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
actmE a : a \in D -> actm to a =1 to^~ a.	Proof. by move=> Da; rewrite actmEfun. Qed.	Lemma	actmE	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", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setactE S a : to^* S a = [set to x a \| x in S].	Proof. by []. Qed.	Lemma	setactE	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
mem_setact S a x : x \in S -> to x a \in to^* S a.	Proof. exact: imset_f. Qed.	Lemma	mem_setact	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "imset_f", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_setact S a : #\|to^* S a\| = #\|S\|.	Proof. by apply: card_imset; apply: act_inj. Qed.	Lemma	card_setact	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", "apply", "card_imset", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setact_is_action : is_action D to^*.	Proof. split=> [a R S eqRS \| a b Da Db S]; last first. by rewrite /setact /= -imset_comp; apply: eq_imset => x; apply: actMin. apply/setP=> x; apply/idP/idP=> /(mem_setact a). by rewrite eqRS => /imsetP[y Sy /act_inj->]. by rewrite -eqRS => /imsetP[y Sy /act_inj->]. Qed.	Lemma	setact_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", "actMin", "act_inj", "apply", "eq_imset", "imsetP", "imset_comp", "is_action", "last", "mem_setact", "setP", "setact", "split", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_action	:= Action setact_is_action.	Canonical	set_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" ]	[ "setact_is_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbitE A x : orbit to A x = to x @: A.	Proof. by []. Qed.	Lemma	orbitE	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", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbitP A x y : reflect (exists2 a, a \in A & to x a = y) (y \in orbit to A x).	Proof. by apply: (iffP imsetP) => [] [a]; exists a. Qed.	Lemma	orbitP	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", "imsetP", "orbit", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_orbit A x a : a \in A -> to x a \in orbit to A x.	Proof. exact: imset_f. Qed.	Lemma	mem_orbit	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "imset_f", "orbit", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixP A x : reflect (forall a, a \in A -> to x a = x) (x \in 'Fix_to(A)).	Proof. rewrite inE; apply: (iffP subsetP) => [xfix a /xfix \| xfix a Aa]. by rewrite inE => /eqP. by rewrite inE xfix. Qed.	Lemma	afixP	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", "inE", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixS A B : A \subset B -> 'Fix_to(B) \subset 'Fix_to(A).	Proof. by move=> sAB; apply/subsetP=> u /[!inE]; apply: subset_trans. Qed.	Lemma	afixS	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", "inE", "subsetP", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixU A B : 'Fix_to(A :\|: B) = 'Fix_to(A) :&: 'Fix_to(B).	Proof. by apply/setP=> x; rewrite !inE subUset. Qed.	Lemma	afixU	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", "inE", "setP", "subUset" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix1P a x : reflect (to x a = x) (x \in 'Fix_to[a]).	Proof. by rewrite inE sub1set inE; apply: eqP. Qed.	Lemma	afix1P	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", "inE", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabIdom S : 'C_D(S \| to) = 'C(S \| to).	Proof. by rewrite setIA setIid. Qed.	Lemma	astabIdom	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
astab_dom S : {subset 'C(S \| to) <= D}.	Proof. by move=> a /setIP[]. Qed.	Lemma	astab_dom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setIP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_act S a x : a \in 'C(S \| to) -> x \in S -> to x a = x.	Proof. rewrite 2!inE => /andP[_ cSa] Sx; apply/eqP. by have /[1!inE] := subsetP cSa x Sx. Qed.	Lemma	astab_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" ]	[ "apply", "inE", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabS S1 S2 : S1 \subset S2 -> 'C(S2 \| to) \subset 'C(S1 \| to).	Proof. by move=> sS12; apply/subsetP=> x /[!inE] /andP[->]; apply: subset_trans. Qed.	Lemma	astabS	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "S1", "S2", "apply", "inE", "subsetP", "subset_trans", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsIdom S : 'N_D(S \| to) = 'N(S \| to).	Proof. by rewrite setIA setIid. Qed.	Lemma	astabsIdom	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
astabs_dom S : {subset 'N(S \| to) <= D}.	Proof. by move=> a /setIdP[]. Qed.	Lemma	astabs_dom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setIdP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_act S a x : a \in 'N(S \| to) -> (to x a \in S) = (x \in S).	Proof. rewrite 2!inE subEproper properEcard => /andP[_]. rewrite (card_preimset _ (act_inj _)) ltnn andbF orbF => /eqP{2}->. by rewrite inE. Qed.	Lemma	astabs_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" ]	[ "act_inj", "card_preimset", "inE", "ltnn", "properEcard", "subEproper", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_sub S : 'C(S \| to) \subset 'N(S \| to).	Proof. apply/subsetP=> a cSa; rewrite !inE (astab_dom cSa). by apply/subsetP=> x Sx; rewrite inE (astab_act cSa). Qed.	Lemma	astab_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" ]	[ "apply", "astab_act", "astab_dom", "inE", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsC S : 'N(~: S \| to) = 'N(S \| to).	Proof. apply/setP=> a; apply/idP/idP=> nSa; rewrite !inE (astabs_dom nSa). by rewrite -setCS -preimsetC; apply/subsetP=> x; rewrite inE astabs_act. by rewrite preimsetC setCS; apply/subsetP=> x; rewrite inE astabs_act. Qed.	Lemma	astabsC	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", "astabs_act", "astabs_dom", "inE", "preimsetC", "setCS", "setP", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsI S T : 'N(S \| to) :&: 'N(T \| to) \subset 'N(S :&: T \| to).	Proof. apply/subsetP=> a; rewrite !inE -!andbA preimsetI => /and4P[-> nSa _ nTa] /=. by rewrite setISS. Qed.	Lemma	astabsI	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", "inE", "preimsetI", "setISS", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_setact S a : a \in 'N(S \| to) -> to^* S a = S.	Proof. move=> nSa; apply/eqP; rewrite eqEcard card_setact leqnn andbT. by apply/subsetP=> _ /imsetP[x Sx ->]; rewrite astabs_act. Qed.	Lemma	astabs_setact	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", "astabs_act", "card_setact", "eqEcard", "imsetP", "leqnn", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1_set S : 'C[S \| set_action] = 'N(S \| to).	Proof. apply/setP=> a; apply/idP/idP=> nSa. case/setIdP: nSa => Da; rewrite !inE Da sub1set inE => /eqP defS. by apply/subsetP=> x Sx; rewrite inE -defS mem_setact. by rewrite !inE (astabs_dom nSa) sub1set inE /= astabs_setact. Qed.	Lemma	astab1_set	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_dom", "astabs_setact", "inE", "mem_setact", "setIdP", "setP", "set_action", "sub1set", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_set1 x : 'N([set x] \| to) = 'C[x \| to].	Proof. apply/eqP; rewrite eqEsubset astab_sub andbC setIS //. by apply/subsetP=> a; rewrite ?(inE,sub1set). Qed.	Lemma	astabs_set1	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", "astab_sub", "eqEsubset", "inE", "setIS", "sub1set", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_dom A S : [acts A, on S \| to] -> A \subset D.	Proof. by move=> nSA; rewrite (subset_trans nSA) ?subsetIl. Qed.	Lemma	acts_dom	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", "subsetIl", "subset_trans", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_act A S : [acts A, on S \| to] -> {acts A, on S \| to}.	Proof. by move=> nAS a Aa x; rewrite astabs_act ?(subsetP nAS). Qed.	Lemma	acts_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" ]	[ "astabs_act", "on", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabCin A S : A \subset D -> (A \subset 'C(S \| to)) = (S \subset 'Fix_to(A)).	Proof. move=> sAD; apply/subsetP/subsetP=> [sAC x xS \| sSF a aA]. by apply/afixP=> a aA; apply: astab_act (sAC _ aA) xS. rewrite !inE (subsetP sAD _ aA); apply/subsetP=> x xS. by move/afixP/(_ _ aA): (sSF _ xS) => /[1!inE] ->. Qed.	Lemma	astabCin	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afixP", "apply", "astab_act", "inE", "sAD", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(AactS : [acts A, on S \| to]) (AactT : [acts A, on T \| to]).		Hypotheses	AactS	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
