phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
map_of_mono : mono_map >-> map.		Coercion	map_of_mono	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "map", "mono_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pack_iso F Fcont Fgrp Fsub	:= @IsoMap F Fgrp Fsub Fcont.	Definition	pack_iso	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
clone_iso (F : object_map)	:= fun Fgrp Fsub Fcont (isoF := @IsoMap F Fgrp Fsub Fcont) => fun isoF0 & phant_id (apply isoF0) F & phant_id isoF isoF0 => isoF.	Definition	clone_iso	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "object_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
clone (F : object_map)	:= fun isoF & phant_id (apply isoF) F => fun (funF0 : map) & phant_id (apply funF0) F => fun Fcont (funF := @Map isoF Fcont) & phant_id funF0 funF => funF.	Definition	clone	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "map", "object_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
clone_pmap (F : object_map)	:= fun (funF : map) & phant_id (apply funF) F => fun (pfunF0 : pmap) & phant_id (apply pfunF0) F => fun Fher (pfunF := @Pmap funF Fher) & phant_id pfunF0 pfunF => pfunF.	Definition	clone_pmap	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "map", "object_map", "pmap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
clone_mono (F : object_map)	:= fun (funF : map) & phant_id (apply funF) F => fun (mfunF0 : mono_map) & phant_id (apply mfunF0) F => fun Fmon (mfunF := @MonoMap funF Fmon) & phant_id mfunF0 mfunF => mfunF.	Definition	clone_mono	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "map", "mono_map", "object_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
continuous_is_iso_continuous : continuous >-> iso_continuous.		Coercion	continuous_is_iso_continuous	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "continuous", "iso_continuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcontinuous_is_continuous : pcontinuous >-> continuous.		Coercion	pcontinuous_is_continuous	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "continuous", "pcontinuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcontinuous_is_hereditary : pcontinuous >-> hereditary.		Coercion	pcontinuous_is_hereditary	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "hereditary", "pcontinuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'igFun' 'by' Fsub & Fcont ]"	:= (pack_iso (continuous_is_iso_continuous Fcont) (fun gT G => groupP _) Fsub) (format "[ 'igFun' 'by' Fsub & Fcont ]") : form_scope.	Notation	[ 'igFun' 'by' Fsub & Fcont ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "continuous_is_iso_continuous", "gT", "groupP", "pack_iso" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'igFun' 'by' Fsub & ! Fcont ]"	:= (pack_iso Fcont (fun gT G => groupP _) Fsub) (format "[ 'igFun' 'by' Fsub & ! Fcont ]") : form_scope.	Notation	[ 'igFun' 'by' Fsub & ! Fcont ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "groupP", "pack_iso" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'igFun' 'of' F ]"	:= (@clone_iso F _ _ _ _ id id) (format "[ 'igFun' 'of' F ]") : form_scope.	Notation	[ 'igFun' 'of' F ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "clone_iso", "id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'gFun' 'by' Fcont ]"	:= (Map Fcont) (format "[ 'gFun' 'by' Fcont ]") : form_scope.	Notation	[ 'gFun' 'by' Fcont ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'gFun' 'of' F ]"	:= (@clone F _ id _ id _ id) (format "[ 'gFun' 'of' F ]") : form_scope.	Notation	[ 'gFun' 'of' F ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "clone", "id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'pgFun' 'by' Fher ]"	:= (Pmap Fher) (format "[ 'pgFun' 'by' Fher ]") : form_scope.	Notation	[ 'pgFun' 'by' Fher ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'pgFun' 'of' F ]"	:= (@clone_pmap F _ id _ id _ id) (format "[ 'pgFun' 'of' F ]") : form_scope.	Notation	[ 'pgFun' 'of' F ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "clone_pmap", "id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'mgFun' 'by' Fmon ]"	:= (MonoMap Fmon) (format "[ 'mgFun' 'by' Fmon ]") : form_scope.	Notation	[ 'mgFun' 'by' Fmon ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'mgFun' 'of' F ]"	:= (@clone_mono F _ id _ id _ id) (format "[ 'mgFun' 'of' F ]") : form_scope.	Notation	[ 'mgFun' 'of' F ]	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "clone_mono", "id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"F1 \o F2"	:= (GFunctor.comp F1 F2) : gFun_scope.	Notation	F1 \o F2	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "comp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"F1 %% F2"	:= (GFunctor.modulo F1 F2) : gFun_scope.	Notation	F1 %% F2	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "modulo" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFgroupset : group_set (F gT G).	Proof. by case: F. Qed.	Lemma	gFgroupset	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group_set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFgroup	:= Group gFgroupset.	Canonical	gFgroup	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFgroupset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFmod_group (F1 : GFunctor.iso_map) (F2 : GFunctor.object_map) (gT : finGroupType) (G : {group gT})	:= [group of (F1 %% F2)%gF gT G].	Canonical	gFmod_group	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "gT", "group", "iso_map", "object_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFsub gT (G : {group gT}) : F gT G \subset G.	Proof. by case: F gT G. Qed.	Lemma	gFsub	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFsub_trans gT (G : {group gT}) (A : {pred gT}) : G \subset A -> F gT G \subset A.	Proof. exact/subset_trans/gFsub. Qed.	Lemma	gFsub_trans	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFsub", "gT", "group", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gF1 gT : F gT 1 = 1.	Proof. exact/trivgP/gFsub. Qed.	Lemma	gF1	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFsub", "gT", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFiso_cont : GFunctor.iso_continuous F.	Proof. by case F. Qed.	Lemma	gFiso_cont	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "iso_continuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFchar gT (G : {group gT}) : F gT G \char G.	Proof. apply/andP; split => //; first by apply: gFsub. apply/forall_inP=> f Af; rewrite -{2}(im_autm Af) -(autmE Af). by rewrite -morphimEsub ?gFsub ?gFiso_cont ?injm_autm. Qed.	Lemma	gFchar	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "autmE", "char", "forall_inP", "gFiso_cont", "gFsub", "gT", "group", "im_autm", "injm_autm", "morphimEsub", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFnorm gT (G : {group gT}) : G \subset 'N(F gT G).	Proof. exact/char_norm/gFchar. Qed.	