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	source_url stringclasses 1 value	commit stringclasses 1 value
cSH : H \subset 'C(S \| to).		Hypothesis	cSH	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fixSH : S \subset 'Fix_to(D :&: H).	Proof. by rewrite -astabCin ?subsetIl // subIset ?cSH ?orbT. Qed.	Let	fixSH	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabCin", "cSH", "subIset", "subsetIl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_mod : 'N(S \| mod_action) = 'N(S \| to) / H.	Proof. apply/setP=> Ha; apply/idP/morphimP=> [nSa \| [a nHa nSa ->]]. case/morphimP: (astabs_dom nSa) => a nHa Da defHa. exists a => //; rewrite !inE Da; apply/subsetP=> x Sx; rewrite !inE. by have:= Sx; rewrite -(astabs_act x nSa) defHa /= modactE ?(subsetP fixSH). have Da := astabs_dom nSa; rewrite !inE mem_quot...	Lemma	astabs_mod	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "apply", "astabs_act", "astabs_dom", "fixSH", "inE", "mem_quotient", "mod_action", "modactE", "morphimP", "setP", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_mod : 'C(S \| mod_action) = 'C(S \| to) / H.	Proof. apply/setP=> Ha; apply/idP/morphimP=> [cSa \| [a nHa cSa ->]]. case/morphimP: (astab_dom cSa) => a nHa Da defHa. exists a => //; rewrite !inE Da; apply/subsetP=> x Sx; rewrite !inE. by rewrite -{2}[x](astab_act cSa) // defHa /= modactE ?(subsetP fixSH). have Da := astab_dom cSa; rewrite !inE mem_quotient //...	Lemma	astab_mod	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "apply", "astab_act", "astab_dom", "fixSH", "inE", "mem_quotient", "mod_action", "modactE", "morphimP", "setP", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_mod G S : H \subset 'C(S \| to) -> G \subset 'N_D(H) -> 'Fix_(S \| mod_action)(G / H) = 'Fix_(S \| to)(G).	Proof. move=> cSH /subsetIP[sGD nHG]. apply/eqP; rewrite eqEsubset !subsetI !subsetIl /= -!astabCin ?quotientS //. have cfixH F: H \subset 'C(S :&: F \| to). by rewrite (subset_trans cSH) // astabS ?subsetIl. rewrite andbC astab_mod ?quotientS //=; first by rewrite astabCin ?subsetIr. by rewrite -(quotientSGK nHG) //=...	Lemma	afix_mod	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabCin", "astabS", "astab_mod", "cSH", "eqEsubset", "mod_action", "nHG", "quotientS", "quotientSGK", "sGD", "subsetI", "subsetIP", "subsetIl", "subsetIr", "subset_trans", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modact_faithful G S : [faithful G / 'C_G(S \| to), on S \| mod_action 'C_G(S \| to)].	Proof. rewrite /faithful astab_mod ?subsetIr //=. by rewrite -quotientIG ?subsetIr ?trivg_quotient. Qed.	Lemma	modact_faithful	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab_mod", "faithful", "mod_action", "on", "quotientIG", "subsetIr", "to", "trivg_quotient" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to %% H"	:= (mod_action to H) : action_scope.	Notation	to %% H	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "mod_action", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actperm a	:= perm (act_inj to a).	Definition	actperm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act_inj", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actpermM : {in D &, {morph actperm : a b / a * b}}.	Proof. by move=> a b Da Db; apply/permP=> x; rewrite permM !permE actMin. Qed.	Lemma	actpermM	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actMin", "actperm", "apply", "permE", "permM", "permP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actperm_morphism	:= Morphism actpermM.	Canonical	actperm_morphism	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actpermM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actpermE a x : actperm a x = to x a.	Proof. by rewrite permE. Qed.	Lemma	actpermE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm", "permE", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actpermK x a : aperm x (actperm a) = to x a.	Proof. exact: actpermE. Qed.	Lemma	actpermK	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm", "actpermE", "aperm", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_actperm : 'ker actperm = 'C(setT \| to).	Proof. congr (_ :&: _); apply/setP=> a /[!inE]/=. apply/eqP/subsetP=> [a1 x _ \| a1]; first by rewrite inE -actpermE a1 perm1. by apply/permP=> x; apply/eqP; have:= a1 x; rewrite !inE actpermE perm1 => ->. Qed.	Lemma	ker_actperm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "a1", "actperm", "actpermE", "apply", "inE", "ker", "perm1", "permP", "setP", "setT", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
faithful_isom (A : {group aT}) S (nSA : actby_cond A S to) : [faithful A, on S \| to] -> isom A (actperm <[nSA]> @* A) (actperm <[nSA]>).	Proof. by move=> ffulAS; apply/isomP; rewrite ker_actperm astab_actby setIT. Qed.	Lemma	faithful_isom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aT", "actby_cond", "actperm", "apply", "astab_actby", "faithful", "group", "isom", "isomP", "ker_actperm", "on", "setIT", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ractpermE : actperm (to \ sAD) =1 actperm to.	Proof. by move=> a; apply/permP=> x; rewrite !permE. Qed.	Lemma	ractpermE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm", "apply", "permE", "permP", "sAD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_ract B : 'Fix_(to \ sAD)(B) = 'Fix_to(B).	Proof. by []. Qed.	Lemma	afix_ract	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "sAD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_ract S : 'C(S \| to \ sAD) = 'C_A(S \| to).	Proof. by rewrite setIA (setIidPl sAD). Qed.	Lemma	astab_ract	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "sAD", "setIA", "setIidPl", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_ract S : 'N(S \| to \ sAD) = 'N_A(S \| to).	Proof. by rewrite setIA (setIidPl sAD). Qed.	Lemma	astabs_ract	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "sAD", "setIA", "setIidPl", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_ract (B : {set aT}) S : [acts B, on S \| to \ sAD] = (B \subset A) && [acts B, on S \| to].	Proof. by rewrite astabs_ract subsetI. Qed.	Lemma	acts_ract	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aT", "astabs_ract", "on", "sAD", "subsetI", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mact x a	:= phi a x.	Definition	mact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mact_is_action : is_action D mact.	Proof. split=> [a x y \| x a b Da Db]; first exact: perm_inj. by rewrite /mact morphM //= permM. Qed.	Lemma	mact_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "is_action", "mact", "morphM", "permM", "perm_inj", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_action	:= Action mact_is_action.	Canonical	morph_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "mact_is_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mactE x a : morph_action x a = phi a x.	Proof. by []. Qed.	Lemma	mactE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "morph_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_faithful : 'injm phi -> [faithful D, on setT \| morph_action].	Proof. move/injmP=> phi_inj; apply/subsetP=> a /setIP[Da /astab_act a1]. apply/set1P/phi_inj => //; apply/permP=> x. by rewrite morph1 perm1 -mactE a1 ?inE. Qed.	