">gT</span> <span class="id" title="var">hT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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">phi</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.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#hT"><span class="id" title="variable">hT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>),<br/>
<a class="idref" href="mathcomp.solvable.gfunctor.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>).<br/>
<br/>
</div>
<div class="doc">
Functoriality on the Grp groupoid (arrows are restricted to isos).
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="GFunctor.iso_continuous"><span class="id" title="definition">iso_continuous</span></a> :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">gT</span> <span class="id" title="var">hT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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">phi</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.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#hT"><span class="id" title="variable">hT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>),<br/>
<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.solvable.gfunctor.html#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.solvable.gfunctor.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="GFunctor.continuous_is_iso_continuous"><span class="id" title="lemma">continuous_is_iso_continuous</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous"><span class="id" title="definition">continuous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.iso_continuous"><span class="id" title="definition">iso_continuous</span></a>.<br/>
<br/>
</div>
<div class="doc">
Functoriality on Grp with partial morphisms.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="GFunctor.pcontinuous"><span class="id" title="definition">pcontinuous</span></a> :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">gT</span> <span class="id" title="var">hT</span> (<span class="id" title="var">G</span> <span class="id" title="var">D</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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">phi</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.solvable.gfunctor.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#hT"><span class="id" title="variable">hT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>),<br/>
<a class="idref" href="mathcomp.solvable.gfunctor.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="GFunctor.pcontinuous_is_continuous"><span class="id" title="lemma">pcontinuous_is_continuous</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous"><span class="id" title="definition">pcontinuous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous"><span class="id" title="definition">continuous</span></a>.<br/>
<br/>
</div>
<div class="doc">
Heredity with respect to inclusion
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="GFunctor.hereditary"><span class="id" title="definition">hereditary</span></a> :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">gT</span> (<span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>), <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="GFunctor.pcontinuous_is_hereditary"><span class="id" title="lemma">pcontinuous_is_hereditary</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous"><span class="id" title="definition">pcontinuous</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.hereditary"><span class="id" title="definition">hereditary</span></a>.<br/>
<br/>
</div>
<div class="doc">
Monotonicity with respect to inclusion
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="GFunctor.monotonic"><span class="id" title="definition">monotonic</span></a> :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">gT</span> (<span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>), <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>.<br/>
<br/>
</div>
<div class="doc">
Self-expanding composition, and modulo
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Variables</span> (<a name="GFunctor.Definitions.k"><span class="id" title="variable">k</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>) (<a name="GFunctor.Definitions.F1"><span class="id" title="variable">F1</span></a> <a name="GFunctor.Definitions.F2"><span class="id" title="variable">F2</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="GFunctor.comp_head"><span class="id" title="definition">comp_head</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a> := <span class="id" title="keyword">fun</span> <span class="id" title="var">gT</span> <span class="id" title="var">A</span> ⇒ <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> := <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.k"><span class="id" title="variable">k</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F1"><span class="id" title="variable">F1</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F2"><span class="id" title="variable">F2</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#A"><span class="id" title="variable">A</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="GFunctor.modulo"><span class="id" title="definition">modulo</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a> :=<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">gT</span> <span class="id" title="var">A</span> ⇒ <a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F2"><span class="id" title="variable">F2</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#A"><span class="id" title="variable">A</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F1"><span class="id" title="variable">F1</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.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.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions.F2"><span class="id" title="variable">F2</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Definitions"><span class="id" title="section">Definitions</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="GFunctor.ClassDefinitions"><span class="id" title="section">ClassDefinitions</span></a>.<br/>
<br/>
<span class="id" title="keyword">Structure</span> <a name="GFunctor.iso_map"><span class="id" title="record">iso_map</span></a> := <a name="GFunctor.IsoMap"><span class="id" title="constructor">IsoMap</span></a> {<br/>
<a name="GFunctor.apply"><span class="id" title="projection">apply</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a>;<br/>