astabU : 'C(S :\|: T \| to) = 'C(S \| to) :&: 'C(T \| to).	Proof. by apply/setP=> a; rewrite !inE subUset; case: (a \in D). Qed.	Lemma	astabU	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", "inE", "setP", "subUset", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsU : 'N(S \| to) :&: 'N(T \| to) \subset 'N(S :\|: T \| to).	Proof. by rewrite -(astabsC S) -(astabsC T) -(astabsC (S :\|: T)) setCU astabsI. Qed.	Lemma	astabsU	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabsC", "astabsI", "setCU", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsD : 'N(S \| to) :&: 'N(T \| to) \subset 'N(S :\: T\| to).	Proof. by rewrite setDE -(astabsC T) astabsI. Qed.	Lemma	astabsD	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabsC", "astabsI", "setDE", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actsI : [acts A, on S :&: T \| to].	Proof. by apply: subset_trans (astabsI S T); rewrite subsetI AactS. Qed.	Lemma	actsI	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "AactS", "apply", "astabsI", "on", "subsetI", "subset_trans", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actsU : [acts A, on S :\|: T \| to].	Proof. by apply: subset_trans astabsU; rewrite subsetI AactS. Qed.	Lemma	actsU	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "AactS", "apply", "astabsU", "on", "subsetI", "subset_trans", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actsD : [acts A, on S :\: T \| to].	Proof. by apply: subset_trans astabsD; rewrite subsetI AactS. Qed.	Lemma	actsD	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "AactS", "apply", "astabsD", "on", "subsetI", "subset_trans", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_in_orbit A S x y : [acts A, on S \| to] -> y \in orbit to A x -> x \in S -> y \in S.	Proof. by move=> nSA/imsetP[a Aa ->{y}] Sx; rewrite (astabs_act _ (subsetP nSA a Aa)). Qed.	Lemma	acts_in_orbit	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabs_act", "imsetP", "on", "orbit", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_faithful A B S : B \subset A -> [faithful A, on S \| to] -> [faithful B, on S \| to].	Proof. by move=> sAB; apply: subset_trans; apply: setSI. Qed.	Lemma	subset_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" ]	[ "apply", "faithful", "on", "setSI", "subset_trans", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
reindex_astabs a F : a \in 'N(S \| to) -> \big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).	Proof. move=> nSa; rewrite (reindex_inj (act_inj a)); apply: eq_bigl => x. exact: astabs_act. Qed.	Lemma	reindex_astabs	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", "apply", "astabs_act", "eq_bigl", "reindex_inj", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
reindex_acts A a F : [acts A, on S \| to] -> a \in A -> \big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).	Proof. by move=> nSA /(subsetP nSA); apply: reindex_astabs. Qed.	Lemma	reindex_acts	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", "on", "reindex_astabs", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to ^*"	:= (set_action to) : action_scope.	Notation	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" ]	[ "set_action", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
act1 x : to x 1 = x.	Proof. by apply: (act_inj to 1); rewrite -actMin ?mulg1. Qed.	Lemma	act1	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", "apply", "mulg1", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actKin : {in D, right_loop inv to}.	Proof. by move=> a Da /= x; rewrite -actMin ?groupV // mulgV act1. Qed.	Lemma	actKin	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", "act1", "actMin", "groupV", "inv", "mulgV", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actKVin : {in D, rev_right_loop inv to}.	Proof. by move=> a Da /= x; rewrite -{2}(invgK a) actKin ?groupV. Qed.	Lemma	actKVin	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", "actKin", "groupV", "inv", "invgK", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setactVin S a : a \in D -> to^* S a^-1 = to^~ a @^-1: S.	Proof. by move=> Da; apply: can2_imset_pre; [apply: actKVin \| apply: actKin]. Qed.	Lemma	setactVin	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", "actKVin", "actKin", "apply", "can2_imset_pre", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actXin x a i : a \in D -> to x (a ^+ i) = iter i (to^~ a) x.	Proof. move=> Da; elim: i => /= [\|i <-]; first by rewrite act1. by rewrite expgSr actMin ?groupX. Qed.	Lemma	actXin	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", "act1", "actMin", "expgSr", "groupX", "iter", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix1 : 'Fix_to(1) = setT.	Proof. by apply/setP=> x; rewrite !inE sub1set inE act1 eqxx. Qed.	Lemma	afix1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act1", "apply", "eqxx", "inE", "setP", "setT", "sub1set" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixD1 G : 'Fix_to(G^#) = 'Fix_to(G).	Proof. by rewrite -{2}(setD1K (group1 G)) afixU afix1 setTI. Qed.	Lemma	afixD1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afix1", "afixU", "group1", "setD1K", "setTI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_refl G x : x \in orbit to G x.	Proof. by rewrite -{1}[x]act1 mem_orbit. Qed.	Lemma	orbit_refl	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act1", "mem_orbit", "orbit", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_rel A	:= (fun x y => x \in orbit to A y).	Notation	orbit_rel	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", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
contra_orbit G x y : x \notin orbit to G y -> x != y.	Proof. by apply: contraNneq => ->; apply: orbit_refl. Qed.	Lemma	contra_orbit	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", "contraNneq", "orbit", "orbit_refl", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_in_sym G : G \subset D -> symmetric (orbit_rel G).	Proof. move=> sGD; apply: symmetric_from_pre => x y /imsetP[a Ga]. by move/(canLR (actKin (subsetP sGD a Ga))) <-; rewrite mem_orbit ?groupV. Qed.	Lemma	orbit_in_sym	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actKin", "apply", "groupV", "imsetP", "mem_orbit", "orbit_rel", "sGD", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_in_trans G : G \subset D -> transitive (orbit_rel G).	Proof. move=> sGD _ _ z /imsetP[a Ga ->] /imsetP[b Gb ->]. by rewrite -actMin ?mem_orbit ?groupM // (subsetP sGD). Qed.	Lemma	orbit_in_trans	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", "groupM", "imsetP", "mem_orbit", "orbit_rel", "sGD", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_in_eqP G x y : G \subset D -> reflect (orbit to G x = orbit to G y) (x \in orbit to G y).	Proof. move=> sGD; apply: (iffP idP) => [yGx\|<-]; last exact: orbit_refl. by apply/setP=> z; apply/idP/idP=> /orbit_in_trans-> //; rewrite orbit_in_sym. Qed.	Lemma	orbit_in_eqP	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", "last", "orbit", "orbit_in_sym", "orbit_in_trans", "orbit_refl", "sGD", "setP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_in_transl G x y z : G \subset D -> y \in orbit to G x -> (y \in orbit to G z) = (x \in orbit to G z).	Proof. by move=> sGD Gxy; rewrite !(orbit_in_sym sGD _ z) (orbit_in_eqP y x sGD Gxy). Qed.	Lemma	orbit_in_transl	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", "orbit_in_eqP", "orbit_in_sym", "sGD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_act_in x a G : G \subset D -> a \in G -> orbit to G (to x a) = orbit to G x.	Proof. by move=> sGD /mem_orbit/orbit_in_eqP->. Qed.	Lemma	orbit_act_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" ]	[ "mem_orbit", "orbit", "orbit_in_eqP", "sGD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_actr_in x a G y : G \subset D -> a \in G -> (to y a \in orbit to G x) = (y \in orbit to G x).	Proof. by move=> sGD /mem_orbit/orbit_in_transl->. Qed.	Lemma	orbit_actr_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" ]	[ "mem_orbit", "orbit", "orbit_in_transl", "sGD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_inv_in A x y : A \subset D -> (y \in orbit to A^-1 x) = (x \in orbit to A y).	Proof. move/subsetP=> sAD; apply/imsetP/imsetP=> [] [a Aa ->]. by exists a^-1; rewrite -?mem_invg ?actKin // -groupV sAD -?mem_invg. by exists a^-1; rewrite ?memV_invg ?actKin // sAD. Qed.	Lemma	orbit_inv_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" ]	[ "actKin", "apply", "groupV", "imsetP", "memV_invg", "mem_invg", "orbit", "sAD", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_lcoset_in A a x : A \subset D -> a \in D -> orbit to (a *: A) x = orbit to A (to x a).	Proof. move/subsetP=> sAD Da; apply/setP=> y; apply/imsetP/imsetP=> [] [b Ab ->{y}]. by exists (a^-1 * b); rewrite -?actMin ?mulKVg // ?sAD -?mem_lcoset. by exists (a * b); rewrite ?mem_mulg ?set11 ?actMin // sAD. Qed.	Lemma	orbit_lcoset_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" ]	[ "Da", "actMin", "apply", "imsetP", "mem_lcoset", "mem_mulg", "mulKVg", "orbit", "sAD", "set11", "setP", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_rcoset_in A a x y : A \subset D -> a \in D -> (to y a \in orbit to (A :* a) x) = (y \in orbit to A x).	