Lemma	injm_faithful	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "a1", "apply", "astab_act", "faithful", "inE", "injmP", "mactE", "morph1", "morph_action", "on", "perm1", "permP", "set1P", "setIP", "setT", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_mact a : actperm morph_action a = phi a.	Proof. by apply/permP=> x; rewrite permE. Qed.	Lemma	perm_mact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm", "apply", "morph_action", "permE", "permP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< phi >>"	:= (morph_action phi) : action_scope.	Notation	<< phi >>	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "morph_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_act x e	:= to x (f e).	Definition	comp_act	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_is_action : is_action (f @*^-1 D) comp_act.	Proof. split=> [e \| x e1 e2]; first exact: act_inj. move=> /morphpreP[Be1 Dfe1] /morphpreP[Be2 Dfe2]. by rewrite /comp_act morphM ?actMin. Qed.	Lemma	comp_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actMin", "act_inj", "comp_act", "is_action", "morphM", "morphpreP", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_action	:= Action comp_is_action.	Canonical	comp_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "comp_is_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_actE x e : comp_action x e = to x (f e).	Proof. by []. Qed.	Lemma	comp_actE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "comp_action", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_comp (A : {set gT}) : A \subset B -> 'Fix_comp_action(A) = 'Fix_to(f @* A).	Proof. move=> sAB; apply/setP=> x; rewrite !inE /morphim (setIidPr sAB). apply/subsetP/subsetP; first by move=> + _ /imsetP[a + ->] => /[apply]/[!inE]. by move=> + a Aa => /(_ (f a)); rewrite !inE imset_f// => ->. Qed.	Lemma	afix_comp	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "gT", "imsetP", "imset_f", "inE", "morphim", "setIidPr", "setP", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_comp S : 'C(S \| comp_action) = f @*^-1 'C(S \| to).	Proof. by apply/setP=> x; rewrite !inE -andbA. Qed.	Lemma	astab_comp	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "comp_action", "inE", "setP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_comp S : 'N(S \| comp_action) = f @*^-1 'N(S \| to).	Proof. by apply/setP=> x; rewrite !inE -andbA. Qed.	Lemma	astabs_comp	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "comp_action", "inE", "setP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to \o f"	:= (comp_action to f) : action_scope.	Notation	to \o f	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "comp_action", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gT	:= {perm rT}.	Notation	gT	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aperm_is_action : is_action setT (@aperm rT).	Proof. by apply: is_total_action => [x\|x a b]; rewrite apermE (perm1, permM). Qed.	Lemma	aperm_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aperm", "apermE", "apply", "is_action", "is_total_action", "perm1", "permM", "setT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_action	:= Action aperm_is_action.	Canonical	perm_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aperm_is_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbitE a : porbit a = orbit perm_action <[a]>%g.	Proof. by rewrite unlock. Qed.	Lemma	porbitE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "orbit", "perm_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_act1P a : reflect (forall x, aperm x a = x) (a == 1).	Proof. apply: (iffP eqP) => [-> x \| a1]; first exact: act1. by apply/permP=> x; rewrite -apermE a1 perm1. Qed.	Lemma	perm_act1P	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "a1", "act1", "aperm", "apermE", "apply", "perm1", "permP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_faithful A : [faithful A, on setT \| perm_action].	Proof. apply/subsetP=> a /setIP[Da crTa]. by apply/set1P; apply/permP=> x; rewrite -apermE perm1 (astabP crTa) ?inE. Qed.	Lemma	perm_faithful	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "apermE", "apply", "astabP", "faithful", "inE", "on", "perm1", "permP", "perm_action", "set1P", "setIP", "setT", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actperm_id p : actperm perm_action p = p.	Proof. by apply/permP=> x; rewrite permE. Qed.	Lemma	actperm_id	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm", "apply", "permE", "permP", "perm_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'P"	:= (perm_action _) : action_scope.	Notation	'P	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "perm_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_morphim_actperm (A : {set aT}) : A \subset D -> orbit 'P (actperm to @* A) =1 orbit to A.	Proof. move=> sAD x; rewrite morphimEsub // /orbit -imset_comp. by apply: eq_imset => a //=; rewrite actpermK. Qed.	Lemma	orbit_morphim_actperm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "aT", "actperm", "actpermK", "apply", "eq_imset", "imset_comp", "morphimEsub", "orbit", "sAD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
porbit_actperm (a : aT) : a \in D -> porbit (actperm to a) =1 orbit to <[a]>.	Proof. move=> Da x. by rewrite porbitE -orbit_morphim_actperm ?cycle_subG ?morphim_cycle. Qed.	Lemma	porbit_actperm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "aT", "actperm", "cycle_subG", "morphim_cycle", "orbit", "orbit_morphim_actperm", "porbitE", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_perm	:= actperm (<[subxx 'N(S \| 'P)]>).	Definition	restr_perm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm", "subxx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_perm_morphism	:= [morphism of restr_perm].	Canonical	restr_perm_morphism	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "morphism", "restr_perm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_perm_on p : perm_on S (restr_perm p).	Proof. apply/subsetP=> x; apply: contraR => notSx. by rewrite permE /= /actby (negPf notSx). Qed.	Lemma	restr_perm_on	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby", "apply", "permE", "perm_on", "restr_perm", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
triv_restr_perm p : p \notin 'N(S \| 'P) -> restr_perm p = 1.	Proof. move=> not_nSp; apply/permP=> x. by rewrite !permE /= /actby (negPf not_nSp) andbF. Qed.	Lemma	triv_restr_perm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby", "apply", "permE", "permP", "restr_perm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_permE : {in 'N(S \| 'P) & S, forall p, restr_perm p =1 p}.	Proof. by move=> y x nSp Sx; rewrite /= actpermE actbyE. Qed.	Lemma	restr_permE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actbyE", "actpermE", "restr_perm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_restr_perm : 'ker restr_perm = 'C(S \| 'P).	Proof. by rewrite ker_actperm astab_actby setIT (setIidPr (astab_sub _ _)). Qed.	Lemma	ker_restr_perm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab_actby", "astab_sub", "ker", "ker_actperm", "restr_perm", "setIT", "setIidPr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_restr_perm p : restr_perm p @: S = S.	Proof. exact: im_perm_on (restr_perm_on p). Qed.	Lemma	im_restr_perm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "im_perm_on", "restr_perm", "restr_perm_on" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_perm_commute s : commute (restr_perm s) s.	Proof. have [sC\|/triv_restr_perm->] := boolP (s \in 'N(S \| 'P)); last first. exact: (commute_sym (commute1 _)). apply/permP => x; have /= xsS := astabsP sC x; rewrite !permM. have [xS\|xNS] := boolP (x \in S); first by rewrite ?(restr_permE) ?xsS. by rewrite !(out_perm (restr_perm_on _)) ?xsS. Qed.	Lemma	restr_perm_commute	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabsP", "commute", "commute1", "commute_sym", "last", "out_perm", "permM", "permP", "restr_perm", "restr_permE", "restr_perm_on", "triv_restr_perm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