<span class="id" title="var">_</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.group_valued"><span class="id" title="definition">group_valued</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#apply"><span class="id" title="method">apply</span></a>;<br/>
<span class="id" title="var">_</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.closed"><span class="id" title="definition">closed</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#apply"><span class="id" title="method">apply</span></a>;<br/>
<span class="id" title="var">_</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.iso_continuous"><span class="id" title="definition">iso_continuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#apply"><span class="id" title="method">apply</span></a><br/>
}.<br/>
<br/>
<span class="id" title="keyword">Structure</span> <a name="GFunctor.map"><span class="id" title="record">map</span></a> := <a name="GFunctor.Map"><span class="id" title="constructor">Map</span></a> { <a name="GFunctor.iso_of_map"><span class="id" title="projection">iso_of_map</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.iso_map"><span class="id" title="record">iso_map</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous"><span class="id" title="definition">continuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#iso_of_map"><span class="id" title="method">iso_of_map</span></a> }.<br/>
<br/>
<span class="id" title="keyword">Structure</span> <a name="GFunctor.pmap"><span class="id" title="record">pmap</span></a> := <a name="GFunctor.Pmap"><span class="id" title="constructor">Pmap</span></a> { <a name="GFunctor.map_of_pmap"><span class="id" title="projection">map_of_pmap</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map"><span class="id" title="record">map</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.hereditary"><span class="id" title="definition">hereditary</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#map_of_pmap"><span class="id" title="method">map_of_pmap</span></a> }.<br/>
<br/>
<span class="id" title="keyword">Structure</span> <a name="GFunctor.mono_map"><span class="id" title="record">mono_map</span></a> := <a name="GFunctor.MonoMap"><span class="id" title="constructor">MonoMap</span></a> { <a name="GFunctor.map_of_mono"><span class="id" title="projection">map_of_mono</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map"><span class="id" title="record">map</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.monotonic"><span class="id" title="definition">monotonic</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#map_of_mono"><span class="id" title="method">map_of_mono</span></a> }.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="GFunctor.pack_iso"><span class="id" title="definition">pack_iso</span></a> <span class="id" title="var">F</span> <span class="id" title="var">Fcont</span> <span class="id" title="var">Fgrp</span> <span class="id" title="var">Fsub</span> := @<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.IsoMap"><span class="id" title="constructor">IsoMap</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fgrp"><span class="id" title="variable">Fgrp</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fsub"><span class="id" title="variable">Fsub</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fcont"><span class="id" title="variable">Fcont</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="GFunctor.clone_iso"><span class="id" title="definition">clone_iso</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a>) :=<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">Fgrp</span> <span class="id" title="var">Fsub</span> <span class="id" title="var">Fcont</span> (<span class="id" title="var">isoF</span> := @<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.IsoMap"><span class="id" title="constructor">IsoMap</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fgrp"><span class="id" title="variable">Fgrp</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fsub"><span class="id" title="variable">Fsub</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fcont"><span class="id" title="variable">Fcont</span></a>) ⇒<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">isoF0</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">apply</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#isoF0"><span class="id" title="variable">isoF0</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#isoF"><span class="id" title="variable">isoF</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#isoF0"><span class="id" title="variable">isoF0</span></a> ⇒ <a class="idref" href="mathcomp.solvable.gfunctor.html#isoF"><span class="id" title="variable">isoF</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="GFunctor.clone"><span class="id" title="definition">clone</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a>) :=<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">isoF</span> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">apply</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#isoF"><span class="id" title="variable">isoF</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> ⇒<br/>
<span class="id" title="keyword">fun</span> (<span class="id" title="var">funF0</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map"><span class="id" title="record">map</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">apply</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#funF0"><span class="id" title="variable">funF0</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> ⇒<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">Fcont</span> (<span class="id" title="var">funF</span> := @<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Map"><span class="id" title="constructor">Map</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#isoF"><span class="id" title="variable">isoF</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fcont"><span class="id" title="variable">Fcont</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#funF0"><span class="id" title="variable">funF0</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#funF"><span class="id" title="variable">funF</span></a> ⇒ <a class="idref" href="mathcomp.solvable.gfunctor.html#funF"><span class="id" title="variable">funF</span></a>.<br/>
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<span class="id" title="keyword">Definition</span> <a name="GFunctor.clone_pmap"><span class="id" title="definition">clone_pmap</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a>) :=<br/>