Proof. move=> sAD Da; rewrite -orbit_inv_in ?mul_subG ?sub1set // invMg. by rewrite invg_set1 orbit_lcoset_in ?inv_subG ?groupV ?actKin ?orbit_inv_in. Qed.	Lemma	orbit_rcoset_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" ]	[ "Da", "actKin", "groupV", "invMg", "inv_subG", "invg_set1", "mul_subG", "orbit", "orbit_inv_in", "orbit_lcoset_in", "sAD", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_conjsg_in A a x y : A \subset D -> a \in D -> (to y a \in orbit to (A :^ a) (to x a)) = (y \in orbit to A x).	Proof. move=> sAD Da; rewrite conjsgE. by rewrite orbit_lcoset_in ?groupV ?mul_subG ?sub1set ?actKin ?orbit_rcoset_in. Qed.	Lemma	orbit_conjsg_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" ]	[ "Da", "actKin", "conjsgE", "groupV", "mul_subG", "orbit", "orbit_lcoset_in", "orbit_rcoset_in", "sAD", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit1P G x : reflect (orbit to G x = [set x]) (x \in 'Fix_to(G)).	Proof. apply: (iffP afixP) => [xfix \| xfix a Ga]. apply/eqP; rewrite eq_sym eqEsubset sub1set -{1}[x]act1 imset_f //=. by apply/subsetP=> y; case/imsetP=> a Ga ->; rewrite inE xfix. by apply/set1P; rewrite -xfix imset_f. Qed.	Lemma	orbit1P	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act1", "afixP", "apply", "eqEsubset", "eq_sym", "imsetP", "imset_f", "inE", "orbit", "set1P", "sub1set", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_orbit1 G x : #\|orbit to G x\| = 1%N -> orbit to G x = [set x].	Proof. move=> orb1; apply/eqP; rewrite eq_sym eqEcard {}orb1 cards1. by rewrite sub1set orbit_refl. Qed.	Lemma	card_orbit1	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", "cards1", "eqEcard", "eq_sym", "orbit", "orbit_refl", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_partition G S : [acts G, on S \| to] -> partition (orbit to G @: S) S.	Proof. move=> actsGS; have sGD := acts_dom actsGS. have eqiG: {in S & &, equivalence_rel [rel x y \| y \in orbit to G x]}. by move=> x y z * /=; rewrite orbit_refl; split=> // /orbit_in_eqP->. congr (partition _ _): (equivalence_partitionP eqiG). apply: eq_in_imset => x Sx; apply/setP=> y. by rewrite inE /= andb_idl /...	Lemma	orbit_partition	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_dom", "acts_in_orbit", "apply", "eq_in_imset", "equivalence_partitionP", "inE", "on", "orbit", "orbit_in_eqP", "orbit_refl", "partition", "rel", "sGD", "setP", "split", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_transversal A S	:= transversal (orbit to A @: S) S.	Definition	orbit_transversal	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", "to", "transversal" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_transversalP G S (P := orbit to G @: S) (X := orbit_transversal G S) : [acts G, on S \| to] -> [/\ is_transversal X P S, X \subset S, {in X &, forall x y, (y \in orbit to G x) = (x == y)} & forall x, x \in S -> exists2 a, a \in G & to x a \in X].	Proof. move/orbit_partition; rewrite -/P => partP. have [/eqP defS tiP _] := and3P partP. have trXP: is_transversal X P S := transversalP partP. have sXS: X \subset S := transversal_sub trXP. split=> // [x y Xx Xy /= \| x Sx]. have Sx := subsetP sXS x Xx. rewrite -(inj_in_eq (pblock_inj trXP)) // eq_pblock ?defS //....	Lemma	orbit_transversalP	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "def_pblock", "eq_pblock", "imsetP", "imset_f", "inj_in_eq", "is_transversal", "mem_pblock", "on", "orbit", "orbitP", "orbit_partition", "orbit_refl", "orbit_transversal", "pblock", "pblock_inj", "pblock_transversal", "split", "subsetP", "tiP", "to", "transversalP", "transv...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_astab S : group_set 'C(S \| to).	Proof. apply/group_setP; split=> [\|a b cSa cSb]. by rewrite !inE group1; apply/subsetP=> x _; rewrite inE act1. rewrite !inE groupM ?(@astab_dom _ _ _ to S) //; apply/subsetP=> x Sx. by rewrite inE actMin ?(@astab_dom _ _ _ to S) ?(astab_act _ Sx). Qed.	Lemma	group_set_astab	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act1", "actMin", "apply", "astab_act", "astab_dom", "group1", "groupM", "group_set", "group_setP", "inE", "split", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_group S	:= group (group_set_astab S).	Canonical	astab_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" ]	[ "group", "group_set_astab" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_gen_in A : A \subset D -> 'Fix_to(<<A>>) = 'Fix_to(A).	Proof. move=> sAD; apply/eqP; rewrite eqEsubset afixS ?sub_gen //=. by rewrite -astabCin gen_subG ?astabCin. Qed.	Lemma	afix_gen_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" ]	[ "afixS", "apply", "astabCin", "eqEsubset", "gen_subG", "sAD", "sub_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afix_cycle_in a : a \in D -> 'Fix_to(<[a]>) = 'Fix_to[a].	Proof. by move=> Da; rewrite afix_gen_in ?sub1set. Qed.	Lemma	afix_cycle_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" ]	[ "Da", "afix_gen_in", "sub1set" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixYin A B : A \subset D -> B \subset D -> 'Fix_to(A <*> B) = 'Fix_to(A) :&: 'Fix_to(B).	Proof. by move=> sAD sBD; rewrite afix_gen_in ?afixU // subUset sAD. Qed.	Lemma	afixYin	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", "afix_gen_in", "sAD", "subUset" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
afixMin G H : G \subset D -> H \subset D -> 'Fix_to(G * H) = 'Fix_to(G) :&: 'Fix_to(H).	Proof. by move=> sGD sHD; rewrite -afix_gen_in ?mul_subG // genM_join afixYin. Qed.	Lemma	afixMin	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afixYin", "afix_gen_in", "genM_join", "mul_subG", "sGD", "sHD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_astab1_in A x : A \subset D -> (A \subset 'C[x \| to]) = (x \in 'Fix_to(A)).	Proof. by move=> sAD; rewrite astabCin ?sub1set. Qed.	Lemma	sub_astab1_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" ]	[ "astabCin", "sAD", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_astabs S : group_set 'N(S \| to).	Proof. apply/group_setP; split=> [\|a b cSa cSb]. by rewrite !inE group1; apply/subsetP=> x Sx; rewrite inE act1. rewrite !inE groupM ?(@astabs_dom _ _ _ to S) //; apply/subsetP=> x Sx. by rewrite inE actMin ?(@astabs_dom _ _ _ to S) ?astabs_act. Qed.	Lemma	group_set_astabs	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act1", "actMin", "apply", "astabs_act", "astabs_dom", "group1", "groupM", "group_set", "group_setP", "inE", "split", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_group S	:= group (group_set_astabs S).	Canonical	astabs_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" ]	[ "group", "group_set_astabs" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_norm S : 'N(S \| to) \subset 'N('C(S \| to)).	Proof. apply/subsetP=> a nSa; rewrite inE sub_conjg; apply/subsetP=> b cSb. have [Da Db] := (astabs_dom nSa, astab_dom cSb). rewrite mem_conjgV !inE groupJ //; apply/subsetP=> x Sx. rewrite inE !actMin ?groupM ?groupV //. by rewrite (astab_act cSb) ?actKVin ?astabs_act ?groupV. Qed.	Lemma	astab_norm	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", "actKVin", "actMin", "apply", "astab_act", "astab_dom", "astabs_act", "astabs_dom", "groupJ", "groupM", "groupV", "inE", "mem_conjgV", "sub_conjg", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_normal S : 'C(S \| to) <\| 'N(S \| to).	Proof. by rewrite /normal astab_sub astab_norm. Qed.	Lemma	astab_normal	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_norm", "astab_sub", "normal", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_sub_orbit G S x : [acts G, on S \| to] -> (orbit to G x \subset S) = (x \in S).	Proof. move/acts_act=> GactS. apply/subsetP/idP=> [\| Sx y]; first by apply; apply: orbit_refl. by case/orbitP=> a Ga <-{y}; rewrite GactS. Qed.	Lemma	acts_sub_orbit	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_act", "apply", "on", "orbit", "orbitP", "orbit_refl", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_orbit G x : G \subset D -> [acts G, on orbit to G x \| to].	Proof. move/subsetP=> sGD; apply/subsetP=> a Ga; rewrite !inE sGD //. apply/subsetP=> _ /imsetP[b Gb ->]. by rewrite inE -actMin ?sGD // imset_f ?groupM. Qed.	Lemma	acts_orbit	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", "apply", "groupM", "imsetP", "imset_f", "inE", "on", "orbit", "sGD", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_subnorm_fix A : [acts 'N_D(A), on 'Fix_to(D :&: A) \| to].	Proof. apply/subsetP=> a nAa; have [Da _] := setIP nAa; rewrite !inE Da. apply/subsetP=> x Cx /[1!inE]; apply/afixP=> b DAb. have [Db _]:= setIP DAb; rewrite -actMin // conjgCV actMin ?groupJ ?groupV //. by rewrite /= (afixP Cx) // memJ_norm // groupV (subsetP (normsGI _ _) _ nAa). Qed.	Lemma	acts_subnorm_fix	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", "afixP", "apply", "conjgCV", "groupJ", "groupV", "inE", "memJ_norm", "normsGI", "on", "setIP", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atrans_orbit G x : [transitive G, on orbit to G x \| to].	Proof. by apply: imset_f; apply: orbit_refl. Qed.	Lemma	atrans_orbit	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", "imset_f", "on", "orbit", "orbit_refl", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sGD : G \subset D.		Hypothesis	sGD	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