SymE : Sym S = 'C(~: S \| 'P).	Proof. apply/setP => s; rewrite inE; apply/idP/astabP => [sS x\|/= S_id]. by rewrite inE /= apermE => /out_perm->. by apply/subsetP => x; move=> /(contra_neqN (S_id _)); rewrite inE negbK. Qed.	Lemma	SymE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Sym", "apermE", "apply", "astabP", "contra_neqN", "inE", "out_perm", "setP", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_in A (B : {set gT})	:= 'N_A(B \| 'P) / 'C_A(B \| 'P).	Definition	Aut_in	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sHG: H \subset G.		Hypothesis	sHG	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_restr_perm a : a \in Aut G -> restr_perm H a \in Aut H.	Proof. move=> AutGa. case nHa: (a \in 'N(H \| 'P)); last by rewrite triv_restr_perm ?nHa ?group1. rewrite inE restr_perm_on; apply/morphicP=> x y Hx Hy /=. by rewrite !restr_permE ?groupM // -(autmE AutGa) morphM ?(subsetP sHG). Qed.	Lemma	Aut_restr_perm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "apply", "autmE", "group1", "groupM", "inE", "last", "morphM", "morphicP", "restr_perm", "restr_permE", "restr_perm_on", "sHG", "subsetP", "triv_restr_perm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_perm_Aut : restr_perm H @* Aut G \subset Aut H.	Proof. by apply/subsetP=> a'; case/morphimP=> a _ AutGa ->{a'}; apply: Aut_restr_perm. Qed.	Lemma	restr_perm_Aut	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Aut_restr_perm", "apply", "morphimP", "restr_perm", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_in_isog : Aut_in (Aut G) H \isog restr_perm H @* Aut G.	Proof. rewrite /Aut_in -ker_restr_perm kerE -morphpreIdom -morphimIdom -kerE /=. by rewrite setIA (setIC _ (Aut G)) first_isog_loc ?subsetIr. Qed.	Lemma	Aut_in_isog	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Aut_in", "first_isog_loc", "isog", "kerE", "ker_restr_perm", "morphimIdom", "morphpreIdom", "restr_perm", "setIA", "setIC", "subsetIr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_sub_fullP : reflect (forall h : {morphism H >-> gT}, 'injm h -> h @* H = H -> exists g : {morphism G >-> gT}, [/\ 'injm g, g @* G = G & {in H, g =1 h}]) (Aut_in (Aut G) H \isog Aut H).	Proof. rewrite (isog_transl _ Aut_in_isog) /=; set rG := _ @* _. apply: (iffP idP) => [iso_rG h injh hH\| AutHinG]. have: aut injh hH \in rG; last case/morphimP=> g nHg AutGg def_g. suffices ->: rG = Aut H by apply: Aut_aut. by apply/eqP; rewrite eqEcard restr_perm_Aut /= (card_isog iso_rG). exists (autm_mor...	Lemma	Aut_sub_fullP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Aut_aut", "Aut_in", "Aut_in_isog", "actbyE", "actpermE", "aperm", "apply", "aut", "autE", "autm_morphism", "card_isog", "eqEcard", "eqEsubset", "eq_Aut", "gT", "im_autm", "inE", "injm_autm", "isog", "isog_refl", "isog_transl", "last", "mem_morphim", "morphimP"...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(injf : 'injm f) (sGD : G \subset D) (sHG : H \subset G).		Hypotheses	injf	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "sGD", "sHG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sHD	:= subset_trans sHG sGD.	Let	sHD	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "sGD", "sHG", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fGisom	:= (Aut_isom injf sGD).	Notation	fGisom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut_isom", "injf", "sGD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fHisom	:= (Aut_isom injf sHD).	Notation	fHisom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut_isom", "injf", "sHD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inH	:= (restr_perm H).	Notation	inH	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "restr_perm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
infH	:= (restr_perm (f @* H)).	Notation	infH	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "restr_perm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_Aut_isom a : a \in Aut G -> (fGisom a \in 'N(f @* H \| 'P)) = (a \in 'N(H \| 'P)).	Proof. move=> AutGa; rewrite !inE sub_morphim_pre // subsetI sHD /= /aperm. rewrite !(sameP setIidPl eqP) !eqEsubset !subsetIl; apply: eq_subset_r => x. rewrite !inE; apply: andb_id2l => Hx; have Gx: x \in G := subsetP sHG x Hx. have Dax: a x \in D by rewrite (subsetP sGD) // Aut_closed. by rewrite Aut_isomE // -!sub1s...	Lemma	astabs_Aut_isom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Aut_closed", "Aut_isomE", "aperm", "apply", "eqEsubset", "eq_subset_r", "fGisom", "inE", "injmSK", "morphim_set1", "sGD", "sHD", "sHG", "setIidPl", "sub1set", "sub_morphim_pre", "subsetI", "subsetIl", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_restr_perm a : a \in Aut G -> fHisom (inH a) = infH (fGisom a).	Proof. move=> AutGa; case nHa: (a \in 'N(H \| 'P)); last first. by rewrite !triv_restr_perm ?astabs_Aut_isom ?nHa ?morph1. apply: (eq_Aut (Aut_Aut_isom injf sHD _)) => [\|fx Hfx /=]. by rewrite (Aut_restr_perm (morphimS f sHG)) ?Aut_Aut_isom. have [x Dx Hx def_fx] := morphimP Hfx; have Gx := subsetP sHG x Hx. rewrite...	Lemma	isom_restr_perm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Aut_Aut_isom", "Aut_isomE", "Aut_restr_perm", "Dx", "apply", "astabs_Aut_isom", "eq_Aut", "fGisom", "fHisom", "inH", "infH", "injf", "last", "morph1", "morphimP", "morphimS", "restr_permE", "sHD", "sHG", "subsetP", "triv_restr_perm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restr_perm_isom : isom (inH @* Aut G) (infH @* Aut (f @* G)) fHisom.	Proof. apply: sub_isom; rewrite ?restr_perm_Aut ?injm_Aut_isom //=. rewrite -(im_Aut_isom injf sGD) -!morphim_comp. apply: eq_in_morphim; last exact: isom_restr_perm. (* TODO: investigate why rewrite does not match in the same order ) apply/setP=> a; rewrite in_setI [in RHS]in_setI; apply: andb_id2r => AutGa. ( the m...	Lemma	restr_perm_isom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Aut_restr_perm", "apply", "astabs_Aut_isom", "eq_in_morphim", "fHisom", "im_Aut_isom", "inE", "inH", "in_setI", "infH", "injf", "injm_Aut_isom", "isom", "isom_restr_perm", "last", "morphim_comp", "restr_perm_Aut", "sGD", "sHG", "setP", "sub_isom" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Aut_sub : Aut_in (Aut (f @* G)) (f @* H) \isog Aut_in (Aut G) H.	Proof. do 2!rewrite isog_sym (isog_transl _ (Aut_in_isog _ _)). by rewrite isog_sym (isom_isog _ _ restr_perm_isom) // restr_perm_Aut. Qed.	Lemma	injm_Aut_sub	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Aut_in", "Aut_in_isog", "isog", "isog_sym", "isog_transl", "isom_isog", "restr_perm_Aut", "restr_perm_isom" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Aut_full : (Aut_in (Aut (f @* G)) (f @* H) \isog Aut (f @* H)) = (Aut_in (Aut G) H \isog Aut H).	Proof. by rewrite (isog_transl _ injm_Aut_sub) (isog_transr _ (injm_Aut injf sHD)). Qed.	Lemma	injm_Aut_full	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Aut_in", "injf", "injm_Aut", "injm_Aut_sub", "isog", "isog_transl", "isog_transr", "sHD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actT	:= (action D rT).	Notation	actT	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_groupAction (to : actT)	:= {in D, forall a, actperm to a \in Aut R}.	Definition	is_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "actT", "actperm", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupAction	:= GroupAction {gact :> actT; _ : is_groupAction gact}.	Structure	groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actT", "gact", "is_groupAction" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
