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
pquotient_pgroup : G \subset 'N(K) -> pi.-group (G / K) = pi.-group G.	Proof. by move=> nKG; rewrite pmorphim_pgroup ?ker_coset. Qed.	Lemma	pquotient_pgroup	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "group", "ker_coset", "nKG", "pi", "pmorphim_pgroup" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pquotient_pHall : K <\| G -> K <\| H -> pi.-Hall(G / K) (H / K) = pi.-Hall(G) H.	Proof. case/andP=> sKG nKG; case/andP=> sKH nKH. by rewrite pmorphim_pHall // ker_coset /psubgroup subsetI sKH sKG. Qed.	Lemma	pquotient_pHall	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "ker_coset", "nKG", "nKH", "pi", "pmorphim_pHall", "psubgroup", "sKG", "subsetI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltn_log_quotient : p.-group G -> H :!=: 1 -> H \subset G -> logn p #\|G / H\| < logn p #\|G\|.	Proof. move=> pG ntH sHG; apply: contraLR (ltn_quotient ntH sHG); rewrite -!leqNgt. rewrite {2}(card_pgroup pG) {2}(card_pgroup (morphim_pgroup _ pG)). by case: (posnP p) => [-> //\|]; apply: leq_pexp2l. Qed.	Lemma	ltn_log_quotient	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "card_pgroup", "group", "leqNgt", "leq_pexp2l", "logn", "ltn_quotient", "morphim_pgroup", "pG", "posnP", "sHG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nCG : G \subset 'N(C).		Hypothesis	nCG	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn_quotient_cent_cyclic_pgroup : p.-group C -> cyclic C -> logn p #\|G / 'C_G(C)\| <= (logn p #\|C\|).-1.	Proof. move=> pC cycC; have [-> \| ntC] := eqsVneq C 1. by rewrite cent1T setIT trivg_quotient cards1 logn1. have [p_pr _ [e oC]] := pgroup_pdiv pC ntC. rewrite -ker_conj_aut (card_isog (first_isog_loc _ _)) //. apply: leq_trans (dvdn_leq_log _ _ (cardSg (Aut_conj_aut _ _))) _ => //. rewrite card_Aut_cyclic // oC toti...	Lemma	logn_quotient_cent_cyclic_pgroup	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Aut_conj_aut", "apply", "cardSg", "card_Aut_cyclic", "card_isog", "cards1", "cent1T", "cyclic", "dvdn_leq_log", "eqsVneq", "first_isog_loc", "group", "gtnNdvd", "ker_conj_aut", "leq_trans", "logn", "logn1", "logn_Gauss", "p_pr", "pfactorK", "pgroup_pdiv", "prime_coprime", ...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p'group_quotient_cent_prime : prime p -> #\|C\| %\| p -> p^'.-group (G / 'C_G(C)).	Proof. move=> p_pr pC; have pgC: p.-group C := pnat_dvd pC (pnat_id p_pr). have [_ dv_p] := primeP p_pr; case/pred2P: {dv_p pC}(dv_p _ pC) => [\|pC]. by move/card1_trivg->; rewrite cent1T setIT trivg_quotient pgroup1. have le_oGC := logn_quotient_cent_cyclic_pgroup pgC. rewrite /pgroup -partn_eq1 ?cardG_gt0 // -dvdn1 ...	Lemma	p'group_quotient_cent_prime	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "card1_trivg", "cardG_gt0", "cent1T", "dvdn1", "group", "leq_trans", "logn1", "logn_quotient_cent_cyclic_pgroup", "p_part", "p_pr", "partn_eq1", "pfactorK", "pfactor_dvdn", "pgroup", "pgroup1", "pnat_dvd", "pnat_id", "pred2P", "prime", "primeP", "prime_cyclic", "setIT", "...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore	:= \bigcap_(G \| [max G \| pi.-subgroup(A) G]) G.	Definition	pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "max", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_group : {group gT}	:= Eval hnf in [group of pcore].	Canonical	pcore_group	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "pcore" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''O_' pi ( G )"	:= (pcore pi G) (pi at level 2, format "''O_' pi ( G )") : group_scope.	Notation	''O_' pi ( G )	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "pcore", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''O_' pi ( G )"	:= (pcore_group pi G) : Group_scope.	Notation	''O_' pi ( G )	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "pcore_group", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_mod pi B	:= coset B @*^-1 'O_pi(A / B).	Definition	pcore_mod	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "coset", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_mod_group pi B : {group gT}	:= Eval hnf in [group of pcore_mod pi B].	Canonical	pcore_mod_group	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "pcore_mod", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries	:= foldr pcore_mod 1 (rev pis).	Definition	pseries	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "foldr", "pcore_mod", "rev" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_group_set : group_set pseries.	Proof. by rewrite /pseries; case: rev => [\|pi1 pi1']; apply: groupP. Qed.	Lemma	pseries_group_set	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "groupP", "group_set", "pseries", "rev" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_group : {group gT}	:= group pseries_group_set.	