Lemma	gFnorm	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "char_norm", "gFchar", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFnorms gT (G : {group gT}) : 'N(G) \subset 'N(F gT G).	Proof. exact/char_norms/gFchar. Qed.	Lemma	gFnorms	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "char_norms", "gFchar", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFnormal gT (G : {group gT}) : F gT G <\| G.	Proof. exact/char_normal/gFchar. Qed.	Lemma	gFnormal	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "char_normal", "gFchar", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFchar_trans gT (G H : {group gT}) : H \char G -> F gT H \char G.	Proof. exact/char_trans/gFchar. Qed.	Lemma	gFchar_trans	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "char", "char_trans", "gFchar", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFnormal_trans gT (G H : {group gT}) : H <\| G -> F gT H <\| G.	Proof. exact/char_normal_trans/gFchar. Qed.	Lemma	gFnormal_trans	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "char_normal_trans", "gFchar", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFnorm_trans gT (A : {pred gT}) (G : {group gT}) : A \subset 'N(G) -> A \subset 'N(F gT G).	Proof. by move/subset_trans/(_ (gFnorms G)). Qed.	Lemma	gFnorm_trans	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFnorms", "gT", "group", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injmF_sub gT rT (G D : {group gT}) (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> f @* (F gT G) \subset F rT (f @* G).	Proof. move=> injf sGD; have:= gFiso_cont (injm_restrm sGD injf). by rewrite im_restrm morphim_restrm (setIidPr _) ?gFsub. Qed.	Lemma	injmF_sub	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFiso_cont", "gFsub", "gT", "group", "im_restrm", "injf", "injm_restrm", "morphim_restrm", "morphism", "sGD", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injmF gT rT (G D : {group gT}) (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> f @* (F gT G) = F rT (f @* G).	Proof. move=> injf sGD; have [sfGD injf'] := (morphimS f sGD, injm_invm injf). apply/esym/eqP; rewrite eqEsubset -(injmSK injf') ?gFsub_trans //. by rewrite !(subset_trans (injmF_sub _ _)) ?morphim_invm // gFsub_trans. Qed.	Lemma	injmF	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "eqEsubset", "gFsub_trans", "gT", "group", "injf", "injmF_sub", "injmSK", "injm_invm", "morphimS", "morphim_invm", "morphism", "sGD", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFisom gT rT (G D : {group gT}) R (f : {morphism D >-> rT}) : G \subset D -> isom G (gval R) f -> isom (F gT G) (F rT R) f.	Proof. case/(restrmP f)=> g [gf _ _ _]; rewrite -{f}gf => /isomP[injg <-]. by rewrite sub_isom ?gFsub ?injmF. Qed.	Lemma	gFisom	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFsub", "gT", "group", "injmF", "isom", "isomP", "morphism", "restrmP", "sub_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFisog gT rT (G : {group gT}) (R : {group rT}) : G \isog R -> F gT G \isog F rT R.	Proof. by case/isogP=> f injf <-; rewrite -injmF // sub_isog ?gFsub. Qed.	Lemma	gFisog	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFsub", "gT", "group", "injf", "injmF", "isog", "isogP", "sub_isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFcont : GFunctor.continuous F.	Proof. by case F. Qed.	Lemma	gFcont	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "continuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimF gT rT (G D : {group gT}) (f : {morphism D >-> rT}) : G \subset D -> f @* (F gT G) \subset F rT (f @* G).	Proof. move=> sGD; rewrite -(setIidPr (gFsub F G)). by rewrite -{3}(setIid G) -!(morphim_restrm sGD) gFcont. Qed.	Lemma	morphimF	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFcont", "gFsub", "gT", "group", "morphim_restrm", "morphism", "sGD", "setIid", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFhereditary : GFunctor.hereditary F.	Proof. by case F. Qed.	Lemma	gFhereditary	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "hereditary" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFunctorI gT (G H : {group gT}) : F gT G :&: H = F gT G :&: F gT (G :&: H).	Proof. rewrite -{1}(setIidPr (gFsub F G)) setIAC setIC. rewrite -(setIidPr (gFhereditary (subsetIl G H))). by rewrite setIC -setIA (setIidPr (gFsub F (G :&: H))). Qed.	Lemma	gFunctorI	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFhereditary", "gFsub", "gT", "group", "setIA", "setIAC", "setIC", "setIidPr", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmorphimF : GFunctor.pcontinuous F.	Proof. move=> gT rT G D f; rewrite -morphimIdom -(setIidPl (gFsub F G)) setICA. apply: (subset_trans (morphimS f (gFhereditary (subsetIr D G)))). by rewrite (subset_trans (morphimF F _ _ )) ?morphimIdom ?subsetIl. Qed.	Lemma	pmorphimF	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "gFhereditary", "gFsub", "gT", "morphimF", "morphimIdom", "morphimS", "pcontinuous", "setICA", "setIidPl", "subsetIl", "subsetIr", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFid gT (G : {group gT}) : F gT (F gT G) = F gT G.	Proof. apply/eqP; rewrite eqEsubset gFsub. by move/gFhereditary: (gFsub F G); rewrite setIid /=. Qed.	Lemma	gFid	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "eqEsubset", "gFhereditary", "gFsub", "gT", "group", "setIid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFmod_closed : GFunctor.closed (F1 %% F2).	Proof. by move=> gT G; rewrite sub_cosetpre_quo ?gFsub ?gFnormal. Qed.	Lemma	gFmod_closed	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "closed", "gFnormal", "gFsub", "gT", "sub_cosetpre_quo" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFmod_cont : GFunctor.continuous (F1 %% F2).	Proof. move=> gT rT G f; have nF2 := gFnorm F2. have sDF: G \subset 'dom (coset (F2 _ G)) by rewrite nF2. have sDFf: G \subset 'dom (coset (F2 _ (f @* G)) \o f). by rewrite -sub_morphim_pre ?subsetIl // nF2. pose K := 'ker (restrm sDFf (coset (F2 _ (f @* G)) \o f)). have sFK: 'ker (restrm sDF (coset (F2 _ G))) \subse...	Lemma	gFmod_cont	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "apply", "continuous", "coset", "cosetpreK", "dom", "factm", "gFmod_closed", "gFnorm", "gFsub", "gT", "ker", "ker_comp", "ker_coset", "ker_restrm", "morphimF", "morphimIG", "morphimIdom", "morphimS", "morphim_comp", "morphim_factm", "morphim_restrm", "normG"...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFmod_igFun	:= [igFun by gFmod_closed & gFmod_cont].	Canonical	gFmod_igFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFmod_closed", "gFmod_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFmod_gFun	:= [gFun by gFmod_cont].	Canonical	gFmod_gFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFmod_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFmod_hereditary : GFunctor.hereditary (F1 %% F2).	Proof. move=> gT H G sHG; set FGH := _ :&: H; have nF2H := gFnorm F2 H. rewrite -sub_quotient_pre; first exact: subset_trans (subsetIr _ _) _. pose rH := restrm nF2H (coset (F2 _ H)); pose rHM := [morphism of rH]. have rnorm_simpl: rHM @* H = H / F2 _ H by rewrite morphim_restrm setIid. have nF2G := subset_trans sHG (g...	