clone_groupAction to	:= let: GroupAction _ toA := to return {type of GroupAction for to} -> _ in fun k => k toA : groupAction.	Definition	clone_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "groupAction", "to", "type" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'groupAction' 'of' to ]"	:= (clone_groupAction (@GroupAction _ _ _ _ to)) (format "[ 'groupAction' 'of' to ]") : form_scope.	Notation	[ 'groupAction' 'of' to ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "clone_groupAction", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gact_range & groupAction D R	:= R.	Definition	gact_range	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "groupAction" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent to A	:= 'Fix_(R \| to)(D :&: A).	Definition	gacent	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_on_group A S to	:= [acts A, on S \| to] /\ S \subset R.	Definition	acts_on_group	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "on", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actby_cond_group A S to : acts_on_group A S to -> actby_cond A S to	:= @proj1 _ _.	Coercion	actby_cond_group	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby_cond", "acts_on_group", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_irreducibly A S to	:= [min S of G \| G :!=: 1 & [acts A, on G \| to]].	Definition	acts_irreducibly	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "min", "on", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( \| to ) ( A )"	:= (gacent to A) : group_scope.	Notation	''C_' ( \| to ) ( A )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacent", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( G \| to ) ( A )"	:= (G :&: 'C_(\|to)(A)) : group_scope.	Notation	''C_' ( G \| to ) ( A )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( \| to ) [ a ]"	:= 'C_(\|to)([set a]) : group_scope.	Notation	''C_' ( \| to ) [ a ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( G \| to ) [ a ]"	:= 'C_(G \| to)([set a]) : group_scope.	Notation	''C_' ( G \| to ) [ a ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'acts' A , 'on' 'group' G \| to }"	:= (acts_on_group A G to) (format "{ 'acts' A , 'on' 'group' G \| to }") : type_scope.	Notation	{ 'acts' A , 'on' 'group' G \| to }	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_on_group", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actperm_Aut : is_groupAction R to.	Proof. by case: to. Qed.	Lemma	actperm_Aut	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "is_groupAction", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_actperm_Aut : actperm to @* D \subset Aut R.	Proof. by apply/subsetP=> _ /morphimP[a _ Da ->]; apply: actperm_Aut. Qed.	Lemma	im_actperm_Aut	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "Da", "actperm", "actperm_Aut", "apply", "morphimP", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gact_out x a : a \in D -> x \notin R -> to x a = x.	Proof. by move=> Da Rx; rewrite -actpermE (out_Aut _ Rx) ?actperm_Aut. Qed.	Lemma	gact_out	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actpermE", "actperm_Aut", "out_Aut", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gactM : {in D, forall a, {in R &, {morph to^~ a : x y / x * y}}}.	Proof. move=> a Da /= x y; rewrite -!(actpermE to); apply: morphicP x y. by rewrite Aut_morphic ?actperm_Aut. Qed.	Lemma	gactM	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut_morphic", "Da", "actpermE", "actperm_Aut", "apply", "morphicP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actmM a : {in R &, {morph actm to a : x y / x * y}}.	Proof. by rewrite /actm; case: ifP => //; apply: gactM. Qed.	Lemma	actmM	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actm", "apply", "gactM", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
act_morphism a	:= Morphism (actmM a).	Canonical	act_morphism	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actmM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_actm : {in D, forall a (S : {set rT}), S \subset R -> actm to a @* S = to^* S a}.	Proof. by move=> a Da /= S sSR; rewrite /morphim /= actmEfun ?(setIidPr _). Qed.	Lemma	morphim_actm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actm", "actmEfun", "morphim", "sSR", "setIidPr", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentIdom : 'C_(\|to)(D :&: A) = 'C_(\|to)(A).	Proof. by rewrite /gacent setIA setIid. Qed.	Lemma	gacentIdom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacent", "setIA", "setIid", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentIim : 'C_(R \| to)(A) = 'C_(\|to)(A).	Proof. by rewrite setIA setIid. Qed.	Lemma	gacentIim	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setIA", "setIid", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentS : A \subset B -> 'C_(\|to)(B) \subset 'C_(\|to)(A).	Proof. by move=> sAB; rewrite !(setIS, afixS). Qed.	Lemma	gacentS	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afixS", "setIS", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentU : 'C_(\|to)(A :\|: B) = 'C_(\|to)(A) :&: 'C_(\|to)(B).	Proof. by rewrite -setIIr -afixU -setIUr. Qed.	Lemma	gacentU	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afixU", "setIIr", "setIUr", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(Da : a \in D) (sAD : A \subset D) (sSR : S \subset R).		Hypotheses	Da	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "sAD", "sSR" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentE : 'C_(\|to)(A) = 'Fix_(R \| to)(A).	Proof. by rewrite -{2}(setIidPr sAD). Qed.	Lemma	gacentE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "sAD", "setIidPr", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent1E : 'C_(\|to)[a] = 'Fix_(R \| to)[a].	Proof. by rewrite /gacent [D :&: _](setIidPr _) ?sub1set. Qed.	Lemma	gacent1E	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacent", "setIidPr", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subgacentE : 'C_(S \| to)(A) = 'Fix_(S \| to)(A).	Proof. by rewrite gacentE setIA (setIidPl sSR). Qed.	Lemma	subgacentE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacentE", "sSR", "setIA", "setIidPl", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