ssGD	:= subsetP sGD.	Let	ssGD	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", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amove_act a : a \in G -> amove to G x (to x a) = 'C_G[x \| to] :* a.	Proof. move=> Ga; apply/setP=> b; have Da := ssGD Ga. rewrite mem_rcoset !(inE, sub1set) !groupMr ?groupV //. by case Gb: (b \in G); rewrite //= actMin ?groupV ?ssGD ?(canF_eq (actKVin Da)). Qed.	Lemma	amove_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" ]	[ "Da", "actKVin", "actMin", "amove", "apply", "canF_eq", "groupMr", "groupV", "inE", "mem_rcoset", "setP", "ssGD", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amove_orbit : amove to G x @: orbit to G x = rcosets 'C_G[x \| to] G.	Proof. apply/setP => Ha; apply/imsetP/rcosetsP=> [[y] \| [a Ga ->]]. by case/imsetP=> b Gb -> ->{Ha y}; exists b => //; rewrite amove_act. by rewrite -amove_act //; exists (to x a); first apply: mem_orbit. Qed.	Lemma	amove_orbit	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "amove", "amove_act", "apply", "imsetP", "mem_orbit", "orbit", "rcosets", "rcosetsP", "setP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amoveK : {in orbit to G x, cancel (amove to G x) (fun Ca => to x (repr Ca))}.	Proof. move=> _ /orbitP[a Ga <-]; rewrite amove_act //= -[G :&: _]/(gval _). case: repr_rcosetP => b; rewrite !(inE, sub1set)=> /and3P[Gb _ xbx]. by rewrite actMin ?ssGD ?(eqP xbx). Qed.	Lemma	amoveK	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", "amove", "amove_act", "inE", "orbit", "orbitP", "repr", "repr_rcosetP", "ssGD", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbit_stabilizer : orbit to G x = [set to x (repr Ca) \| Ca in rcosets 'C_G[x \| to] G].	Proof. rewrite -amove_orbit -imset_comp /=; apply/setP=> z. by apply/idP/imsetP=> [xGz \| [y xGy ->]]; first exists z; rewrite /= ?amoveK. Qed.	Lemma	orbit_stabilizer	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "amoveK", "amove_orbit", "apply", "imsetP", "imset_comp", "orbit", "rcosets", "repr", "setP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
act_reprK : {in rcosets 'C_G[x \| to] G, cancel (to x \o repr) (amove to G x)}.	Proof. move=> _ /rcosetsP[a Ga ->] /=; rewrite amove_act ?rcoset_repr //. rewrite -[G :&: _]/(gval _); case: repr_rcosetP => b /setIP[Gb _]. exact: groupM. Qed.	Lemma	act_reprK	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "amove", "amove_act", "groupM", "rcoset_repr", "rcosets", "rcosetsP", "repr", "repr_rcosetP", "setIP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_orbit_in G x : G \subset D -> #\|orbit to G x\| = #\|G : 'C_G[x \| to]\|.	Proof. move=> sGD; rewrite orbit_stabilizer 1?card_in_imset //. exact: can_in_inj (act_reprK _). Qed.	Lemma	card_orbit_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" ]	[ "act_reprK", "card_in_imset", "orbit", "orbit_stabilizer", "sGD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_orbit_in_stab G x : G \subset D -> (#\|orbit to G x\| * #\|'C_G[x \| to]\|)%N = #\|G\|.	Proof. by move=> sGD; rewrite mulnC card_orbit_in ?Lagrange ?subsetIl. Qed.	Lemma	card_orbit_in_stab	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Lagrange", "card_orbit_in", "mulnC", "orbit", "sGD", "subsetIl", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_sum_card_orbit G S : [acts G, on S \| to] -> \sum_(T in orbit to G @: S) #\|T\| = #\|S\|.	Proof. by move/orbit_partition/card_partition. Qed.	Lemma	acts_sum_card_orbit	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "card_partition", "on", "orbit", "orbit_partition", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_setact_in S a : a \in D -> 'C(to^* S a \| to) = 'C(S \| to) :^ a.	Proof. move=> Da; apply/setP=> b; rewrite mem_conjg !inE -mem_conjg conjGid //. apply: andb_id2l => Db; rewrite sub_imset_pre; apply: eq_subset_r => x. by rewrite !inE !actMin ?groupM ?groupV // invgK (canF_eq (actKVin Da)). Qed.	Lemma	astab_setact_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" ]	[ "Da", "actKVin", "actMin", "apply", "canF_eq", "conjGid", "eq_subset_r", "groupM", "groupV", "inE", "invgK", "mem_conjg", "setP", "sub_imset_pre", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1_act_in x a : a \in D -> 'C[to x a \| to] = 'C[x \| to] :^ a.	Proof. by move=> Da; rewrite -astab_setact_in // /setact imset_set1. Qed.	Lemma	astab1_act_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" ]	[ "Da", "astab_setact_in", "imset_set1", "setact", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_Cauchy G S : [acts G, on S \| to] -> \sum_(a in G) #\|'Fix_(S \| to)[a]\| = (#\|orbit to G @: S\| * #\|G\|)%N.	Proof. move=> GactS; have sGD := acts_dom GactS. transitivity (\sum_(a in G) \sum_(x in 'Fix_(S \| to)[a]) 1%N). by apply: eq_bigr => a _; rewrite -sum1_card. rewrite (exchange_big_dep [in S]) /= => [a x _\|]; first by case/setIP. rewrite (set_partition_big _ (orbit_partition GactS)) -sum_nat_const /=. apply: eq_bigr =...	Theorem	Frobenius_Cauchy	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_act", "acts_dom", "apply", "astab1_act_in", "cardJg", "card_orbit_in_stab", "conjGid", "conjIg", "eq_bigl", "eq_bigr", "exchange_big_dep", "imsetP", "inE", "on", "orbit", "orbit_in_sym", "orbit_partition", "sGD", "setIA", "setIP", "setIidPl", "set_partition_big", "s...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atrans_dvd_index_in G S : G \subset D -> [transitive G, on S \| to] -> #\|S\| %\| #\|G : 'C_G(S \| to)\|.	Proof. move=> sGD /imsetP[x Sx {1}->]; rewrite card_orbit_in //. by rewrite indexgS // setIS // astabS // sub1set. Qed.	Lemma	atrans_dvd_index_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" ]	[ "astabS", "card_orbit_in", "imsetP", "indexgS", "on", "sGD", "setIS", "sub1set", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atrans_dvd_in G S : G \subset D -> [transitive G, on S \| to] -> #\|S\| %\| #\|G\|.	Proof. move=> sGD transG; apply: dvdn_trans (atrans_dvd_index_in sGD transG) _. exact: dvdn_indexg. Qed.	Lemma	atrans_dvd_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" ]	[ "apply", "atrans_dvd_index_in", "dvdn_indexg", "dvdn_trans", "on", "sGD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atransPin G S : G \subset D -> [transitive G, on S \| to] -> forall x, x \in S -> orbit to G x = S.	Proof. by move=> sGD /imsetP[y _ ->] x; apply/orbit_in_eqP. Qed.	Lemma	atransPin	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", "imsetP", "on", "orbit", "orbit_in_eqP", "sGD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atransP2in G S : G \subset D -> [transitive G, on S \| to] -> {in S &, forall x y, exists2 a, a \in G & y = to x a}.	Proof. by move=> sGD transG x y /(atransPin sGD transG) <- /imsetP. Qed.	Lemma	atransP2in	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "atransPin", "imsetP", "on", "sGD", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atrans_acts_in G S : G \subset D -> [transitive G, on S \| to] -> [acts G, on S \| to].	Proof. move=> sGD transG; apply/subsetP=> a Ga; rewrite !inE (subsetP sGD) //. by apply/subsetP=> x /(atransPin sGD transG) <-; rewrite inE imset_f. Qed.	Lemma	atrans_acts_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" ]	[ "apply", "atransPin", "imset_f", "inE", "on", "sGD", "subsetP", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subgroup_transitivePin G H S x : x \in S -> H \subset G -> G \subset D -> [transitive G, on S \| to] -> reflect ('C_G[x \| to] * H = G) [transitive H, on S \| to].	Proof. move=> Sx sHG sGD trG; have sHD := subset_trans sHG sGD. apply: (iffP idP) => [trH \| defG]. rewrite group_modr //; apply/setIidPl/subsetP=> a Ga. have Sxa: to x a \in S by rewrite (acts_act (atrans_acts_in sGD trG)). have [b Hb xab]:= atransP2in sHD trH Sxa Sx. have Da := subsetP sGD a Ga; have Db := sub...	Lemma	subgroup_transitivePin	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", "acts_act", "apply", "astab_act", "astab_dom", "atransP2in", "atransPin", "atrans_acts_in", "defG", "groupM", "groupV", "group_modr", "imset2P", "imsetP", "inE", "last", "mem_mulg", "mulgK", "on", "sGD", "sHD", "sHG", "setIP", "setIidPl", "setP", "...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