<span class="id" title="keyword">fun</span> (<span class="id" title="var">funF</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map"><span class="id" title="record">map</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">apply</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#funF"><span class="id" title="variable">funF</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> ⇒<br/>
<span class="id" title="keyword">fun</span> (<span class="id" title="var">pfunF0</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pmap"><span class="id" title="record">pmap</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">apply</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#pfunF0"><span class="id" title="variable">pfunF0</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> ⇒<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">Fher</span> (<span class="id" title="var">pfunF</span> := @<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Pmap"><span class="id" title="constructor">Pmap</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#funF"><span class="id" title="variable">funF</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fher"><span class="id" title="variable">Fher</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#pfunF0"><span class="id" title="variable">pfunF0</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#pfunF"><span class="id" title="variable">pfunF</span></a> ⇒ <a class="idref" href="mathcomp.solvable.gfunctor.html#pfunF"><span class="id" title="variable">pfunF</span></a>.<br/>
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<span class="id" title="keyword">Definition</span> <a name="GFunctor.clone_mono"><span class="id" title="definition">clone_mono</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.object_map"><span class="id" title="definition">object_map</span></a>) :=<br/>
<span class="id" title="keyword">fun</span> (<span class="id" title="var">funF</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map"><span class="id" title="record">map</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">apply</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#funF"><span class="id" title="variable">funF</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> ⇒<br/>
<span class="id" title="keyword">fun</span> (<span class="id" title="var">mfunF0</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.mono_map"><span class="id" title="record">mono_map</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">apply</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#mfunF0"><span class="id" title="variable">mfunF0</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a> ⇒<br/>
<span class="id" title="keyword">fun</span> <span class="id" title="var">Fmon</span> (<span class="id" title="var">mfunF</span> := @<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.MonoMap"><span class="id" title="constructor">MonoMap</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#funF"><span class="id" title="variable">funF</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#Fmon"><span class="id" title="variable">Fmon</span></a>) & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#mfunF0"><span class="id" title="variable">mfunF0</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#mfunF"><span class="id" title="variable">mfunF</span></a> ⇒ <a class="idref" href="mathcomp.solvable.gfunctor.html#mfunF"><span class="id" title="variable">mfunF</span></a>.<br/>
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<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.ClassDefinitions"><span class="id" title="section">ClassDefinitions</span></a>.<br/>
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<span class="id" title="keyword">Module</span> <a name="GFunctor.Exports"><span class="id" title="module">Exports</span></a>.<br/>
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<span class="id" title="keyword">Identity</span> <span class="id" title="keyword">Coercion</span> <span class="id" title="var">fun_of_object_map</span> : <span class="id" title="var">object_map</span> >-> <span class="id" title="var">Funclass</span>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">apply</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">iso_map</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.apply"><span class="id" title="projection">object_map</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.iso_of_map"><span class="id" title="projection">iso_of_map</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.iso_of_map"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.iso_of_map"><span class="id" title="projection">map</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.iso_of_map"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.iso_of_map"><span class="id" title="projection">iso_map</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_pmap"><span class="id" title="projection">map_of_pmap</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_pmap"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_pmap"><span class="id" title="projection">pmap</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_pmap"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_pmap"><span class="id" title="projection">map</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_mono"><span class="id" title="projection">map_of_mono</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_mono"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_mono"><span class="id" title="projection">mono_map</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_mono"><span class="id" title="projection">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.map_of_mono"><span class="id" title="projection">map</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous_is_iso_continuous"><span class="id" title="lemma">continuous_is_iso_continuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous_is_iso_continuous"><span class="id" title="lemma">:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous_is_iso_continuous"><span class="id" title="lemma">continuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous_is_iso_continuous"><span class="id" title="lemma">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous_is_iso_continuous"><span class="id" title="lemma">iso_continuous</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_continuous"><span