cSH : H \subset 'C(S | to).

Hypothesis

cSH

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fixSH : S \subset 'Fix_to(D :&: H).

Proof. by rewrite -astabCin ?subsetIl // subIset ?cSH ?orbT. Qed.

Let

fixSH

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabCin", "cSH", "subIset", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_mod : 'N(S | mod_action) = 'N(S | to) / H.

Proof. apply/setP=> Ha; apply/idP/morphimP=> [nSa | [a nHa nSa ->]]. case/morphimP: (astabs_dom nSa) => a nHa Da defHa. exists a => //; rewrite !inE Da; apply/subsetP=> x Sx; rewrite !inE. by have:= Sx; rewrite -(astabs_act x nSa) defHa /= modactE ?(subsetP fixSH). have Da := astabs_dom nSa; rewrite !inE mem_quot...

Lemma

astabs_mod

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "apply", "astabs_act", "astabs_dom", "fixSH", "inE", "mem_quotient", "mod_action", "modactE", "morphimP", "setP", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_mod : 'C(S | mod_action) = 'C(S | to) / H.

Proof. apply/setP=> Ha; apply/idP/morphimP=> [cSa | [a nHa cSa ->]]. case/morphimP: (astab_dom cSa) => a nHa Da defHa. exists a => //; rewrite !inE Da; apply/subsetP=> x Sx; rewrite !inE. by rewrite -{2}[x](astab_act cSa) // defHa /= modactE ?(subsetP fixSH). have Da := astab_dom cSa; rewrite !inE mem_quotient //...

Lemma

astab_mod

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "apply", "astab_act", "astab_dom", "fixSH", "inE", "mem_quotient", "mod_action", "modactE", "morphimP", "setP", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_mod G S : H \subset 'C(S | to) -> G \subset 'N_D(H) -> 'Fix_(S | mod_action)(G / H) = 'Fix_(S | to)(G).

Proof. move=> cSH /subsetIP[sGD nHG]. apply/eqP; rewrite eqEsubset !subsetI !subsetIl /= -!astabCin ?quotientS //. have cfixH F: H \subset 'C(S :&: F | to). by rewrite (subset_trans cSH) // astabS ?subsetIl. rewrite andbC astab_mod ?quotientS //=; first by rewrite astabCin ?subsetIr. by rewrite -(quotientSGK nHG) //=...

Lemma

afix_mod

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabCin", "astabS", "astab_mod", "cSH", "eqEsubset", "mod_action", "nHG", "quotientS", "quotientSGK", "sGD", "subsetI", "subsetIP", "subsetIl", "subsetIr", "subset_trans", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modact_faithful G S : [faithful G / 'C_G(S | to), on S | mod_action 'C_G(S | to)].

Proof. rewrite /faithful astab_mod ?subsetIr //=. by rewrite -quotientIG ?subsetIr ?trivg_quotient. Qed.

Lemma

modact_faithful

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab_mod", "faithful", "mod_action", "on", "quotientIG", "subsetIr", "to", "trivg_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to %% H"

:= (mod_action to H) : action_scope.

Notation

to %% H

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "mod_action", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actperm a

:= perm (act_inj to a).

Definition

actperm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act_inj", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actpermM : {in D &, {morph actperm : a b / a * b}}.

Proof. by move=> a b Da Db; apply/permP=> x; rewrite permM !permE actMin. Qed.

Lemma

actpermM

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actMin", "actperm", "apply", "permE", "permM", "permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actperm_morphism

:= Morphism actpermM.

Canonical

actperm_morphism

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actpermM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actpermE a x : actperm a x = to x a.

Proof. by rewrite permE. Qed.

Lemma

actpermE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm", "permE", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actpermK x a : aperm x (actperm a) = to x a.

Proof. exact: actpermE. Qed.

Lemma

actpermK

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm", "actpermE", "aperm", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_actperm : 'ker actperm = 'C(setT | to).

Proof. congr (_ :&: _); apply/setP=> a /[!inE]/=. apply/eqP/subsetP=> [a1 x _ | a1]; first by rewrite inE -actpermE a1 perm1. by apply/permP=> x; apply/eqP; have:= a1 x; rewrite !inE actpermE perm1 => ->. Qed.

Lemma

ker_actperm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "a1", "actperm", "actpermE", "apply", "inE", "ker", "perm1", "permP", "setP", "setT", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

faithful_isom (A : {group aT}) S (nSA : actby_cond A S to) : [faithful A, on S | to] -> isom A (actperm <[nSA]> @* A) (actperm <[nSA]>).

Proof. by move=> ffulAS; apply/isomP; rewrite ker_actperm astab_actby setIT. Qed.

Lemma

faithful_isom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aT", "actby_cond", "actperm", "apply", "astab_actby", "faithful", "group", "isom", "isomP", "ker_actperm", "on", "setIT", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ractpermE : actperm (to \ sAD) =1 actperm to.

Proof. by move=> a; apply/permP=> x; rewrite !permE. Qed.

Lemma

ractpermE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm", "apply", "permE", "permP", "sAD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_ract B : 'Fix_(to \ sAD)(B) = 'Fix_to(B).

Proof. by []. Qed.

Lemma

afix_ract

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "sAD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_ract S : 'C(S | to \ sAD) = 'C_A(S | to).

Proof. by rewrite setIA (setIidPl sAD). Qed.

Lemma

astab_ract

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "sAD", "setIA", "setIidPl", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_ract S : 'N(S | to \ sAD) = 'N_A(S | to).

Proof. by rewrite setIA (setIidPl sAD). Qed.

Lemma

astabs_ract

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "sAD", "setIA", "setIidPl", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_ract (B : {set aT}) S : [acts B, on S | to \ sAD] = (B \subset A) && [acts B, on S | to].

Proof. by rewrite astabs_ract subsetI. Qed.

Lemma

acts_ract

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aT", "astabs_ract", "on", "sAD", "subsetI", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mact x a

:= phi a x.

Definition

mact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mact_is_action : is_action D mact.

Proof. split=> [a x y | x a b Da Db]; first exact: perm_inj. by rewrite /mact morphM //= permM. Qed.

Lemma

mact_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "is_action", "mact", "morphM", "permM", "perm_inj", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_action

:= Action mact_is_action.

Canonical

morph_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "mact_is_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mactE x a : morph_action x a = phi a x.

Proof. by []. Qed.

Lemma

mactE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "morph_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_faithful : 'injm phi -> [faithful D, on setT | morph_action].