actmE a : a \in D -> actm to a =1 to^~ a.

Proof. by move=> Da; rewrite actmEfun. Qed.

Lemma

actmE

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", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setactE S a : to^* S a = [set to x a | x in S].

Proof. by []. Qed.

Lemma

setactE

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

mem_setact S a x : x \in S -> to x a \in to^* S a.

Proof. exact: imset_f. Qed.

Lemma

mem_setact

finite_group

finite_group/action.v

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

[ "imset_f", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_setact S a : #|to^* S a| = #|S|.

Proof. by apply: card_imset; apply: act_inj. Qed.

Lemma

card_setact

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", "apply", "card_imset", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setact_is_action : is_action D to^*.

Proof. split=> [a R S eqRS | a b Da Db S]; last first. by rewrite /setact /= -imset_comp; apply: eq_imset => x; apply: actMin. apply/setP=> x; apply/idP/idP=> /(mem_setact a). by rewrite eqRS => /imsetP[y Sy /act_inj->]. by rewrite -eqRS => /imsetP[y Sy /act_inj->]. Qed.

Lemma

setact_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", "actMin", "act_inj", "apply", "eq_imset", "imsetP", "imset_comp", "is_action", "last", "mem_setact", "setP", "setact", "split", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_action

:= Action setact_is_action.

Canonical

set_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" ]

[ "setact_is_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbitE A x : orbit to A x = to x @: A.

Proof. by []. Qed.

Lemma

orbitE

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", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbitP A x y : reflect (exists2 a, a \in A & to x a = y) (y \in orbit to A x).

Proof. by apply: (iffP imsetP) => [] [a]; exists a. Qed.

Lemma

orbitP

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", "imsetP", "orbit", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_orbit A x a : a \in A -> to x a \in orbit to A x.

Proof. exact: imset_f. Qed.

Lemma

mem_orbit

finite_group

finite_group/action.v

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

[ "imset_f", "orbit", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixP A x : reflect (forall a, a \in A -> to x a = x) (x \in 'Fix_to(A)).

Proof. rewrite inE; apply: (iffP subsetP) => [xfix a /xfix | xfix a Aa]. by rewrite inE => /eqP. by rewrite inE xfix. Qed.

Lemma

afixP

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", "inE", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixS A B : A \subset B -> 'Fix_to(B) \subset 'Fix_to(A).

Proof. by move=> sAB; apply/subsetP=> u /[!inE]; apply: subset_trans. Qed.

Lemma

afixS

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", "inE", "subsetP", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixU A B : 'Fix_to(A :|: B) = 'Fix_to(A) :&: 'Fix_to(B).

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

Lemma

afixU

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", "inE", "setP", "subUset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix1P a x : reflect (to x a = x) (x \in 'Fix_to[a]).

Proof. by rewrite inE sub1set inE; apply: eqP. Qed.

Lemma

afix1P

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", "inE", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabIdom S : 'C_D(S | to) = 'C(S | to).

Proof. by rewrite setIA setIid. Qed.

Lemma

astabIdom

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

astab_dom S : {subset 'C(S | to) <= D}.

Proof. by move=> a /setIP[]. Qed.

Lemma

astab_dom

finite_group

finite_group/action.v

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

[ "setIP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_act S a x : a \in 'C(S | to) -> x \in S -> to x a = x.

Proof. rewrite 2!inE => /andP[_ cSa] Sx; apply/eqP. by have /[1!inE] := subsetP cSa x Sx. Qed.

Lemma

astab_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" ]

[ "apply", "inE", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabS S1 S2 : S1 \subset S2 -> 'C(S2 | to) \subset 'C(S1 | to).

Proof. by move=> sS12; apply/subsetP=> x /[!inE] /andP[->]; apply: subset_trans. Qed.

Lemma

astabS

finite_group

finite_group/action.v

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

[ "S1", "S2", "apply", "inE", "subsetP", "subset_trans", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsIdom S : 'N_D(S | to) = 'N(S | to).

Proof. by rewrite setIA setIid. Qed.

Lemma

astabsIdom

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

astabs_dom S : {subset 'N(S | to) <= D}.

Proof. by move=> a /setIdP[]. Qed.

Lemma

astabs_dom

finite_group

finite_group/action.v

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

[ "setIdP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_act S a x : a \in 'N(S | to) -> (to x a \in S) = (x \in S).

Proof. rewrite 2!inE subEproper properEcard => /andP[_]. rewrite (card_preimset _ (act_inj _)) ltnn andbF orbF => /eqP{2}->. by rewrite inE. Qed.

Lemma

astabs_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" ]

[ "act_inj", "card_preimset", "inE", "ltnn", "properEcard", "subEproper", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_sub S : 'C(S | to) \subset 'N(S | to).

Proof. apply/subsetP=> a cSa; rewrite !inE (astab_dom cSa). by apply/subsetP=> x Sx; rewrite inE (astab_act cSa). Qed.

Lemma

astab_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" ]

[ "apply", "astab_act", "astab_dom", "inE", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsC S : 'N(~: S | to) = 'N(S | to).

Proof. apply/setP=> a; apply/idP/idP=> nSa; rewrite !inE (astabs_dom nSa). by rewrite -setCS -preimsetC; apply/subsetP=> x; rewrite inE astabs_act. by rewrite preimsetC setCS; apply/subsetP=> x; rewrite inE astabs_act. Qed.

Lemma

astabsC

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", "astabs_act", "astabs_dom", "inE", "preimsetC", "setCS", "setP", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsI S T : 'N(S | to) :&: 'N(T | to) \subset 'N(S :&: T | to).

Proof. apply/subsetP=> a; rewrite !inE -!andbA preimsetI => /and4P[-> nSa _ nTa] /=. by rewrite setISS. Qed.

Lemma

astabsI

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", "inE", "preimsetI", "setISS", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_setact S a : a \in 'N(S | to) -> to^* S a = S.

Proof. move=> nSa; apply/eqP; rewrite eqEcard card_setact leqnn andbT. by apply/subsetP=> _ /imsetP[x Sx ->]; rewrite astabs_act. Qed.

Lemma

astabs_setact

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", "astabs_act", "card_setact", "eqEcard", "imsetP", "leqnn", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1_set S : 'C[S | set_action] = 'N(S | to).

Proof. apply/setP=> a; apply/idP/idP=> nSa. case/setIdP: nSa => Da; rewrite !inE Da sub1set inE => /eqP defS. by apply/subsetP=> x Sx; rewrite inE -defS mem_setact. by rewrite !inE (astabs_dom nSa) sub1set inE /= astabs_setact. Qed.

Lemma

astab1_set

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_dom", "astabs_setact", "inE", "mem_setact", "setIdP", "setP", "set_action", "sub1set", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_set1 x : 'N([set x] | to) = 'C[x | to].

Proof. apply/eqP; rewrite eqEsubset astab_sub andbC setIS //. by apply/subsetP=> a; rewrite ?(inE,sub1set). Qed.

Lemma

astabs_set1

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", "astab_sub", "eqEsubset", "inE", "setIS", "sub1set", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_dom A S : [acts A, on S | to] -> A \subset D.

Proof. by move=> nSA; rewrite (subset_trans nSA) ?subsetIl. Qed.

Lemma

acts_dom

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", "subsetIl", "subset_trans", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_act A S : [acts A, on S | to] -> {acts A, on S | to}.

Proof. by move=> nAS a Aa x; rewrite astabs_act ?(subsetP nAS). Qed.

Lemma

acts_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" ]

[ "astabs_act", "on", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabCin A S : A \subset D -> (A \subset 'C(S | to)) = (S \subset 'Fix_to(A)).

Proof. move=> sAD; apply/subsetP/subsetP=> [sAC x xS | sSF a aA]. by apply/afixP=> a aA; apply: astab_act (sAC _ aA) xS. rewrite !inE (subsetP sAD _ aA); apply/subsetP=> x xS. by move/afixP/(_ _ aA): (sSF _ xS) => /[1!inE] ->. Qed.

Lemma

astabCin

finite_group

finite_group/action.v

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

[ "afixP", "apply", "astab_act", "inE", "sAD", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(AactS : [acts A, on S | to]) (AactT : [acts A, on T | to]).

Hypotheses

AactS

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

astabU : 'C(S :|: T | to) = 'C(S | to) :&: 'C(T | to).

Proof. by apply/setP=> a; rewrite !inE subUset; case: (a \in D). Qed.

Lemma

astabU

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", "inE", "setP", "subUset", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsU : 'N(S | to) :&: 'N(T | to) \subset 'N(S :|: T | to).

Proof. by rewrite -(astabsC S) -(astabsC T) -(astabsC (S :|: T)) setCU astabsI. Qed.

Lemma

astabsU

finite_group

finite_group/action.v

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

[ "astabsC", "astabsI", "setCU", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsD : 'N(S | to) :&: 'N(T | to) \subset 'N(S :\: T| to).

Proof. by rewrite setDE -(astabsC T) astabsI. Qed.

Lemma

astabsD

finite_group

finite_group/action.v

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

[ "astabsC", "astabsI", "setDE", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actsI : [acts A, on S :&: T | to].

Proof. by apply: subset_trans (astabsI S T); rewrite subsetI AactS. Qed.

Lemma

actsI

finite_group

finite_group/action.v

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

[ "AactS", "apply", "astabsI", "on", "subsetI", "subset_trans", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actsU : [acts A, on S :|: T | to].