Lemma	gFmod_hereditary	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "apply", "coset", "cosetpreK", "gFcont", "gFhereditary", "gFnorm", "gFsub", "gT", "hereditary", "ker", "ker_coset", "ker_restrm", "morphimS", "morphim_factm", "morphim_restrm", "morphism", "quotientE", "quotientI", "quotientS", "rG", "rH", "restrm", "sHG",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFmod_pgFun	:= [pgFun by gFmod_hereditary].	Canonical	gFmod_pgFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFmod_hereditary" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFunctorS (F : GFunctor.mono_map) : GFunctor.monotonic F.	Proof. by case: F. Qed.	Lemma	gFunctorS	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "mono_map", "monotonic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFcomp_closed : GFunctor.closed (F1 \o F2).	Proof. by move=> gT G; rewrite !gFsub_trans. Qed.	Lemma	gFcomp_closed	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "closed", "gFsub_trans", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFcomp_cont : GFunctor.continuous (F1 \o F2).	Proof. move=> gT rT G phi; rewrite (subset_trans (morphimF _ _ (gFsub _ _))) //. by rewrite (subset_trans (gFunctorS F1 (gFcont F2 phi))). Qed.	Lemma	gFcomp_cont	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "continuous", "gFcont", "gFsub", "gFunctorS", "gT", "morphimF", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFcomp_igFun	:= [igFun by gFcomp_closed & gFcomp_cont].	Canonical	gFcomp_igFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFcomp_closed", "gFcomp_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFcomp_gFun	:=[gFun by gFcomp_cont].	Canonical	gFcomp_gFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFcomp_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFcompS : GFunctor.monotonic (F1 \o F2).	Proof. by move=> gT H G sHG; rewrite !gFunctorS. Qed.	Lemma	gFcompS	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "gFunctorS", "gT", "monotonic", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFcomp_mgFun	:= [mgFun by gFcompS].	Canonical	gFcomp_mgFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gFcompS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idGfun gT	:= @id {set gT}.	Definition	idGfun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idGfun_closed : GFunctor.closed idGfun.	Proof. by []. Qed.	Lemma	idGfun_closed	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "closed", "idGfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idGfun_cont : GFunctor.continuous idGfun.	Proof. by []. Qed.	Lemma	idGfun_cont	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "continuous", "idGfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idGfun_monotonic : GFunctor.monotonic idGfun.	Proof. by []. Qed.	Lemma	idGfun_monotonic	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "idGfun", "monotonic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bgFunc_id	:= [igFun by idGfun_closed & idGfun_cont].	Canonical	bgFunc_id	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "idGfun_closed", "idGfun_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gFunc_id	:= [gFun by idGfun_cont].	Canonical	gFunc_id	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "idGfun_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mgFunc_id	:= [mgFun by idGfun_monotonic].	Canonical	mgFunc_id	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "idGfun_monotonic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivGfun gT & {set gT}	:= [1 gT].	Definition	trivGfun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivGfun_cont : GFunctor.pcontinuous trivGfun.	Proof. by move=> gT rT D G f; rewrite morphim1. Qed.	Lemma	trivGfun_cont	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "morphim1", "pcontinuous", "trivGfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivGfun_igFun	:= [igFun by sub1G & trivGfun_cont].	Canonical	trivGfun_igFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "sub1G", "trivGfun_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivGfun_gFun	:= [gFun by trivGfun_cont].	Canonical	trivGfun_gFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "trivGfun_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivGfun_pgFun	:= [pgFun by trivGfun_cont].	Canonical	trivGfun_pgFun	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "trivGfun_cont" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupT	:= (group_of gT).	Notation	groupT	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "gT", "group_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subnormal A B	:= (A \subset B) && (iter #\|B\| (fun N => generated (class_support A N)) B == A).	Definition	subnormal	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "class_support", "iter" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invariant_factor A B C	:= [&& A \subset 'N(B), A \subset 'N(C) & B <\| C].	Definition	invariant_factor	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_rel_of (r : rel {set gT})	:= [rel H G : groupT \| r H G].	Definition	group_rel_of	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "gT", "groupT", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
stable_factor A V U	:= ([~: U, A] \subset V) && (V <\| U).	Definition	stable_factor	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
central_factor A V U	:= [&& [~: U, A] \subset V, V \subset U & U \subset A].	Definition	central_factor	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[]	this orders allows and3P to be used	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maximal A B	:= [max A of G \| G \proper B].	Definition	maximal	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "max", "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maximal_eq A B	:= (A == B) \|\| maximal A B.	Definition	maximal_eq	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "maximal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxnormal A B U	:= [max A of G \| G \proper B & U \subset 'N(G)].	Definition	maxnormal	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "max", "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minnormal A B	:= [min A of G \| G :!=: 1 & B \subset 'N(G)].	Definition	minnormal	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
simple A	:= minnormal A A.	Definition	simple	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "minnormal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