Proof. move/injmP=> phi_inj; apply/subsetP=> a /setIP[Da /astab_act a1]. apply/set1P/phi_inj => //; apply/permP=> x. by rewrite morph1 perm1 -mactE a1 ?inE. Qed.

Lemma

injm_faithful

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "a1", "apply", "astab_act", "faithful", "inE", "injmP", "mactE", "morph1", "morph_action", "on", "perm1", "permP", "set1P", "setIP", "setT", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_mact a : actperm morph_action a = phi a.

Proof. by apply/permP=> x; rewrite permE. Qed.

Lemma

perm_mact

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm", "apply", "morph_action", "permE", "permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<< phi >>"

:= (morph_action phi) : action_scope.

Notation

<< phi >>

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "morph_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_act x e

:= to x (f e).

Definition

comp_act

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_is_action : is_action (f @*^-1 D) comp_act.

Proof. split=> [e | x e1 e2]; first exact: act_inj. move=> /morphpreP[Be1 Dfe1] /morphpreP[Be2 Dfe2]. by rewrite /comp_act morphM ?actMin. Qed.

Lemma

comp_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actMin", "act_inj", "comp_act", "is_action", "morphM", "morphpreP", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_action

:= Action comp_is_action.

Canonical

comp_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "comp_is_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_actE x e : comp_action x e = to x (f e).

Proof. by []. Qed.

Lemma

comp_actE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "comp_action", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_comp (A : {set gT}) : A \subset B -> 'Fix_comp_action(A) = 'Fix_to(f @* A).

Proof. move=> sAB; apply/setP=> x; rewrite !inE /morphim (setIidPr sAB). apply/subsetP/subsetP; first by move=> + _ /imsetP[a + ->] => /[apply]/[!inE]. by move=> + a Aa => /(_ (f a)); rewrite !inE imset_f// => ->. Qed.

Lemma

afix_comp

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "gT", "imsetP", "imset_f", "inE", "morphim", "setIidPr", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_comp S : 'C(S | comp_action) = f @*^-1 'C(S | to).

Proof. by apply/setP=> x; rewrite !inE -andbA. Qed.

Lemma

astab_comp

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "comp_action", "inE", "setP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_comp S : 'N(S | comp_action) = f @*^-1 'N(S | to).

Proof. by apply/setP=> x; rewrite !inE -andbA. Qed.

Lemma

astabs_comp

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "comp_action", "inE", "setP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to \o f"

:= (comp_action to f) : action_scope.

Notation

to \o f

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "comp_action", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gT

:= {perm rT}.

Notation

gT

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aperm_is_action : is_action setT (@aperm rT).

Proof. by apply: is_total_action => [x|x a b]; rewrite apermE (perm1, permM). Qed.

Lemma

aperm_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aperm", "apermE", "apply", "is_action", "is_total_action", "perm1", "permM", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_action

:= Action aperm_is_action.

Canonical

perm_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aperm_is_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbitE a : porbit a = orbit perm_action <[a]>%g.

Proof. by rewrite unlock. Qed.

Lemma

porbitE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "orbit", "perm_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_act1P a : reflect (forall x, aperm x a = x) (a == 1).

Proof. apply: (iffP eqP) => [-> x | a1]; first exact: act1. by apply/permP=> x; rewrite -apermE a1 perm1. Qed.

Lemma

perm_act1P

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "a1", "act1", "aperm", "apermE", "apply", "perm1", "permP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_faithful A : [faithful A, on setT | perm_action].

Proof. apply/subsetP=> a /setIP[Da crTa]. by apply/set1P; apply/permP=> x; rewrite -apermE perm1 (astabP crTa) ?inE. Qed.

Lemma

perm_faithful

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "apermE", "apply", "astabP", "faithful", "inE", "on", "perm1", "permP", "perm_action", "set1P", "setIP", "setT", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actperm_id p : actperm perm_action p = p.

Proof. by apply/permP=> x; rewrite permE. Qed.

Lemma

actperm_id

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm", "apply", "permE", "permP", "perm_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'P"

:= (perm_action _) : action_scope.

Notation

'P

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "perm_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_morphim_actperm (A : {set aT}) : A \subset D -> orbit 'P (actperm to @* A) =1 orbit to A.

Proof. move=> sAD x; rewrite morphimEsub // /orbit -imset_comp. by apply: eq_imset => a //=; rewrite actpermK. Qed.

Lemma

orbit_morphim_actperm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "aT", "actperm", "actpermK", "apply", "eq_imset", "imset_comp", "morphimEsub", "orbit", "sAD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

porbit_actperm (a : aT) : a \in D -> porbit (actperm to a) =1 orbit to <[a]>.

Proof. move=> Da x. by rewrite porbitE -orbit_morphim_actperm ?cycle_subG ?morphim_cycle. Qed.

Lemma

porbit_actperm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "aT", "actperm", "cycle_subG", "morphim_cycle", "orbit", "orbit_morphim_actperm", "porbitE", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_perm

:= actperm (<[subxx 'N(S | 'P)]>).

Definition

restr_perm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_perm_morphism

:= [morphism of restr_perm].

Canonical

restr_perm_morphism

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "morphism", "restr_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_perm_on p : perm_on S (restr_perm p).

Proof. apply/subsetP=> x; apply: contraR => notSx. by rewrite permE /= /actby (negPf notSx). Qed.

Lemma

restr_perm_on

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby", "apply", "permE", "perm_on", "restr_perm", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

triv_restr_perm p : p \notin 'N(S | 'P) -> restr_perm p = 1.

Proof. move=> not_nSp; apply/permP=> x. by rewrite !permE /= /actby (negPf not_nSp) andbF. Qed.

Lemma

triv_restr_perm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby", "apply", "permE", "permP", "restr_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_permE : {in 'N(S | 'P) & S, forall p, restr_perm p =1 p}.

Proof. by move=> y x nSp Sx; rewrite /= actpermE actbyE. Qed.

Lemma

restr_permE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actbyE", "actpermE", "restr_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_restr_perm : 'ker restr_perm = 'C(S | 'P).

Proof. by rewrite ker_actperm astab_actby setIT (setIidPr (astab_sub _ _)). Qed.

Lemma

ker_restr_perm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab_actby", "astab_sub", "ker", "ker_actperm", "restr_perm", "setIT", "setIidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_restr_perm p : restr_perm p @: S = S.

Proof. exact: im_perm_on (restr_perm_on p). Qed.

Lemma

im_restr_perm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "im_perm_on", "restr_perm", "restr_perm_on" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_perm_commute s : commute (restr_perm s) s.

Proof. have [sC|/triv_restr_perm->] := boolP (s \in 'N(S | 'P)); last first. exact: (commute_sym (commute1 _)). apply/permP => x; have /= xsS := astabsP sC x; rewrite !permM. have [xS|xNS] := boolP (x \in S); first by rewrite ?(restr_permE) ?xsS. by rewrite !(out_perm (restr_perm_on _)) ?xsS. Qed.