Proof. by apply: subset_trans astabsU; rewrite subsetI AactS. Qed.

Lemma

actsU

finite_group

finite_group/action.v

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

[ "AactS", "apply", "astabsU", "on", "subsetI", "subset_trans", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actsD : [acts A, on S :\: T | to].

Proof. by apply: subset_trans astabsD; rewrite subsetI AactS. Qed.

Lemma

actsD

finite_group

finite_group/action.v

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

[ "AactS", "apply", "astabsD", "on", "subsetI", "subset_trans", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_in_orbit A S x y : [acts A, on S | to] -> y \in orbit to A x -> x \in S -> y \in S.

Proof. by move=> nSA/imsetP[a Aa ->{y}] Sx; rewrite (astabs_act _ (subsetP nSA a Aa)). Qed.

Lemma

acts_in_orbit

finite_group

finite_group/action.v

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

[ "astabs_act", "imsetP", "on", "orbit", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_faithful A B S : B \subset A -> [faithful A, on S | to] -> [faithful B, on S | to].

Proof. by move=> sAB; apply: subset_trans; apply: setSI. Qed.

Lemma

subset_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" ]

[ "apply", "faithful", "on", "setSI", "subset_trans", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

reindex_astabs a F : a \in 'N(S | to) -> \big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).

Proof. move=> nSa; rewrite (reindex_inj (act_inj a)); apply: eq_bigl => x. exact: astabs_act. Qed.

Lemma

reindex_astabs

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", "apply", "astabs_act", "eq_bigl", "reindex_inj", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

reindex_acts A a F : [acts A, on S | to] -> a \in A -> \big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).

Proof. by move=> nSA /(subsetP nSA); apply: reindex_astabs. Qed.

Lemma

reindex_acts

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", "on", "reindex_astabs", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to ^*"

:= (set_action to) : action_scope.

Notation

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" ]

[ "set_action", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

act1 x : to x 1 = x.

Proof. by apply: (act_inj to 1); rewrite -actMin ?mulg1. Qed.

Lemma

act1

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", "apply", "mulg1", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actKin : {in D, right_loop inv to}.

Proof. by move=> a Da /= x; rewrite -actMin ?groupV // mulgV act1. Qed.

Lemma

actKin

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", "act1", "actMin", "groupV", "inv", "mulgV", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actKVin : {in D, rev_right_loop inv to}.

Proof. by move=> a Da /= x; rewrite -{2}(invgK a) actKin ?groupV. Qed.

Lemma

actKVin

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", "actKin", "groupV", "inv", "invgK", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setactVin S a : a \in D -> to^* S a^-1 = to^~ a @^-1: S.

Proof. by move=> Da; apply: can2_imset_pre; [apply: actKVin | apply: actKin]. Qed.

Lemma

setactVin

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", "actKVin", "actKin", "apply", "can2_imset_pre", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actXin x a i : a \in D -> to x (a ^+ i) = iter i (to^~ a) x.

Proof. move=> Da; elim: i => /= [|i <-]; first by rewrite act1. by rewrite expgSr actMin ?groupX. Qed.

Lemma

actXin

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", "act1", "actMin", "expgSr", "groupX", "iter", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix1 : 'Fix_to(1) = setT.

Proof. by apply/setP=> x; rewrite !inE sub1set inE act1 eqxx. Qed.

Lemma

afix1

finite_group

finite_group/action.v

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

[ "act1", "apply", "eqxx", "inE", "setP", "setT", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixD1 G : 'Fix_to(G^#) = 'Fix_to(G).

Proof. by rewrite -{2}(setD1K (group1 G)) afixU afix1 setTI. Qed.

Lemma

afixD1

finite_group

finite_group/action.v

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

[ "afix1", "afixU", "group1", "setD1K", "setTI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_refl G x : x \in orbit to G x.

Proof. by rewrite -{1}[x]act1 mem_orbit. Qed.

Lemma

orbit_refl

finite_group

finite_group/action.v

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

[ "act1", "mem_orbit", "orbit", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_rel A

:= (fun x y => x \in orbit to A y).

Notation

orbit_rel

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", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

contra_orbit G x y : x \notin orbit to G y -> x != y.

Proof. by apply: contraNneq => ->; apply: orbit_refl. Qed.

Lemma

contra_orbit

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", "contraNneq", "orbit", "orbit_refl", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_in_sym G : G \subset D -> symmetric (orbit_rel G).

Proof. move=> sGD; apply: symmetric_from_pre => x y /imsetP[a Ga]. by move/(canLR (actKin (subsetP sGD a Ga))) <-; rewrite mem_orbit ?groupV. Qed.

Lemma

orbit_in_sym

finite_group

finite_group/action.v

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

[ "actKin", "apply", "groupV", "imsetP", "mem_orbit", "orbit_rel", "sGD", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_in_trans G : G \subset D -> transitive (orbit_rel G).

Proof. move=> sGD _ _ z /imsetP[a Ga ->] /imsetP[b Gb ->]. by rewrite -actMin ?mem_orbit ?groupM // (subsetP sGD). Qed.

Lemma

orbit_in_trans

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", "groupM", "imsetP", "mem_orbit", "orbit_rel", "sGD", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_in_eqP G x y : G \subset D -> reflect (orbit to G x = orbit to G y) (x \in orbit to G y).

Proof. move=> sGD; apply: (iffP idP) => [yGx|<-]; last exact: orbit_refl. by apply/setP=> z; apply/idP/idP=> /orbit_in_trans-> //; rewrite orbit_in_sym. Qed.

Lemma

orbit_in_eqP

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", "last", "orbit", "orbit_in_sym", "orbit_in_trans", "orbit_refl", "sGD", "setP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_in_transl G x y z : G \subset D -> y \in orbit to G x -> (y \in orbit to G z) = (x \in orbit to G z).

Proof. by move=> sGD Gxy; rewrite !(orbit_in_sym sGD _ z) (orbit_in_eqP y x sGD Gxy). Qed.

Lemma

orbit_in_transl

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", "orbit_in_eqP", "orbit_in_sym", "sGD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_act_in x a G : G \subset D -> a \in G -> orbit to G (to x a) = orbit to G x.

Proof. by move=> sGD /mem_orbit/orbit_in_eqP->. Qed.

Lemma

orbit_act_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" ]

[ "mem_orbit", "orbit", "orbit_in_eqP", "sGD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_actr_in x a G y : G \subset D -> a \in G -> (to y a \in orbit to G x) = (y \in orbit to G x).

Proof. by move=> sGD /mem_orbit/orbit_in_transl->. Qed.

Lemma

orbit_actr_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" ]

[ "mem_orbit", "orbit", "orbit_in_transl", "sGD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_inv_in A x y : A \subset D -> (y \in orbit to A^-1 x) = (x \in orbit to A y).

Proof. move/subsetP=> sAD; apply/imsetP/imsetP=> [] [a Aa ->]. by exists a^-1; rewrite -?mem_invg ?actKin // -groupV sAD -?mem_invg. by exists a^-1; rewrite ?memV_invg ?actKin // sAD. Qed.

Lemma

orbit_inv_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" ]

[ "actKin", "apply", "groupV", "imsetP", "memV_invg", "mem_invg", "orbit", "sAD", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_lcoset_in A a x : A \subset D -> a \in D -> orbit to (a *: A) x = orbit to A (to x a).

Proof. move/subsetP=> sAD Da; apply/setP=> y; apply/imsetP/imsetP=> [] [b Ab ->{y}]. by exists (a^-1 * b); rewrite -?actMin ?mulKVg // ?sAD -?mem_lcoset. by exists (a * b); rewrite ?mem_mulg ?set11 ?actMin // sAD. Qed.

Lemma

orbit_lcoset_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" ]

[ "Da", "actMin", "apply", "imsetP", "mem_lcoset", "mem_mulg", "mulKVg", "orbit", "sAD", "set11", "setP", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_rcoset_in A a x y : A \subset D -> a \in D -> (to y a \in orbit to (A :* a) x) = (y \in orbit to A x).

Proof. move=> sAD Da; rewrite -orbit_inv_in ?mul_subG ?sub1set // invMg. by rewrite invg_set1 orbit_lcoset_in ?inv_subG ?groupV ?actKin ?orbit_inv_in. Qed.

Lemma

orbit_rcoset_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" ]

[ "Da", "actKin", "groupV", "invMg", "inv_subG", "invg_set1", "mul_subG", "orbit", "orbit_inv_in", "orbit_lcoset_in", "sAD", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_conjsg_in A a x y : A \subset D -> a \in D -> (to y a \in orbit to (A :^ a) (to x a)) = (y \in orbit to A x).

Proof. move=> sAD Da; rewrite conjsgE. by rewrite orbit_lcoset_in ?groupV ?mul_subG ?sub1set ?actKin ?orbit_rcoset_in. Qed.

Lemma

orbit_conjsg_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" ]

[ "Da", "actKin", "conjsgE", "groupV", "mul_subG", "orbit", "orbit_lcoset_in", "orbit_rcoset_in", "sAD", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit1P G x : reflect (orbit to G x = [set x]) (x \in 'Fix_to(G)).