chief_factor A V U	:= maxnormal V U A && (U <\| A).	Definition	chief_factor	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "maxnormal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"H <\|<\| G"	:= (subnormal H G) (at level 70, no associativity) : group_scope.	Notation	H <\|<\| G	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "subnormal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A .-invariant"	:= (invariant_factor A) (format "A .-invariant") : group_rel_scope.	Notation	A .-invariant	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "invariant_factor" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A .-stable"	:= (stable_factor A) (format "A .-stable") : group_rel_scope.	Notation	A .-stable	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "stable_factor" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A .-central"	:= (central_factor A) (format "A .-central") : group_rel_scope.	Notation	A .-central	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "central_factor" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"G .-chief"	:= (chief_factor G) (format "G .-chief") : group_rel_scope.	Notation	G .-chief	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "chief_factor" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"r .-series"	:= (path (rel_of_simpl (group_rel_of r))) (format "r .-series") : group_scope.	Notation	r .-series	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "group_rel_of", "path" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setIgr H G	:= (G :&: H)%G.	Let	setIgr	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_setIgr G H : G \subset H -> G = setIgr H G.	Proof. by move/setIidPl/group_inj. Qed.	Let	sub_setIgr	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "group_inj", "setIgr", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
path_setIgr H G s : normal.-series H s -> normal.-series (setIgr G H) (map (setIgr G) s).	Proof. elim: s H => //= K s IHs H /andP[/andP[sHK nHK] Ksn]. by rewrite /normal setSI ?normsIG ?IHs. Qed.	Let	path_setIgr	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "map", "nHK", "normal", "normsIG", "sHK", "setIgr", "setSI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subnormalP H G : reflect (exists2 s, normal.-series H s & last H s = G) (H <\|<\| G).	Proof. apply: (iffP andP) => [[sHG snHG] \| [s Hsn <-{G}]]. move: #\|G\| snHG => m; elim: m => [\|m IHm] in G sHG *. by exists [::]; last by apply/eqP; rewrite eq_sym. rewrite iterSr => /IHm[\|s Hsn defG]. by rewrite sub_gen // class_supportEr (bigD1 1) //= conjsg1 subsetUl. exists (rcons s G); rewrite ?last_r...	Lemma	subnormalP	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "apply", "bigD1", "cat_path", "cats1", "class_supportEr", "class_support_id", "class_support_norm", "class_support_subG", "class_support_sub_norm", "conjsg1", "defG", "eq_sym", "genGid", "gen_subG", "group", "iter", "iterD", "iterSr", "last", "lastP", "last_map", "last_rcon...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subnormal_refl G : G <\|<\| G.	Proof. by apply/subnormalP; exists [::]. Qed.	Lemma	subnormal_refl	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "apply", "subnormalP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subnormal_trans K H G : H <\|<\| K -> K <\|<\| G -> H <\|<\| G.	Proof. case/subnormalP=> [s1 Hs1 <-] /subnormalP[s2 Hs12 <-]. by apply/subnormalP; exists (s1 ++ s2); rewrite ?last_cat // cat_path Hs1. Qed.	Lemma	subnormal_trans	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "apply", "cat_path", "last_cat", "s1", "s2", "subnormalP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_subnormal H G : H <\| G -> H <\|<\| G.	Proof. by move=> nsHG; apply/subnormalP; exists [:: G]; rewrite //= nsHG. Qed.	Lemma	normal_subnormal	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "apply", "nsHG", "subnormalP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setI_subnormal G H K : K \subset G -> H <\|<\| G -> H :&: K <\|<\| K.	Proof. move=> sKG /subnormalP[s Hs defG]; apply/subnormalP. exists (map (setIgr K) s); first exact: path_setIgr. rewrite (last_map (setIgr K)) defG. by apply: val_inj; rewrite /= (setIidPr sKG). Qed.	Lemma	setI_subnormal	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "apply", "defG", "last_map", "map", "path_setIgr", "sKG", "setIgr", "setIidPr", "subnormalP", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subnormal_sub G H : H <\|<\| G -> H \subset G.	Proof. by case/andP. Qed.	Lemma	subnormal_sub	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invariant_subnormal A G H : A \subset 'N(G) -> A \subset 'N(H) -> H <\|<\| G -> exists2 s, (A.-invariant).-series H s & last H s = G.	Proof. move=> nGA nHA /andP[]; move: #\|G\| => m. elim: m => [\|m IHm] in G nGA * => sHG. by rewrite eq_sym; exists [::]; last apply/eqP. rewrite iterSr; set K := <<_>>. have nKA: A \subset 'N(K) by rewrite norms_gen ?norms_class_support. have sHK: H \subset K by rewrite sub_gen ?sub_class_support. case/IHm=> // s Hsn d...	Lemma	invariant_subnormal	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "apply", "class_support_norm", "class_support_subG", "eq_sym", "gen_subG", "invariant", "iterSr", "last", "last_rcons", "norms_class_support", "norms_gen", "rcons", "rcons_path", "sHG", "sHK", "sub_class_support", "sub_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subnormalEsupport G H : H <\|<\| G -> H :=: G \/ <<class_support H G>> \proper G.	Proof. case/andP=> sHG; set K := <<_>> => /eqP <-. have: K \subset G by rewrite gen_subG class_support_subG. rewrite subEproper; case/predU1P=> [defK\|]; [left \| by right]. by elim: #\|G\| => //= _ ->. Qed.	Lemma	subnormalEsupport	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "class_support", "class_support_subG", "gen_subG", "predU1P", "proper", "sHG", "subEproper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subnormalEr G H : H <\|<\| G -> H :=: G \/ (exists K : {group gT}, [/\ H <\|<\| K, K <\| G & K \proper G]).	Proof. case/subnormalP=> s Hs <-{G}. elim/last_ind: s Hs => [\|s G IHs]; first by left. rewrite last_rcons -cats1 cat_path /= andbT; set K := last H s. case/andP=> Hs nsKG; have /[1!subEproper] := normal_sub nsKG. case/predU1P=> [<- \| prKG]; [exact: IHs \| right; exists K; split=> //]. by apply/subnormalP; exists s. Qed.	Lemma	subnormalEr	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "apply", "cat_path", "cats1", "gT", "group", "last", "last_ind", "last_rcons", "normal_sub", "nsKG", "predU1P", "proper", "split", "subEproper", "subnormalP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subnormalEl G H : H <\|<\| G -> H :=: G \/ (exists K : {group gT}, [/\ H <\| K, K <\|<\| G & H \proper K]).	Proof. case/subnormalP=> s Hs <-{G}; elim: s H Hs => /= [\|K s IHs] H; first by left. case/andP=> nsHK Ks; have /[1!subEproper] := normal_sub nsHK. case/predU1P=> [-> \| prHK]; [exact: IHs \| right; exists K; split=> //]. by apply/subnormalP; exists s. Qed.	Lemma	subnormalEl	solvable	solvable/gseries.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]	[ "apply", "gT", "group", "normal_sub", "predU1P", "proper", "split", "subEproper", "subnormalP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_of_mono : mono_map >-> map.