class="id" title="lemma">pcontinuous_is_continuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_continuous"><span class="id" title="lemma">:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_continuous"><span class="id" title="lemma">pcontinuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_continuous"><span class="id" title="lemma">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_continuous"><span class="id" title="lemma">continuous</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_hereditary"><span class="id" title="lemma">pcontinuous_is_hereditary</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_hereditary"><span class="id" title="lemma">:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_hereditary"><span class="id" title="lemma">pcontinuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_hereditary"><span class="id" title="lemma">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous_is_hereditary"><span class="id" title="lemma">hereditary</span></a>.<br/>
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<span class="id" title="keyword">Notation</span> <a name="a1ae11930941a680f6750f6723874923"><span class="id" title="notation">"</span></a>[ 'igFun' 'by' Fsub & Fcont ]" :=<br/>
(<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pack_iso"><span class="id" title="definition">pack_iso</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.continuous_is_iso_continuous"><span class="id" title="lemma">continuous_is_iso_continuous</span></a> <span class="id" title="var">Fcont</span>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">gT</span> <span class="id" title="var">G</span> ⇒ <a class="idref" href="mathcomp.fingroup.fingroup.html#groupP"><span class="id" title="lemma">groupP</span></a> <span class="id" title="var">_</span>) <span class="id" title="var">Fsub</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'igFun' 'by' Fsub & Fcont ]") : <span class="id" title="var">form_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="6a21e731147765f65bc5f2e922126efc"><span class="id" title="notation">"</span></a>[ 'igFun' 'by' Fsub & ! Fcont ]" :=<br/>
(<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pack_iso"><span class="id" title="definition">pack_iso</span></a> <span class="id" title="var">Fcont</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">gT</span> <span class="id" title="var">G</span> ⇒ <a class="idref" href="mathcomp.fingroup.fingroup.html#groupP"><span class="id" title="lemma">groupP</span></a> <span class="id" title="var">_</span>) <span class="id" title="var">Fsub</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'igFun' 'by' Fsub & ! Fcont ]") : <span class="id" title="var">form_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="0d3cf0514afe70d7a29245834117002f"><span class="id" title="notation">"</span></a>[ 'igFun' 'of' F ]" := (@<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.clone_iso"><span class="id" title="definition">clone_iso</span></a> <span class="id" title="var">F</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.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.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'igFun' 'of' F ]") : <span class="id" title="var">form_scope</span>.<br/>
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<span class="id" title="keyword">Notation</span> <a name="f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">"</span></a>[ 'gFun' 'by' Fcont ]" := (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Map"><span class="id" title="constructor">Map</span></a> <span class="id" title="var">Fcont</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'gFun' 'by' Fcont ]") : <span class="id" title="var">form_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="607d805a80d936daf34262a28a285db3"><span class="id" title="notation">"</span></a>[ 'gFun' 'of' F ]" := (@<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">F</span> <span class="id" title="var">_</span> <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> <span class="id" title="var">_</span> <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> <span class="id" title="var">_</span> <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/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'gFun' 'of' F ]") : <span class="id" title="var">form_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">"</span></a>[ 'pgFun' 'by' Fher ]" := (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Pmap"><span class="id" title="constructor">Pmap</span></a> <span class="id" title="var">Fher</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'pgFun' 'by' Fher ]") : <span class="id" title="var">form_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="ee155ab488843b0ecc37c27fc7341f64"><span class="id" title="notation">"</span></a>[ 'pgFun' 'of' F ]" := (@<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.clone_pmap"><span class="id" title="definition">clone_pmap</span></a> <span class="id" title="var">F</span> <span class="id" title="var">_</span> <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> <span class="id" title="var">_</span> <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> <span class="id" title="var">_</span> <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/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'pgFun' 'of' F ]") : <span class="id" title="var">form_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">"</span></a>[ 'mgFun' 'by' Fmon ]" := (<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.MonoMap"><span class="id" title="constructor">MonoMap</span></a> <span class="id" title="var">Fmon</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'mgFun' 'by' Fmon ]") : <span class="id" title="var">form_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="617ba6355e0c6813ccd42cf7ac127f20"><span class="id" title="notation">"</span></a>[ 'mgFun' 'of' F ]" := (@<a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.clone_mono"><span class="id" title="definition">clone_mono</span></a> <span class="id" title="var">F</span> <span class="id" title="var">_</span> <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> <span class="id" title="var">_</span> <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> <span class="id" title="var">_</span> <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/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'mgFun' 'of' F ]") : <span class="id" title="var">form_scope</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.Exports"><span class="id" title="module">Exports</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor"><span class="id" title="module">GFunctor</span></a>.<br/>