Proof. apply: (iffP afixP) => [xfix | xfix a Ga]. apply/eqP; rewrite eq_sym eqEsubset sub1set -{1}[x]act1 imset_f //=. by apply/subsetP=> y; case/imsetP=> a Ga ->; rewrite inE xfix. by apply/set1P; rewrite -xfix imset_f. Qed.

Lemma

orbit1P

finite_group

finite_group/action.v

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

[ "act1", "afixP", "apply", "eqEsubset", "eq_sym", "imsetP", "imset_f", "inE", "orbit", "set1P", "sub1set", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_orbit1 G x : #|orbit to G x| = 1%N -> orbit to G x = [set x].

Proof. move=> orb1; apply/eqP; rewrite eq_sym eqEcard {}orb1 cards1. by rewrite sub1set orbit_refl. Qed.

Lemma

card_orbit1

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", "cards1", "eqEcard", "eq_sym", "orbit", "orbit_refl", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_partition G S : [acts G, on S | to] -> partition (orbit to G @: S) S.

Proof. move=> actsGS; have sGD := acts_dom actsGS. have eqiG: {in S & &, equivalence_rel [rel x y | y \in orbit to G x]}. by move=> x y z * /=; rewrite orbit_refl; split=> // /orbit_in_eqP->. congr (partition _ _): (equivalence_partitionP eqiG). apply: eq_in_imset => x Sx; apply/setP=> y. by rewrite inE /= andb_idl /...

Lemma

orbit_partition

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_dom", "acts_in_orbit", "apply", "eq_in_imset", "equivalence_partitionP", "inE", "on", "orbit", "orbit_in_eqP", "orbit_refl", "partition", "rel", "sGD", "setP", "split", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_transversal A S

:= transversal (orbit to A @: S) S.

Definition

orbit_transversal

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", "to", "transversal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_transversalP G S (P := orbit to G @: S) (X := orbit_transversal G S) : [acts G, on S | to] -> [/\ is_transversal X P S, X \subset S, {in X &, forall x y, (y \in orbit to G x) = (x == y)} & forall x, x \in S -> exists2 a, a \in G & to x a \in X].

Proof. move/orbit_partition; rewrite -/P => partP. have [/eqP defS tiP _] := and3P partP. have trXP: is_transversal X P S := transversalP partP. have sXS: X \subset S := transversal_sub trXP. split=> // [x y Xx Xy /= | x Sx]. have Sx := subsetP sXS x Xx. rewrite -(inj_in_eq (pblock_inj trXP)) // eq_pblock ?defS //....

Lemma

orbit_transversalP

finite_group

finite_group/action.v

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

[ "def_pblock", "eq_pblock", "imsetP", "imset_f", "inj_in_eq", "is_transversal", "mem_pblock", "on", "orbit", "orbitP", "orbit_partition", "orbit_refl", "orbit_transversal", "pblock", "pblock_inj", "pblock_transversal", "split", "subsetP", "tiP", "to", "transversalP", "transv...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_astab S : group_set 'C(S | to).

Proof. apply/group_setP; split=> [|a b cSa cSb]. by rewrite !inE group1; apply/subsetP=> x _; rewrite inE act1. rewrite !inE groupM ?(@astab_dom _ _ _ to S) //; apply/subsetP=> x Sx. by rewrite inE actMin ?(@astab_dom _ _ _ to S) ?(astab_act _ Sx). Qed.

Lemma

group_set_astab

finite_group

finite_group/action.v

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

[ "act1", "actMin", "apply", "astab_act", "astab_dom", "group1", "groupM", "group_set", "group_setP", "inE", "split", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_group S

:= group (group_set_astab S).

Canonical

astab_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" ]

[ "group", "group_set_astab" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_gen_in A : A \subset D -> 'Fix_to(<<A>>) = 'Fix_to(A).

Proof. move=> sAD; apply/eqP; rewrite eqEsubset afixS ?sub_gen //=. by rewrite -astabCin gen_subG ?astabCin. Qed.

Lemma

afix_gen_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" ]

[ "afixS", "apply", "astabCin", "eqEsubset", "gen_subG", "sAD", "sub_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afix_cycle_in a : a \in D -> 'Fix_to(<[a]>) = 'Fix_to[a].

Proof. by move=> Da; rewrite afix_gen_in ?sub1set. Qed.

Lemma

afix_cycle_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" ]

[ "Da", "afix_gen_in", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixYin A B : A \subset D -> B \subset D -> 'Fix_to(A <*> B) = 'Fix_to(A) :&: 'Fix_to(B).

Proof. by move=> sAD sBD; rewrite afix_gen_in ?afixU // subUset sAD. Qed.

Lemma

afixYin

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", "afix_gen_in", "sAD", "subUset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

afixMin G H : G \subset D -> H \subset D -> 'Fix_to(G * H) = 'Fix_to(G) :&: 'Fix_to(H).

Proof. by move=> sGD sHD; rewrite -afix_gen_in ?mul_subG // genM_join afixYin. Qed.

Lemma

afixMin

finite_group

finite_group/action.v

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

[ "afixYin", "afix_gen_in", "genM_join", "mul_subG", "sGD", "sHD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_astab1_in A x : A \subset D -> (A \subset 'C[x | to]) = (x \in 'Fix_to(A)).

Proof. by move=> sAD; rewrite astabCin ?sub1set. Qed.

Lemma

sub_astab1_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" ]

[ "astabCin", "sAD", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_astabs S : group_set 'N(S | to).

Proof. apply/group_setP; split=> [|a b cSa cSb]. by rewrite !inE group1; apply/subsetP=> x Sx; rewrite inE act1. rewrite !inE groupM ?(@astabs_dom _ _ _ to S) //; apply/subsetP=> x Sx. by rewrite inE actMin ?(@astabs_dom _ _ _ to S) ?astabs_act. Qed.

Lemma

group_set_astabs

finite_group

finite_group/action.v

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

[ "act1", "actMin", "apply", "astabs_act", "astabs_dom", "group1", "groupM", "group_set", "group_setP", "inE", "split", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_group S

:= group (group_set_astabs S).

Canonical

astabs_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" ]

[ "group", "group_set_astabs" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_norm S : 'N(S | to) \subset 'N('C(S | to)).

Proof. apply/subsetP=> a nSa; rewrite inE sub_conjg; apply/subsetP=> b cSb. have [Da Db] := (astabs_dom nSa, astab_dom cSb). rewrite mem_conjgV !inE groupJ //; apply/subsetP=> x Sx. rewrite inE !actMin ?groupM ?groupV //. by rewrite (astab_act cSb) ?actKVin ?astabs_act ?groupV. Qed.

Lemma

astab_norm

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", "actKVin", "actMin", "apply", "astab_act", "astab_dom", "astabs_act", "astabs_dom", "groupJ", "groupM", "groupV", "inE", "mem_conjgV", "sub_conjg", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_normal S : 'C(S | to) <| 'N(S | to).

Proof. by rewrite /normal astab_sub astab_norm. Qed.

Lemma

astab_normal

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_norm", "astab_sub", "normal", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_sub_orbit G S x : [acts G, on S | to] -> (orbit to G x \subset S) = (x \in S).

Proof. move/acts_act=> GactS. apply/subsetP/idP=> [| Sx y]; first by apply; apply: orbit_refl. by case/orbitP=> a Ga <-{y}; rewrite GactS. Qed.

Lemma

acts_sub_orbit

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_act", "apply", "on", "orbit", "orbitP", "orbit_refl", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_orbit G x : G \subset D -> [acts G, on orbit to G x | to].

Proof. move/subsetP=> sGD; apply/subsetP=> a Ga; rewrite !inE sGD //. apply/subsetP=> _ /imsetP[b Gb ->]. by rewrite inE -actMin ?sGD // imset_f ?groupM. Qed.

Lemma

acts_orbit

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", "apply", "groupM", "imsetP", "imset_f", "inE", "on", "orbit", "sGD", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_subnorm_fix A : [acts 'N_D(A), on 'Fix_to(D :&: A) | to].

Proof. apply/subsetP=> a nAa; have [Da _] := setIP nAa; rewrite !inE Da. apply/subsetP=> x Cx /[1!inE]; apply/afixP=> b DAb. have [Db _]:= setIP DAb; rewrite -actMin // conjgCV actMin ?groupJ ?groupV //. by rewrite /= (afixP Cx) // memJ_norm // groupV (subsetP (normsGI _ _) _ nAa). Qed.

Lemma

acts_subnorm_fix

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", "afixP", "apply", "conjgCV", "groupJ", "groupV", "inE", "memJ_norm", "normsGI", "on", "setIP", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atrans_orbit G x : [transitive G, on orbit to G x | to].

Proof. by apply: imset_f; apply: orbit_refl. Qed.

Lemma

atrans_orbit

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", "imset_f", "on", "orbit", "orbit_refl", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sGD : G \subset D.

Hypothesis

sGD

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

ssGD

:= subsetP sGD.