Coercion

map_of_mono

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "map", "mono_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pack_iso F Fcont Fgrp Fsub

:= @IsoMap F Fgrp Fsub Fcont.

Definition

pack_iso

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

clone_iso (F : object_map)

:= fun Fgrp Fsub Fcont (isoF := @IsoMap F Fgrp Fsub Fcont) => fun isoF0 & phant_id (apply isoF0) F & phant_id isoF isoF0 => isoF.

Definition

clone_iso

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "object_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

clone (F : object_map)

:= fun isoF & phant_id (apply isoF) F => fun (funF0 : map) & phant_id (apply funF0) F => fun Fcont (funF := @Map isoF Fcont) & phant_id funF0 funF => funF.

Definition

clone

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "map", "object_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

clone_pmap (F : object_map)

:= fun (funF : map) & phant_id (apply funF) F => fun (pfunF0 : pmap) & phant_id (apply pfunF0) F => fun Fher (pfunF := @Pmap funF Fher) & phant_id pfunF0 pfunF => pfunF.

Definition

clone_pmap

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "map", "object_map", "pmap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

clone_mono (F : object_map)

:= fun (funF : map) & phant_id (apply funF) F => fun (mfunF0 : mono_map) & phant_id (apply mfunF0) F => fun Fmon (mfunF := @MonoMap funF Fmon) & phant_id mfunF0 mfunF => mfunF.

Definition

clone_mono

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "map", "mono_map", "object_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

continuous_is_iso_continuous : continuous >-> iso_continuous.

Coercion

continuous_is_iso_continuous

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "continuous", "iso_continuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcontinuous_is_continuous : pcontinuous >-> continuous.

Coercion

pcontinuous_is_continuous

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "continuous", "pcontinuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcontinuous_is_hereditary : pcontinuous >-> hereditary.

Coercion

pcontinuous_is_hereditary

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "hereditary", "pcontinuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'igFun' 'by' Fsub & Fcont ]"

:= (pack_iso (continuous_is_iso_continuous Fcont) (fun gT G => groupP _) Fsub) (format "[ 'igFun' 'by' Fsub & Fcont ]") : form_scope.

Notation

[ 'igFun' 'by' Fsub & Fcont ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "continuous_is_iso_continuous", "gT", "groupP", "pack_iso" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'igFun' 'by' Fsub & ! Fcont ]"

:= (pack_iso Fcont (fun gT G => groupP _) Fsub) (format "[ 'igFun' 'by' Fsub & ! Fcont ]") : form_scope.

Notation

[ 'igFun' 'by' Fsub & ! Fcont ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "groupP", "pack_iso" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'igFun' 'of' F ]"

:= (@clone_iso F _ _ _ _ id id) (format "[ 'igFun' 'of' F ]") : form_scope.

Notation

[ 'igFun' 'of' F ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "clone_iso", "id" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'gFun' 'by' Fcont ]"

:= (Map Fcont) (format "[ 'gFun' 'by' Fcont ]") : form_scope.

Notation

[ 'gFun' 'by' Fcont ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'gFun' 'of' F ]"

:= (@clone F _ id _ id _ id) (format "[ 'gFun' 'of' F ]") : form_scope.

Notation

[ 'gFun' 'of' F ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "clone", "id" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'pgFun' 'by' Fher ]"

:= (Pmap Fher) (format "[ 'pgFun' 'by' Fher ]") : form_scope.

Notation

[ 'pgFun' 'by' Fher ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'pgFun' 'of' F ]"

:= (@clone_pmap F _ id _ id _ id) (format "[ 'pgFun' 'of' F ]") : form_scope.

Notation

[ 'pgFun' 'of' F ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "clone_pmap", "id" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'mgFun' 'by' Fmon ]"

:= (MonoMap Fmon) (format "[ 'mgFun' 'by' Fmon ]") : form_scope.

Notation

[ 'mgFun' 'by' Fmon ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'mgFun' 'of' F ]"

:= (@clone_mono F _ id _ id _ id) (format "[ 'mgFun' 'of' F ]") : form_scope.

Notation

[ 'mgFun' 'of' F ]

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "clone_mono", "id" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"F1 \o F2"

:= (GFunctor.comp F1 F2) : gFun_scope.

Notation

F1 \o F2

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "comp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"F1 %% F2"

:= (GFunctor.modulo F1 F2) : gFun_scope.

Notation

F1 %% F2

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "modulo" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFgroupset : group_set (F gT G).

Proof. by case: F. Qed.

Lemma

gFgroupset

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group_set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFgroup

:= Group gFgroupset.

Canonical

gFgroup

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFgroupset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFmod_group (F1 : GFunctor.iso_map) (F2 : GFunctor.object_map) (gT : finGroupType) (G : {group gT})

:= [group of (F1 %% F2)%gF gT G].

Canonical

gFmod_group

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "gT", "group", "iso_map", "object_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFsub gT (G : {group gT}) : F gT G \subset G.

Proof. by case: F gT G. Qed.

Lemma

gFsub

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFsub_trans gT (G : {group gT}) (A : {pred gT}) : G \subset A -> F gT G \subset A.

Proof. exact/subset_trans/gFsub. Qed.

Lemma

gFsub_trans

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFsub", "gT", "group", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gF1 gT : F gT 1 = 1.

Proof. exact/trivgP/gFsub. Qed.

Lemma

gF1

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFsub", "gT", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFiso_cont : GFunctor.iso_continuous F.

Proof. by case F. Qed.

Lemma

gFiso_cont

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "iso_continuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFchar gT (G : {group gT}) : F gT G \char G.

Proof. apply/andP; split => //; first by apply: gFsub. apply/forall_inP=> f Af; rewrite -{2}(im_autm Af) -(autmE Af). by rewrite -morphimEsub ?gFsub ?gFiso_cont ?injm_autm. Qed.

Lemma

gFchar

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "autmE", "char", "forall_inP", "gFiso_cont", "gFsub", "gT", "group", "im_autm", "injm_autm", "morphimEsub", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFnorm gT (G : {group gT}) : G \subset 'N(F gT G).

Proof. exact/char_norm/gFchar. Qed.

Lemma

gFnorm

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "char_norm", "gFchar", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFnorms gT (G : {group gT}) : 'N(G) \subset 'N(F gT G).