<span class="id" title="keyword">Export</span> <span class="id" title="var">GFunctor.Exports</span>.<br/>
<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="5feaca9261529b1a79b9f50bc9e0bd05"><span class="id" title="notation">"</span></a>F1 \o F2" := (<a class="idref" href="mathcomp.solvable.gfunctor.html#comp_head"><span class="id" title="definition">GFunctor.comp_head</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> <span class="id" title="var">F1</span> <span class="id" title="var">F2</span>) : <span class="id" title="var">gFun_scope</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="397612d5557f5f74747d26baf6517ab7"><span class="id" title="notation">"</span></a>F1 %% F2" := (<a class="idref" href="mathcomp.solvable.gfunctor.html#modulo"><span class="id" title="definition">GFunctor.modulo</span></a> <span class="id" title="var">F1</span> <span class="id" title="var">F2</span>) : <span class="id" title="var">gFun_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="FunctorGroup"><span class="id" title="section">FunctorGroup</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="FunctorGroup.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#iso_map"><span class="id" title="record">GFunctor.iso_map</span></a>) (<a name="FunctorGroup.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="FunctorGroup.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.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFgroupset"><span class="id" title="lemma">gFgroupset</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#FunctorGroup.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#FunctorGroup.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#FunctorGroup.G"><span class="id" title="variable">G</span></a>). <br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFgroup</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#Group"><span class="id" title="constructor">Group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFgroupset"><span class="id" title="lemma">gFgroupset</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#FunctorGroup"><span class="id" title="section">FunctorGroup</span></a>.<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFmod_group</span><br/>
(<span class="id" title="var">F1</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#iso_map"><span class="id" title="record">GFunctor.iso_map</span></a>) (<span class="id" title="var">F2</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#object_map"><span class="id" title="definition">GFunctor.object_map</span></a>)<br/>
(<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">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :=<br/>
<a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">of</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#F1"><span class="id" title="variable">F1</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#397612d5557f5f74747d26baf6517ab7"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#F2"><span class="id" title="variable">F2</span></a>)%<span class="id" title="var">gF</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="IsoFunctorTheory"><span class="id" title="section">IsoFunctorTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<span class="id" title="keyword">Variable</span> <a name="IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#iso_map"><span class="id" title="record">GFunctor.iso_map</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFsub"><span class="id" title="lemma">gFsub</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFsub_trans"><span class="id" title="lemma">gFsub_trans</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred_class"><span class="id" title="abbreviation">pred_class</span></a>) :<br/>
<a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#A"><span class="id" title="variable">A</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gF1"><span class="id" title="lemma">gF1</span></a> <span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</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> 1. <br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFiso_cont"><span class="id" title="lemma">gFiso_cont</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#iso_continuous"><span class="id" title="definition">GFunctor.iso_continuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFchar"><span class="id" title="lemma">gFchar</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</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.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFnorm"><span class="id" title="lemma">gFnorm</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.gfunctor.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.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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="gFnorms"><span class="id" title="lemma">gFnorms</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gfunctor.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="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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="gFnormal"><span class="id" title="lemma">gFnormal</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFchar_trans"><span class="id" title="lemma">gFchar_trans</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</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.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</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.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFnormal_trans"><span class="id" title="lemma">gFnormal_trans</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFnorm_trans"><span class="id" title="lemma">gFnorm_trans</span></a> <span class="id" title="var">gT</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#pred_class"><span class="id" title="abbreviation">pred_class</span></a>) (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