Let

ssGD

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", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amove_act a : a \in G -> amove to G x (to x a) = 'C_G[x | to] :* a.

Proof. move=> Ga; apply/setP=> b; have Da := ssGD Ga. rewrite mem_rcoset !(inE, sub1set) !groupMr ?groupV //. by case Gb: (b \in G); rewrite //= actMin ?groupV ?ssGD ?(canF_eq (actKVin Da)). Qed.

Lemma

amove_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" ]

[ "Da", "actKVin", "actMin", "amove", "apply", "canF_eq", "groupMr", "groupV", "inE", "mem_rcoset", "setP", "ssGD", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amove_orbit : amove to G x @: orbit to G x = rcosets 'C_G[x | to] G.

Proof. apply/setP => Ha; apply/imsetP/rcosetsP=> [[y] | [a Ga ->]]. by case/imsetP=> b Gb -> ->{Ha y}; exists b => //; rewrite amove_act. by rewrite -amove_act //; exists (to x a); first apply: mem_orbit. Qed.

Lemma

amove_orbit

finite_group

finite_group/action.v

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

[ "amove", "amove_act", "apply", "imsetP", "mem_orbit", "orbit", "rcosets", "rcosetsP", "setP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amoveK : {in orbit to G x, cancel (amove to G x) (fun Ca => to x (repr Ca))}.

Proof. move=> _ /orbitP[a Ga <-]; rewrite amove_act //= -[G :&: _]/(gval _). case: repr_rcosetP => b; rewrite !(inE, sub1set)=> /and3P[Gb _ xbx]. by rewrite actMin ?ssGD ?(eqP xbx). Qed.

Lemma

amoveK

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", "amove", "amove_act", "inE", "orbit", "orbitP", "repr", "repr_rcosetP", "ssGD", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbit_stabilizer : orbit to G x = [set to x (repr Ca) | Ca in rcosets 'C_G[x | to] G].

Proof. rewrite -amove_orbit -imset_comp /=; apply/setP=> z. by apply/idP/imsetP=> [xGz | [y xGy ->]]; first exists z; rewrite /= ?amoveK. Qed.

Lemma

orbit_stabilizer

finite_group

finite_group/action.v

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

[ "amoveK", "amove_orbit", "apply", "imsetP", "imset_comp", "orbit", "rcosets", "repr", "setP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

act_reprK : {in rcosets 'C_G[x | to] G, cancel (to x \o repr) (amove to G x)}.

Proof. move=> _ /rcosetsP[a Ga ->] /=; rewrite amove_act ?rcoset_repr //. rewrite -[G :&: _]/(gval _); case: repr_rcosetP => b /setIP[Gb _]. exact: groupM. Qed.

Lemma

act_reprK

finite_group

finite_group/action.v

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

[ "amove", "amove_act", "groupM", "rcoset_repr", "rcosets", "rcosetsP", "repr", "repr_rcosetP", "setIP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_orbit_in G x : G \subset D -> #|orbit to G x| = #|G : 'C_G[x | to]|.

Proof. move=> sGD; rewrite orbit_stabilizer 1?card_in_imset //. exact: can_in_inj (act_reprK _). Qed.

Lemma

card_orbit_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" ]

[ "act_reprK", "card_in_imset", "orbit", "orbit_stabilizer", "sGD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_orbit_in_stab G x : G \subset D -> (#|orbit to G x| * #|'C_G[x | to]|)%N = #|G|.

Proof. by move=> sGD; rewrite mulnC card_orbit_in ?Lagrange ?subsetIl. Qed.

Lemma

card_orbit_in_stab

finite_group

finite_group/action.v

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

[ "Lagrange", "card_orbit_in", "mulnC", "orbit", "sGD", "subsetIl", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_sum_card_orbit G S : [acts G, on S | to] -> \sum_(T in orbit to G @: S) #|T| = #|S|.

Proof. by move/orbit_partition/card_partition. Qed.

Lemma

acts_sum_card_orbit

finite_group

finite_group/action.v

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

[ "card_partition", "on", "orbit", "orbit_partition", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_setact_in S a : a \in D -> 'C(to^* S a | to) = 'C(S | to) :^ a.

Proof. move=> Da; apply/setP=> b; rewrite mem_conjg !inE -mem_conjg conjGid //. apply: andb_id2l => Db; rewrite sub_imset_pre; apply: eq_subset_r => x. by rewrite !inE !actMin ?groupM ?groupV // invgK (canF_eq (actKVin Da)). Qed.

Lemma

astab_setact_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" ]

[ "Da", "actKVin", "actMin", "apply", "canF_eq", "conjGid", "eq_subset_r", "groupM", "groupV", "inE", "invgK", "mem_conjg", "setP", "sub_imset_pre", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1_act_in x a : a \in D -> 'C[to x a | to] = 'C[x | to] :^ a.

Proof. by move=> Da; rewrite -astab_setact_in // /setact imset_set1. Qed.

Lemma

astab1_act_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" ]

[ "Da", "astab_setact_in", "imset_set1", "setact", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_Cauchy G S : [acts G, on S | to] -> \sum_(a in G) #|'Fix_(S | to)[a]| = (#|orbit to G @: S| * #|G|)%N.

Proof. move=> GactS; have sGD := acts_dom GactS. transitivity (\sum_(a in G) \sum_(x in 'Fix_(S | to)[a]) 1%N). by apply: eq_bigr => a _; rewrite -sum1_card. rewrite (exchange_big_dep [in S]) /= => [a x _|]; first by case/setIP. rewrite (set_partition_big _ (orbit_partition GactS)) -sum_nat_const /=. apply: eq_bigr =...

Theorem

Frobenius_Cauchy

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_act", "acts_dom", "apply", "astab1_act_in", "cardJg", "card_orbit_in_stab", "conjGid", "conjIg", "eq_bigl", "eq_bigr", "exchange_big_dep", "imsetP", "inE", "on", "orbit", "orbit_in_sym", "orbit_partition", "sGD", "setIA", "setIP", "setIidPl", "set_partition_big", "s...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atrans_dvd_index_in G S : G \subset D -> [transitive G, on S | to] -> #|S| %| #|G : 'C_G(S | to)|.

Proof. move=> sGD /imsetP[x Sx {1}->]; rewrite card_orbit_in //. by rewrite indexgS // setIS // astabS // sub1set. Qed.

Lemma

atrans_dvd_index_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" ]

[ "astabS", "card_orbit_in", "imsetP", "indexgS", "on", "sGD", "setIS", "sub1set", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atrans_dvd_in G S : G \subset D -> [transitive G, on S | to] -> #|S| %| #|G|.

Proof. move=> sGD transG; apply: dvdn_trans (atrans_dvd_index_in sGD transG) _. exact: dvdn_indexg. Qed.

Lemma

atrans_dvd_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" ]

[ "apply", "atrans_dvd_index_in", "dvdn_indexg", "dvdn_trans", "on", "sGD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atransPin G S : G \subset D -> [transitive G, on S | to] -> forall x, x \in S -> orbit to G x = S.

Proof. by move=> sGD /imsetP[y _ ->] x; apply/orbit_in_eqP. Qed.

Lemma

atransPin

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", "imsetP", "on", "orbit", "orbit_in_eqP", "sGD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atransP2in G S : G \subset D -> [transitive G, on S | to] -> {in S &, forall x y, exists2 a, a \in G & y = to x a}.

Proof. by move=> sGD transG x y /(atransPin sGD transG) <- /imsetP. Qed.

Lemma

atransP2in

finite_group

finite_group/action.v

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

[ "atransPin", "imsetP", "on", "sGD", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atrans_acts_in G S : G \subset D -> [transitive G, on S | to] -> [acts G, on S | to].

Proof. move=> sGD transG; apply/subsetP=> a Ga; rewrite !inE (subsetP sGD) //. by apply/subsetP=> x /(atransPin sGD transG) <-; rewrite inE imset_f. Qed.

Lemma

atrans_acts_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" ]

[ "apply", "atransPin", "imset_f", "inE", "on", "sGD", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgroup_transitivePin G H S x : x \in S -> H \subset G -> G \subset D -> [transitive G, on S | to] -> reflect ('C_G[x | to] * H = G) [transitive H, on S | to].

Proof. move=> Sx sHG sGD trG; have sHD := subset_trans sHG sGD. apply: (iffP idP) => [trH | defG]. rewrite group_modr //; apply/setIidPl/subsetP=> a Ga. have Sxa: to x a \in S by rewrite (acts_act (atrans_acts_in sGD trG)). have [b Hb xab]:= atransP2in sHD trH Sxa Sx. have Da := subsetP sGD a Ga; have Db := sub...

Lemma

subgroup_transitivePin

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", "acts_act", "apply", "astab_act", "astab_dom", "atransP2in", "atransPin", "atrans_acts_in", "defG", "groupM", "groupV", "group_modr", "imset2P", "imsetP", "inE", "last", "mem_mulg", "mulgK", "on", "sGD", "sHD", "sHG", "setIP", "setIidPl", "setP", "...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d