Lemma

restr_perm_commute

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabsP", "commute", "commute1", "commute_sym", "last", "out_perm", "permM", "permP", "restr_perm", "restr_permE", "restr_perm_on", "triv_restr_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

SymE : Sym S = 'C(~: S | 'P).

Proof. apply/setP => s; rewrite inE; apply/idP/astabP => [sS x|/= S_id]. by rewrite inE /= apermE => /out_perm->. by apply/subsetP => x; move=> /(contra_neqN (S_id _)); rewrite inE negbK. Qed.

Lemma

SymE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Sym", "apermE", "apply", "astabP", "contra_neqN", "inE", "out_perm", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_in A (B : {set gT})

:= 'N_A(B | 'P) / 'C_A(B | 'P).

Definition

Aut_in

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sHG: H \subset G.

Hypothesis

sHG

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_restr_perm a : a \in Aut G -> restr_perm H a \in Aut H.

Proof. move=> AutGa. case nHa: (a \in 'N(H | 'P)); last by rewrite triv_restr_perm ?nHa ?group1. rewrite inE restr_perm_on; apply/morphicP=> x y Hx Hy /=. by rewrite !restr_permE ?groupM // -(autmE AutGa) morphM ?(subsetP sHG). Qed.

Lemma

Aut_restr_perm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "apply", "autmE", "group1", "groupM", "inE", "last", "morphM", "morphicP", "restr_perm", "restr_permE", "restr_perm_on", "sHG", "subsetP", "triv_restr_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_perm_Aut : restr_perm H @* Aut G \subset Aut H.

Proof. by apply/subsetP=> a'; case/morphimP=> a _ AutGa ->{a'}; apply: Aut_restr_perm. Qed.

Lemma

restr_perm_Aut

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Aut_restr_perm", "apply", "morphimP", "restr_perm", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_in_isog : Aut_in (Aut G) H \isog restr_perm H @* Aut G.

Proof. rewrite /Aut_in -ker_restr_perm kerE -morphpreIdom -morphimIdom -kerE /=. by rewrite setIA (setIC _ (Aut G)) first_isog_loc ?subsetIr. Qed.

Lemma

Aut_in_isog

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Aut_in", "first_isog_loc", "isog", "kerE", "ker_restr_perm", "morphimIdom", "morphpreIdom", "restr_perm", "setIA", "setIC", "subsetIr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_sub_fullP : reflect (forall h : {morphism H >-> gT}, 'injm h -> h @* H = H -> exists g : {morphism G >-> gT}, [/\ 'injm g, g @* G = G & {in H, g =1 h}]) (Aut_in (Aut G) H \isog Aut H).

Proof. rewrite (isog_transl _ Aut_in_isog) /=; set rG := _ @* _. apply: (iffP idP) => [iso_rG h injh hH| AutHinG]. have: aut injh hH \in rG; last case/morphimP=> g nHg AutGg def_g. suffices ->: rG = Aut H by apply: Aut_aut. by apply/eqP; rewrite eqEcard restr_perm_Aut /= (card_isog iso_rG). exists (autm_mor...

Lemma

Aut_sub_fullP

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Aut_aut", "Aut_in", "Aut_in_isog", "actbyE", "actpermE", "aperm", "apply", "aut", "autE", "autm_morphism", "card_isog", "eqEcard", "eqEsubset", "eq_Aut", "gT", "im_autm", "inE", "injm_autm", "isog", "isog_refl", "isog_transl", "last", "mem_morphim", "morphimP"...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(injf : 'injm f) (sGD : G \subset D) (sHG : H \subset G).

Hypotheses

injf

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "sGD", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sHD

:= subset_trans sHG sGD.

Let

sHD

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "sGD", "sHG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fGisom

:= (Aut_isom injf sGD).

Notation

fGisom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut_isom", "injf", "sGD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fHisom

:= (Aut_isom injf sHD).

Notation

fHisom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut_isom", "injf", "sHD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inH

:= (restr_perm H).

Notation

inH

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "restr_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

infH

:= (restr_perm (f @* H)).

Notation

infH

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "restr_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_Aut_isom a : a \in Aut G -> (fGisom a \in 'N(f @* H | 'P)) = (a \in 'N(H | 'P)).

Proof. move=> AutGa; rewrite !inE sub_morphim_pre // subsetI sHD /= /aperm. rewrite !(sameP setIidPl eqP) !eqEsubset !subsetIl; apply: eq_subset_r => x. rewrite !inE; apply: andb_id2l => Hx; have Gx: x \in G := subsetP sHG x Hx. have Dax: a x \in D by rewrite (subsetP sGD) // Aut_closed. by rewrite Aut_isomE // -!sub1s...

Lemma

astabs_Aut_isom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Aut_closed", "Aut_isomE", "aperm", "apply", "eqEsubset", "eq_subset_r", "fGisom", "inE", "injmSK", "morphim_set1", "sGD", "sHD", "sHG", "setIidPl", "sub1set", "sub_morphim_pre", "subsetI", "subsetIl", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_restr_perm a : a \in Aut G -> fHisom (inH a) = infH (fGisom a).

Proof. move=> AutGa; case nHa: (a \in 'N(H | 'P)); last first. by rewrite !triv_restr_perm ?astabs_Aut_isom ?nHa ?morph1. apply: (eq_Aut (Aut_Aut_isom injf sHD _)) => [|fx Hfx /=]. by rewrite (Aut_restr_perm (morphimS f sHG)) ?Aut_Aut_isom. have [x Dx Hx def_fx] := morphimP Hfx; have Gx := subsetP sHG x Hx. rewrite...

Lemma

isom_restr_perm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Aut_Aut_isom", "Aut_isomE", "Aut_restr_perm", "Dx", "apply", "astabs_Aut_isom", "eq_Aut", "fGisom", "fHisom", "inH", "infH", "injf", "last", "morph1", "morphimP", "morphimS", "restr_permE", "sHD", "sHG", "subsetP", "triv_restr_perm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restr_perm_isom : isom (inH @* Aut G) (infH @* Aut (f @* G)) fHisom.

Proof. apply: sub_isom; rewrite ?restr_perm_Aut ?injm_Aut_isom //=. rewrite -(im_Aut_isom injf sGD) -!morphim_comp. apply: eq_in_morphim; last exact: isom_restr_perm. (* TODO: investigate why rewrite does not match in the same order *) apply/setP=> a; rewrite in_setI [in RHS]in_setI; apply: andb_id2r => AutGa. (* the m...