<a class="idref" href="mathcomp.solvable.gfunctor.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.solvable.gfunctor.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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="injmF_sub"><span class="id" title="lemma">injmF_sub</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">G</span> <span class="id" title="var">D</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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">f</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.solvable.gfunctor.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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/>
<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.solvable.gfunctor.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#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.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#rT"><span class="id" title="variable">rT</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#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.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="injmF"><span class="id" title="lemma">injmF</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">G</span> <span class="id" title="var">D</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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">f</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.solvable.gfunctor.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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/>
<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.solvable.gfunctor.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#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.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#rT"><span class="id" title="variable">rT</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#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.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFisom"><span class="id" title="lemma">gFisom</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">G</span> <span class="id" title="var">D</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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">R</span> (<span class="id" title="var">f</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.solvable.gfunctor.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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/>
<a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#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.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#gval"><span class="id" title="projection">gval</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#R"><span class="id" title="variable">R</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#f"><span class="id" title="variable">f</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.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#R"><span class="id" title="variable">R</span></a>) <a class="idref" href="mathcomp.solvable.gfunctor.html#f"><span class="id" title="variable">f</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFisog"><span class="id" title="lemma">gFisog</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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">R</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#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/>
<a class="idref" href="mathcomp.solvable.gfunctor.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.solvable.gfunctor.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.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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.solvable.gfunctor.html#IsoFunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#R"><span class="id" title="variable">R</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#IsoFunctorTheory"><span class="id" title="section">IsoFunctorTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="FunctorTheory"><span class="id" title="section">FunctorTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<span class="id" title="keyword">Variable</span> <a name="FunctorTheory.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#map"><span class="id" title="record">GFunctor.map</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFcont"><span class="id" title="lemma">gFcont</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#continuous"><span class="id" title="definition">GFunctor.continuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#FunctorTheory.F"><span class="id" title="variable">F</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="morphimF"><span class="id" title="lemma">morphimF</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">G</span> <span class="id" title="var">D</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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">f</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.solvable.gfunctor.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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/>
<a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#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.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#FunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#FunctorTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#rT"><span class="id" title="variable">rT</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#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.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#FunctorTheory"><span class="id" title="section">FunctorTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="PartialFunctorTheory"><span class="id" title="section">PartialFunctorTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="PartialFunctorTheory.BasicTheory"><span class="id" title="section">BasicTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#pmap"><span class="id" title="record">GFunctor.pmap</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFhereditary"><span class="id" title="lemma">gFhereditary</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#hereditary"><span class="id" title="definition">GFunctor.hereditary</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFunctorI"><span class="id" title="lemma">gFunctorI</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