Proof. exact/char_norms/gFchar. Qed.

Lemma

gFnorms

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "char_norms", "gFchar", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFnormal gT (G : {group gT}) : F gT G <| G.

Proof. exact/char_normal/gFchar. Qed.

Lemma

gFnormal

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "char_normal", "gFchar", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFchar_trans gT (G H : {group gT}) : H \char G -> F gT H \char G.

Proof. exact/char_trans/gFchar. Qed.

Lemma

gFchar_trans

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "char", "char_trans", "gFchar", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFnormal_trans gT (G H : {group gT}) : H <| G -> F gT H <| G.

Proof. exact/char_normal_trans/gFchar. Qed.

Lemma

gFnormal_trans

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "char_normal_trans", "gFchar", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFnorm_trans gT (A : {pred gT}) (G : {group gT}) : A \subset 'N(G) -> A \subset 'N(F gT G).

Proof. by move/subset_trans/(_ (gFnorms G)). Qed.

Lemma

gFnorm_trans

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFnorms", "gT", "group", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injmF_sub gT rT (G D : {group gT}) (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> f @* (F gT G) \subset F rT (f @* G).

Proof. move=> injf sGD; have:= gFiso_cont (injm_restrm sGD injf). by rewrite im_restrm morphim_restrm (setIidPr _) ?gFsub. Qed.

Lemma

injmF_sub

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFiso_cont", "gFsub", "gT", "group", "im_restrm", "injf", "injm_restrm", "morphim_restrm", "morphism", "sGD", "setIidPr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injmF gT rT (G D : {group gT}) (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> f @* (F gT G) = F rT (f @* G).

Proof. move=> injf sGD; have [sfGD injf'] := (morphimS f sGD, injm_invm injf). apply/esym/eqP; rewrite eqEsubset -(injmSK injf') ?gFsub_trans //. by rewrite !(subset_trans (injmF_sub _ _)) ?morphim_invm // gFsub_trans. Qed.

Lemma

injmF

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "eqEsubset", "gFsub_trans", "gT", "group", "injf", "injmF_sub", "injmSK", "injm_invm", "morphimS", "morphim_invm", "morphism", "sGD", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFisom gT rT (G D : {group gT}) R (f : {morphism D >-> rT}) : G \subset D -> isom G (gval R) f -> isom (F gT G) (F rT R) f.

Proof. case/(restrmP f)=> g [gf _ _ _]; rewrite -{f}gf => /isomP[injg <-]. by rewrite sub_isom ?gFsub ?injmF. Qed.

Lemma

gFisom

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFsub", "gT", "group", "injmF", "isom", "isomP", "morphism", "restrmP", "sub_isom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFisog gT rT (G : {group gT}) (R : {group rT}) : G \isog R -> F gT G \isog F rT R.

Proof. by case/isogP=> f injf <-; rewrite -injmF // sub_isog ?gFsub. Qed.

Lemma

gFisog

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFsub", "gT", "group", "injf", "injmF", "isog", "isogP", "sub_isog" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFcont : GFunctor.continuous F.

Proof. by case F. Qed.

Lemma

gFcont

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "continuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimF gT rT (G D : {group gT}) (f : {morphism D >-> rT}) : G \subset D -> f @* (F gT G) \subset F rT (f @* G).

Proof. move=> sGD; rewrite -(setIidPr (gFsub F G)). by rewrite -{3}(setIid G) -!(morphim_restrm sGD) gFcont. Qed.

Lemma

morphimF

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFcont", "gFsub", "gT", "group", "morphim_restrm", "morphism", "sGD", "setIid", "setIidPr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFhereditary : GFunctor.hereditary F.

Proof. by case F. Qed.

Lemma

gFhereditary

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "hereditary" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFunctorI gT (G H : {group gT}) : F gT G :&: H = F gT G :&: F gT (G :&: H).

Proof. rewrite -{1}(setIidPr (gFsub F G)) setIAC setIC. rewrite -(setIidPr (gFhereditary (subsetIl G H))). by rewrite setIC -setIA (setIidPr (gFsub F (G :&: H))). Qed.

Lemma

gFunctorI

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFhereditary", "gFsub", "gT", "group", "setIA", "setIAC", "setIC", "setIidPr", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmorphimF : GFunctor.pcontinuous F.

Proof. move=> gT rT G D f; rewrite -morphimIdom -(setIidPl (gFsub F G)) setICA. apply: (subset_trans (morphimS f (gFhereditary (subsetIr D G)))). by rewrite (subset_trans (morphimF F _ _ )) ?morphimIdom ?subsetIl. Qed.

Lemma

pmorphimF

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "gFhereditary", "gFsub", "gT", "morphimF", "morphimIdom", "morphimS", "pcontinuous", "setICA", "setIidPl", "subsetIl", "subsetIr", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFid gT (G : {group gT}) : F gT (F gT G) = F gT G.

Proof. apply/eqP; rewrite eqEsubset gFsub. by move/gFhereditary: (gFsub F G); rewrite setIid /=. Qed.

Lemma

gFid

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "eqEsubset", "gFhereditary", "gFsub", "gT", "group", "setIid" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFmod_closed : GFunctor.closed (F1 %% F2).

Proof. by move=> gT G; rewrite sub_cosetpre_quo ?gFsub ?gFnormal. Qed.

Lemma

gFmod_closed

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "closed", "gFnormal", "gFsub", "gT", "sub_cosetpre_quo" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFmod_cont : GFunctor.continuous (F1 %% F2).

Proof. move=> gT rT G f; have nF2 := gFnorm F2. have sDF: G \subset 'dom (coset (F2 _ G)) by rewrite nF2. have sDFf: G \subset 'dom (coset (F2 _ (f @* G)) \o f). by rewrite -sub_morphim_pre ?subsetIl // nF2. pose K := 'ker (restrm sDFf (coset (F2 _ (f @* G)) \o f)). have sFK: 'ker (restrm sDF (coset (F2 _ G))) \subse...

Lemma

gFmod_cont

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "apply", "continuous", "coset", "cosetpreK", "dom", "factm", "gFmod_closed", "gFnorm", "gFsub", "gT", "ker", "ker_comp", "ker_coset", "ker_restrm", "morphimF", "morphimIG", "morphimIdom", "morphimS", "morphim_comp", "morphim_factm", "morphim_restrm", "normG"...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFmod_igFun

:= [igFun by gFmod_closed & gFmod_cont].