Lemma

restr_perm_isom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Aut_restr_perm", "apply", "astabs_Aut_isom", "eq_in_morphim", "fHisom", "im_Aut_isom", "inE", "inH", "in_setI", "infH", "injf", "injm_Aut_isom", "isom", "isom_restr_perm", "last", "morphim_comp", "restr_perm_Aut", "sGD", "sHG", "setP", "sub_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Aut_sub : Aut_in (Aut (f @* G)) (f @* H) \isog Aut_in (Aut G) H.

Proof. do 2!rewrite isog_sym (isog_transl _ (Aut_in_isog _ _)). by rewrite isog_sym (isom_isog _ _ restr_perm_isom) // restr_perm_Aut. Qed.

Lemma

injm_Aut_sub

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Aut_in", "Aut_in_isog", "isog", "isog_sym", "isog_transl", "isom_isog", "restr_perm_Aut", "restr_perm_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Aut_full : (Aut_in (Aut (f @* G)) (f @* H) \isog Aut (f @* H)) = (Aut_in (Aut G) H \isog Aut H).

Proof. by rewrite (isog_transl _ injm_Aut_sub) (isog_transr _ (injm_Aut injf sHD)). Qed.

Lemma

injm_Aut_full

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Aut_in", "injf", "injm_Aut", "injm_Aut_sub", "isog", "isog_transl", "isog_transr", "sHD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actT

:= (action D rT).

Notation

actT

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

is_groupAction (to : actT)

:= {in D, forall a, actperm to a \in Aut R}.

Definition

is_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "actT", "actperm", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupAction

:= GroupAction {gact :> actT; _ : is_groupAction gact}.

Structure

groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actT", "gact", "is_groupAction" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

clone_groupAction to

:= let: GroupAction _ toA := to return {type of GroupAction for to} -> _ in fun k => k toA : groupAction.

Definition

clone_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "groupAction", "to", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'groupAction' 'of' to ]"

:= (clone_groupAction (@GroupAction _ _ _ _ to)) (format "[ 'groupAction' 'of' to ]") : form_scope.

Notation

[ 'groupAction' 'of' to ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "clone_groupAction", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gact_range & groupAction D R

:= R.

Definition

gact_range

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "groupAction" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent to A

:= 'Fix_(R | to)(D :&: A).

Definition

gacent

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_on_group A S to

:= [acts A, on S | to] /\ S \subset R.

Definition

acts_on_group

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "on", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actby_cond_group A S to : acts_on_group A S to -> actby_cond A S to

:= @proj1 _ _.

Coercion

actby_cond_group

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby_cond", "acts_on_group", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_irreducibly A S to

:= [min S of G | G :!=: 1 & [acts A, on G | to]].

Definition

acts_irreducibly

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "min", "on", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( | to ) ( A )"

:= (gacent to A) : group_scope.

Notation

''C_' ( | to ) ( A )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacent", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( G | to ) ( A )"

:= (G :&: 'C_(|to)(A)) : group_scope.

Notation

''C_' ( G | to ) ( A )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( | to ) [ a ]"

:= 'C_(|to)([set a]) : group_scope.

Notation

''C_' ( | to ) [ a ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( G | to ) [ a ]"

:= 'C_(G | to)([set a]) : group_scope.

Notation

''C_' ( G | to ) [ a ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'acts' A , 'on' 'group' G | to }"

:= (acts_on_group A G to) (format "{ 'acts' A , 'on' 'group' G | to }") : type_scope.

Notation

{ 'acts' A , 'on' 'group' G | to }

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_on_group", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actperm_Aut : is_groupAction R to.

Proof. by case: to. Qed.

Lemma

actperm_Aut

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "is_groupAction", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_actperm_Aut : actperm to @* D \subset Aut R.

Proof. by apply/subsetP=> _ /morphimP[a _ Da ->]; apply: actperm_Aut. Qed.

Lemma

im_actperm_Aut

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "Da", "actperm", "actperm_Aut", "apply", "morphimP", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gact_out x a : a \in D -> x \notin R -> to x a = x.

Proof. by move=> Da Rx; rewrite -actpermE (out_Aut _ Rx) ?actperm_Aut. Qed.

Lemma

gact_out

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actpermE", "actperm_Aut", "out_Aut", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gactM : {in D, forall a, {in R &, {morph to^~ a : x y / x * y}}}.

Proof. move=> a Da /= x y; rewrite -!(actpermE to); apply: morphicP x y. by rewrite Aut_morphic ?actperm_Aut. Qed.

Lemma

gactM

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut_morphic", "Da", "actpermE", "actperm_Aut", "apply", "morphicP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actmM a : {in R &, {morph actm to a : x y / x * y}}.

Proof. by rewrite /actm; case: ifP => //; apply: gactM. Qed.

Lemma

actmM

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actm", "apply", "gactM", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

act_morphism a

:= Morphism (actmM a).

Canonical

act_morphism

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actmM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_actm : {in D, forall a (S : {set rT}), S \subset R -> actm to a @* S = to^* S a}.

Proof. by move=> a Da /= S sSR; rewrite /morphim /= actmEfun ?(setIidPr _). Qed.

Lemma

morphim_actm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actm", "actmEfun", "morphim", "sSR", "setIidPr", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentIdom : 'C_(|to)(D :&: A) = 'C_(|to)(A).

Proof. by rewrite /gacent setIA setIid. Qed.

Lemma

gacentIdom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacent", "setIA", "setIid", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentIim : 'C_(R | to)(A) = 'C_(|to)(A).

Proof. by rewrite setIA setIid. Qed.

Lemma

gacentIim

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "setIA", "setIid", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentS : A \subset B -> 'C_(|to)(B) \subset 'C_(|to)(A).

Proof. by move=> sAB; rewrite !(setIS, afixS). Qed.

Lemma

gacentS

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afixS", "setIS", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentU : 'C_(|to)(A :|: B) = 'C_(|to)(A) :&: 'C_(|to)(B).

Proof. by rewrite -setIIr -afixU -setIUr. Qed.

Lemma

gacentU

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afixU", "setIIr", "setIUr", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(Da : a \in D) (sAD : A \subset D) (sSR : S \subset R).

Hypotheses

Da

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "sAD", "sSR" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentE : 'C_(|to)(A) = 'Fix_(R | to)(A).

Proof. by rewrite -{2}(setIidPr sAD). Qed.

Lemma

gacentE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "sAD", "setIidPr", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent1E : 'C_(|to)[a] = 'Fix_(R | to)[a].

Proof. by rewrite /gacent [D :&: _](setIidPr _) ?sub1set. Qed.

Lemma

gacent1E

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacent", "setIidPr", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgacentE : 'C_(S | to)(A) = 'Fix_(S | to)(A).

Proof. by rewrite gacentE setIA (setIidPl sSR). Qed.

Lemma

subgacentE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacentE", "sSR", "setIA", "setIidPl", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d