<a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#H"><span class="id" title="variable">H</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="pmorphimF"><span class="id" title="lemma">pmorphimF</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#pcontinuous"><span class="id" title="definition">GFunctor.pcontinuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFid"><span class="id" title="lemma">gFid</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#G"><span class="id" title="variable">G</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.BasicTheory"><span class="id" title="section">BasicTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="PartialFunctorTheory.Modulo"><span class="id" title="section">Modulo</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="PartialFunctorTheory.Modulo.F1"><span class="id" title="variable">F1</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#pmap"><span class="id" title="record">GFunctor.pmap</span></a>) (<a name="PartialFunctorTheory.Modulo.F2"><span class="id" title="variable">F2</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#map"><span class="id" title="record">GFunctor.map</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFmod_closed"><span class="id" title="lemma">gFmod_closed</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#closed"><span class="id" title="definition">GFunctor.closed</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.Modulo.F1"><span class="id" title="variable">F1</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#397612d5557f5f74747d26baf6517ab7"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.Modulo.F2"><span class="id" title="variable">F2</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFmod_cont"><span class="id" title="lemma">gFmod_cont</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#continuous"><span class="id" title="definition">GFunctor.continuous</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.Modulo.F1"><span class="id" title="variable">F1</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#397612d5557f5f74747d26baf6517ab7"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.Modulo.F2"><span class="id" title="variable">F2</span></a>).<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFmod_igFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">igFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFmod_closed"><span class="id" title="lemma">gFmod_closed</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFmod_cont"><span class="id" title="lemma">gFmod_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFmod_gFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">gFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFmod_cont"><span class="id" title="lemma">gFmod_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.Modulo"><span class="id" title="section">Modulo</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> <a name="PartialFunctorTheory.F1"><span class="id" title="variable">F1</span></a> <a name="PartialFunctorTheory.F2"><span class="id" title="variable">F2</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#pmap"><span class="id" title="record">GFunctor.pmap</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFmod_hereditary"><span class="id" title="lemma">gFmod_hereditary</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#hereditary"><span class="id" title="definition">GFunctor.hereditary</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.F1"><span class="id" title="variable">F1</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#397612d5557f5f74747d26baf6517ab7"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory.F2"><span class="id" title="variable">F2</span></a>).<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFmod_pgFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">pgFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFmod_hereditary"><span class="id" title="lemma">gFmod_hereditary</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#PartialFunctorTheory"><span class="id" title="section">PartialFunctorTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="MonotonicFunctorTheory"><span class="id" title="section">MonotonicFunctorTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFunctorS"><span class="id" title="lemma">gFunctorS</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.solvable.gfunctor.html#mono_map"><span class="id" title="record">GFunctor.mono_map</span></a>) : <a class="idref" href="mathcomp.solvable.gfunctor.html#monotonic"><span class="id" title="definition">GFunctor.monotonic</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#F"><span class="id" title="variable">F</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="MonotonicFunctorTheory.Composition"><span class="id" title="section">Composition</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="MonotonicFunctorTheory.Composition.F1"><span class="id" title="variable">F1</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#mono_map"><span class="id" title="record">GFunctor.mono_map</span></a>) (<a name="MonotonicFunctorTheory.Composition.F2"><span class="id" title="variable">F2</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#map"><span class="id" title="record">GFunctor.map</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFcomp_closed"><span class="id" title="lemma">gFcomp_closed</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#closed"><span class="id" title="definition">GFunctor.closed</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#MonotonicFunctorTheory.Composition.F1"><span class="id" title="variable">F1</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#5feaca9261529b1a79b9f50bc9e0bd05"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#5feaca9261529b1a79b9f50bc9e0bd05"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#MonotonicFunctorTheory.Composition.F2"><span class="id" title="variable">F2</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFcomp_cont"><span class="id" title="lemma">gFcomp_cont</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#continuous"><span class="id" title="definition">GFunctor.continuous</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#MonotonicFunctorTheory.Composition.F1"><span class="id" title="variable">F1</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#5feaca9261529b1a79b9f50bc9e0bd05"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#5feaca9261529b1a79b9f50bc9e0bd05"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#MonotonicFunctorTheory.Composition.F2"><span class="id" title="variable">F2</span></a>).