Canonical

gFmod_igFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFmod_closed", "gFmod_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFmod_gFun

:= [gFun by gFmod_cont].

Canonical

gFmod_gFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFmod_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFmod_hereditary : GFunctor.hereditary (F1 %% F2).

Proof. move=> gT H G sHG; set FGH := _ :&: H; have nF2H := gFnorm F2 H. rewrite -sub_quotient_pre; first exact: subset_trans (subsetIr _ _) _. pose rH := restrm nF2H (coset (F2 _ H)); pose rHM := [morphism of rH]. have rnorm_simpl: rHM @* H = H / F2 _ H by rewrite morphim_restrm setIid. have nF2G := subset_trans sHG (g...

Lemma

gFmod_hereditary

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "apply", "coset", "cosetpreK", "gFcont", "gFhereditary", "gFnorm", "gFsub", "gT", "hereditary", "ker", "ker_coset", "ker_restrm", "morphimS", "morphim_factm", "morphim_restrm", "morphism", "quotientE", "quotientI", "quotientS", "rG", "rH", "restrm", "sHG",...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFmod_pgFun

:= [pgFun by gFmod_hereditary].

Canonical

gFmod_pgFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFmod_hereditary" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFunctorS (F : GFunctor.mono_map) : GFunctor.monotonic F.

Proof. by case: F. Qed.

Lemma

gFunctorS

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "mono_map", "monotonic" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFcomp_closed : GFunctor.closed (F1 \o F2).

Proof. by move=> gT G; rewrite !gFsub_trans. Qed.

Lemma

gFcomp_closed

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "closed", "gFsub_trans", "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFcomp_cont : GFunctor.continuous (F1 \o F2).

Proof. move=> gT rT G phi; rewrite (subset_trans (morphimF _ _ (gFsub _ _))) //. by rewrite (subset_trans (gFunctorS F1 (gFcont F2 phi))). Qed.

Lemma

gFcomp_cont

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "continuous", "gFcont", "gFsub", "gFunctorS", "gT", "morphimF", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFcomp_igFun

:= [igFun by gFcomp_closed & gFcomp_cont].

Canonical

gFcomp_igFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFcomp_closed", "gFcomp_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFcomp_gFun

:=[gFun by gFcomp_cont].

Canonical

gFcomp_gFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFcomp_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFcompS : GFunctor.monotonic (F1 \o F2).

Proof. by move=> gT H G sHG; rewrite !gFunctorS. Qed.

Lemma

gFcompS

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "gFunctorS", "gT", "monotonic", "sHG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFcomp_mgFun

:= [mgFun by gFcompS].

Canonical

gFcomp_mgFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gFcompS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idGfun gT

:= @id {set gT}.

Definition

idGfun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "id" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idGfun_closed : GFunctor.closed idGfun.

Proof. by []. Qed.

Lemma

idGfun_closed

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "closed", "idGfun" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idGfun_cont : GFunctor.continuous idGfun.

Proof. by []. Qed.

Lemma

idGfun_cont

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "continuous", "idGfun" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idGfun_monotonic : GFunctor.monotonic idGfun.

Proof. by []. Qed.

Lemma

idGfun_monotonic

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "idGfun", "monotonic" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bgFunc_id

:= [igFun by idGfun_closed & idGfun_cont].

Canonical

bgFunc_id

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "idGfun_closed", "idGfun_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gFunc_id

:= [gFun by idGfun_cont].

Canonical

gFunc_id

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "idGfun_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mgFunc_id

:= [mgFun by idGfun_monotonic].

Canonical

mgFunc_id

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "idGfun_monotonic" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivGfun gT & {set gT}

:= [1 gT].

Definition

trivGfun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivGfun_cont : GFunctor.pcontinuous trivGfun.

Proof. by move=> gT rT D G f; rewrite morphim1. Qed.

Lemma

trivGfun_cont

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "morphim1", "pcontinuous", "trivGfun" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivGfun_igFun

:= [igFun by sub1G & trivGfun_cont].

Canonical

trivGfun_igFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "sub1G", "trivGfun_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivGfun_gFun

:= [gFun by trivGfun_cont].

Canonical

trivGfun_gFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "trivGfun_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivGfun_pgFun

:= [pgFun by trivGfun_cont].

Canonical

trivGfun_pgFun

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "trivGfun_cont" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupT

:= (group_of gT).

Notation

groupT

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "gT", "group_of" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subnormal A B

:= (A \subset B) && (iter #|B| (fun N => generated (class_support A N)) B == A).

Definition

subnormal

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "class_support", "iter" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invariant_factor A B C

:= [&& A \subset 'N(B), A \subset 'N(C) & B <| C].

Definition

invariant_factor

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_rel_of (r : rel {set gT})

:= [rel H G : groupT | r H G].

Definition

group_rel_of

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "gT", "groupT", "rel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

stable_factor A V U

:= ([~: U, A] \subset V) && (V <| U).

Definition

stable_factor

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

central_factor A V U

:= [&& [~: U, A] \subset V, V \subset U & U \subset A].

Definition

central_factor

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[]

this orders allows and3P to be used

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maximal A B

:= [max A of G | G \proper B].

Definition

maximal

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "max", "proper" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maximal_eq A B

:= (A == B) || maximal A B.

Definition

maximal_eq

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "maximal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxnormal A B U

:= [max A of G | G \proper B & U \subset 'N(G)].

Definition

maxnormal

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "max", "proper" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minnormal A B

:= [min A of G | G :!=: 1 & B \subset 'N(G)].

Definition

minnormal

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "min" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

simple A

:= minnormal A A.

Definition

simple

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "minnormal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

chief_factor A V U

:= maxnormal V U A && (U <| A).

Definition

chief_factor

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "maxnormal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"H <|<| G"

:= (subnormal H G) (at level 70, no associativity) : group_scope.

Notation

H <|<| G

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "subnormal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A .-invariant"

:= (invariant_factor A) (format "A .-invariant") : group_rel_scope.

Notation

A .-invariant

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "invariant_factor" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A .-stable"

:= (stable_factor A) (format "A .-stable") : group_rel_scope.

Notation

A .-stable

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "stable_factor" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A .-central"

:= (central_factor A) (format "A .-central") : group_rel_scope.