<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFcomp_igFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">igFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFcomp_closed"><span class="id" title="lemma">gFcomp_closed</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFcomp_cont"><span class="id" title="lemma">gFcomp_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFcomp_gFun</span> :=<a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">gFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFcomp_cont"><span class="id" title="lemma">gFcomp_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#MonotonicFunctorTheory.Composition"><span class="id" title="section">Composition</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> <a name="MonotonicFunctorTheory.F1"><span class="id" title="variable">F1</span></a> <a name="MonotonicFunctorTheory.F2"><span class="id" title="variable">F2</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#mono_map"><span class="id" title="record">GFunctor.mono_map</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="gFcompS"><span class="id" title="lemma">gFcompS</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#monotonic"><span class="id" title="definition">GFunctor.monotonic</span></a> (<a class="idref" href="mathcomp.solvable.gfunctor.html#MonotonicFunctorTheory.F1"><span class="id" title="variable">F1</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#5feaca9261529b1a79b9f50bc9e0bd05"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#5feaca9261529b1a79b9f50bc9e0bd05"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#MonotonicFunctorTheory.F2"><span class="id" title="variable">F2</span></a>).<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFcomp_mgFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">mgFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gFcompS"><span class="id" title="lemma">gFcompS</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#MonotonicFunctorTheory"><span class="id" title="section">MonotonicFunctorTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="GFunctorExamples"><span class="id" title="section">GFunctorExamples</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="idGfun"><span class="id" title="definition">idGfun</span></a> <span class="id" title="var">gT</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#18cd0ece1c60c28b29a57eea4cb098ca"><span class="id" title="notation">@</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#18cd0ece1c60c28b29a57eea4cb098ca"><span class="id" title="notation">id</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.solvable.gfunctor.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">Lemma</span> <a name="idGfun_closed"><span class="id" title="lemma">idGfun_closed</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#closed"><span class="id" title="definition">GFunctor.closed</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#idGfun"><span class="id" title="definition">idGfun</span></a>. <br/>
<span class="id" title="keyword">Lemma</span> <a name="idGfun_cont"><span class="id" title="lemma">idGfun_cont</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#continuous"><span class="id" title="definition">GFunctor.continuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#idGfun"><span class="id" title="definition">idGfun</span></a>. <br/>
<span class="id" title="keyword">Lemma</span> <a name="idGfun_monotonic"><span class="id" title="lemma">idGfun_monotonic</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#monotonic"><span class="id" title="definition">GFunctor.monotonic</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#idGfun"><span class="id" title="definition">idGfun</span></a>. <br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">bgFunc_id</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">igFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#idGfun_closed"><span class="id" title="lemma">idGfun_closed</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#idGfun_cont"><span class="id" title="lemma">idGfun_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">gFunc_id</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">gFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#idGfun_cont"><span class="id" title="lemma">idGfun_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">mgFunc_id</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">mgFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#idGfun_monotonic"><span class="id" title="lemma">idGfun_monotonic</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#34d675edff20d6d91e43f305650662d9"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="trivGfun"><span class="id" title="definition">trivGfun</span></a> <span class="id" title="var">gT</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.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> := <a class="idref" href="mathcomp.fingroup.fingroup.html#26d0437a0433a7dd4f49130a7fb26acc"><span class="id" title="notation">[1</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#26d0437a0433a7dd4f49130a7fb26acc"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="trivGfun_cont"><span class="id" title="lemma">trivGfun_cont</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#pcontinuous"><span class="id" title="definition">GFunctor.pcontinuous</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#trivGfun"><span class="id" title="definition">trivGfun</span></a>.<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">trivGfun_igFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">igFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#sub1G"><span class="id" title="lemma">sub1G</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#trivGfun_cont"><span class="id" title="lemma">trivGfun_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#a1ae11930941a680f6750f6723874923"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">trivGfun_gFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">gFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#trivGfun_cont"><span class="id" title="lemma">trivGfun_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f83b8fb103ae042d25e7034a8b11b7e7"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">trivGfun_pgFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">pgFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#trivGfun_cont"><span class="id" title="lemma">trivGfun_cont</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#75f5bf98333ce9930f3fc4415f01e6a0"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctorExamples"><span class="id" title="section">GFunctorExamples</span></a>.<br/>
<br/>
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