Notation

A .-central

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "central_factor" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"G .-chief"

:= (chief_factor G) (format "G .-chief") : group_rel_scope.

Notation

G .-chief

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "chief_factor" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"r .-series"

:= (path (rel_of_simpl (group_rel_of r))) (format "r .-series") : group_scope.

Notation

r .-series

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "group_rel_of", "path" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setIgr H G

:= (G :&: H)%G.

Let

setIgr

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_setIgr G H : G \subset H -> G = setIgr H G.

Proof. by move/setIidPl/group_inj. Qed.

Let

sub_setIgr

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "group_inj", "setIgr", "setIidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

path_setIgr H G s : normal.-series H s -> normal.-series (setIgr G H) (map (setIgr G) s).

Proof. elim: s H => //= K s IHs H /andP[/andP[sHK nHK] Ksn]. by rewrite /normal setSI ?normsIG ?IHs. Qed.

Let

path_setIgr

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "map", "nHK", "normal", "normsIG", "sHK", "setIgr", "setSI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subnormalP H G : reflect (exists2 s, normal.-series H s & last H s = G) (H <|<| G).

Proof. apply: (iffP andP) => [[sHG snHG] | [s Hsn <-{G}]]. move: #|G| snHG => m; elim: m => [|m IHm] in G sHG *. by exists [::]; last by apply/eqP; rewrite eq_sym. rewrite iterSr => /IHm[|s Hsn defG]. by rewrite sub_gen // class_supportEr (bigD1 1) //= conjsg1 subsetUl. exists (rcons s G); rewrite ?last_r...

Lemma

subnormalP

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "apply", "bigD1", "cat_path", "cats1", "class_supportEr", "class_support_id", "class_support_norm", "class_support_subG", "class_support_sub_norm", "conjsg1", "defG", "eq_sym", "genGid", "gen_subG", "group", "iter", "iterD", "iterSr", "last", "lastP", "last_map", "last_rcon...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subnormal_refl G : G <|<| G.

Proof. by apply/subnormalP; exists [::]. Qed.

Lemma

subnormal_refl

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "apply", "subnormalP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subnormal_trans K H G : H <|<| K -> K <|<| G -> H <|<| G.

Proof. case/subnormalP=> [s1 Hs1 <-] /subnormalP[s2 Hs12 <-]. by apply/subnormalP; exists (s1 ++ s2); rewrite ?last_cat // cat_path Hs1. Qed.

Lemma

subnormal_trans

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "apply", "cat_path", "last_cat", "s1", "s2", "subnormalP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_subnormal H G : H <| G -> H <|<| G.

Proof. by move=> nsHG; apply/subnormalP; exists [:: G]; rewrite //= nsHG. Qed.

Lemma

normal_subnormal

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "apply", "nsHG", "subnormalP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setI_subnormal G H K : K \subset G -> H <|<| G -> H :&: K <|<| K.

Proof. move=> sKG /subnormalP[s Hs defG]; apply/subnormalP. exists (map (setIgr K) s); first exact: path_setIgr. rewrite (last_map (setIgr K)) defG. by apply: val_inj; rewrite /= (setIidPr sKG). Qed.

Lemma

setI_subnormal

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "apply", "defG", "last_map", "map", "path_setIgr", "sKG", "setIgr", "setIidPr", "subnormalP", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subnormal_sub G H : H <|<| G -> H \subset G.

Proof. by case/andP. Qed.

Lemma

subnormal_sub

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invariant_subnormal A G H : A \subset 'N(G) -> A \subset 'N(H) -> H <|<| G -> exists2 s, (A.-invariant).-series H s & last H s = G.

Proof. move=> nGA nHA /andP[]; move: #|G| => m. elim: m => [|m IHm] in G nGA * => sHG. by rewrite eq_sym; exists [::]; last apply/eqP. rewrite iterSr; set K := <<_>>. have nKA: A \subset 'N(K) by rewrite norms_gen ?norms_class_support. have sHK: H \subset K by rewrite sub_gen ?sub_class_support. case/IHm=> // s Hsn d...

Lemma

invariant_subnormal

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "apply", "class_support_norm", "class_support_subG", "eq_sym", "gen_subG", "invariant", "iterSr", "last", "last_rcons", "norms_class_support", "norms_gen", "rcons", "rcons_path", "sHG", "sHK", "sub_class_support", "sub_gen" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subnormalEsupport G H : H <|<| G -> H :=: G \/ <<class_support H G>> \proper G.

Proof. case/andP=> sHG; set K := <<_>> => /eqP <-. have: K \subset G by rewrite gen_subG class_support_subG. rewrite subEproper; case/predU1P=> [defK|]; [left | by right]. by elim: #|G| => //= _ ->. Qed.

Lemma

subnormalEsupport

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "class_support", "class_support_subG", "gen_subG", "predU1P", "proper", "sHG", "subEproper" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subnormalEr G H : H <|<| G -> H :=: G \/ (exists K : {group gT}, [/\ H <|<| K, K <| G & K \proper G]).

Proof. case/subnormalP=> s Hs <-{G}. elim/last_ind: s Hs => [|s G IHs]; first by left. rewrite last_rcons -cats1 cat_path /= andbT; set K := last H s. case/andP=> Hs nsKG; have /[1!subEproper] := normal_sub nsKG. case/predU1P=> [<- | prKG]; [exact: IHs | right; exists K; split=> //]. by apply/subnormalP; exists s. Qed.

Lemma

subnormalEr

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "apply", "cat_path", "cats1", "gT", "group", "last", "last_ind", "last_rcons", "normal_sub", "nsKG", "predU1P", "proper", "split", "subEproper", "subnormalP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subnormalEl G H : H <|<| G -> H :=: G \/ (exists K : {group gT}, [/\ H <| K, K <|<| G & H \proper K]).

Proof. case/subnormalP=> s Hs <-{G}; elim: s H Hs => /= [|K s IHs] H; first by left. case/andP=> nsHK Ks; have /[1!subEproper] := normal_sub nsHK. case/predU1P=> [-> | prHK]; [exact: IHs | right; exists K; split=> //]. by apply/subnormalP; exists s. Qed.

Lemma

subnormalEl

solvable

solvable/gseries.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "action", "commutator", "center" ]

[ "apply", "gT", "group", "normal_sub", "predU1P", "proper", "split", "subEproper", "subnormalP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d