Canonical	pseries_group	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "pseries_group_set" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ConsPred p	:= (@Cons nat_pred p%N) (only parsing).	Notation	ConsPred	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Cons", "nat_pred" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''O_{' p1 , .. , pn } ( A )"	:= (pseries (ConsPred p1 .. (ConsPred pn [::]) ..) A) (format "''O_{' p1 , .. , pn } ( A )") : group_scope.	Notation	''O_{' p1 , .. , pn } ( A )	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "ConsPred", "pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''O_{' p1 , .. , pn } ( A )"	:= (pseries_group (ConsPred p1 .. (ConsPred pn [::]) ..) A) : Group_scope.	Notation	''O_{' p1 , .. , pn } ( A )	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "ConsPred", "pseries_group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_psubgroup G : pi.-subgroup(G) 'O_pi(G).	Proof. have [M maxM _]: {M \| [max M \| pi.-subgroup(G) M] & 1%G \subset M}. by apply: maxgroup_exists; rewrite /psubgroup sub1G pgroup1. have sOM: 'O_pi(G) \subset M by apply: bigcap_inf. have /andP[piM sMG] := maxgroupp maxM. by rewrite /psubgroup (pgroupS sOM) // (subset_trans sOM). Qed.	Lemma	pcore_psubgroup	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "bigcap_inf", "max", "maxgroup_exists", "maxgroupp", "pgroup1", "pgroupS", "pi", "psubgroup", "sub1G", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_pgroup G : pi.-group 'O_pi(G).	Proof. by case/andP: (pcore_psubgroup G). Qed.	Lemma	pcore_pgroup	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "group", "pcore_psubgroup", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_sub G : 'O_pi(G) \subset G.	Proof. by case/andP: (pcore_psubgroup G). Qed.	Lemma	pcore_sub	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "pcore_psubgroup" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_sub_Hall G H : pi.-Hall(G) H -> 'O_pi(G) \subset H.	Proof. by move/Hall_max=> maxH; apply: bigcap_inf. Qed.	Lemma	pcore_sub_Hall	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "Hall_max", "apply", "bigcap_inf", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_max G H : pi.-group H -> H <\| G -> H \subset 'O_pi(G).	Proof. move=> piH nHG; apply/bigcapsP=> M maxM. exact: normal_sub_max_pgroup piH nHG. Qed.	Lemma	pcore_max	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "bigcapsP", "group", "nHG", "normal_sub_max_pgroup", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_pgroup_id G : pi.-group G -> 'O_pi(G) = G.	Proof. by move=> piG; apply/eqP; rewrite eqEsubset pcore_sub pcore_max. Qed.	Lemma	pcore_pgroup_id	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "eqEsubset", "group", "pcore_max", "pcore_sub", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_normal G : 'O_pi(G) <\| G.	Proof. rewrite /(_ <\| G) pcore_sub; apply/subsetP=> x Gx. rewrite inE; apply/bigcapsP=> M maxM; rewrite sub_conjg. by apply: bigcap_inf; apply: max_pgroupJ; rewrite ?groupV. Qed.	Lemma	pcore_normal	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "bigcap_inf", "bigcapsP", "groupV", "inE", "max_pgroupJ", "pcore_sub", "sub_conjg", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_Hall_pcore H G : pi.-Hall(G) H -> H <\| G -> 'O_pi(G) = H.	Proof. move=> hallH nHG; apply/eqP. rewrite eqEsubset (sub_normal_Hall hallH) ?pcore_sub ?pcore_pgroup //=. by rewrite pcore_max //= (pHall_pgroup hallH). Qed.	Lemma	normal_Hall_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "apply", "eqEsubset", "nHG", "pHall_pgroup", "pcore_max", "pcore_pgroup", "pcore_sub", "pi", "sub_normal_Hall" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_Hall_pcore G H : pi.-Hall(G) 'O_pi(G) -> pi.-Hall(G) H -> H :=: 'O_pi(G).	Proof. move=> hallGpi hallH. exact: uniq_normal_Hall (pcore_normal G) (Hall_max hallH). Qed.	Lemma	eq_Hall_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "Hall_max", "pcore_normal", "pi", "uniq_normal_Hall" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_Hall_pcore G K : pi.-Hall(G) 'O_pi(G) -> K \subset G -> (K \subset 'O_pi(G)) = pi.-group K.	Proof. by move=> hallGpi; apply: sub_normal_Hall (pcore_normal G). Qed.	Lemma	sub_Hall_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "apply", "group", "pcore_normal", "pi", "sub_normal_Hall" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_Hall_pcore G x : pi.-Hall(G) 'O_pi(G) -> x \in G -> (x \in 'O_pi(G)) = pi.-elt x.	Proof. by move=> hallGpi; apply: mem_normal_Hall (pcore_normal G). Qed.	Lemma	mem_Hall_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "apply", "mem_normal_Hall", "pcore_normal", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_Hall_pcoreP H G : pi.-Hall(G) 'O_pi(G) -> reflect ('O_pi(G) ><\| H = G) (pi^'.-Hall(G) H).	Proof. move=> hallGpi; rewrite -(compl_pHall H hallGpi) complgC. exact: sdprod_normal_complP (pcore_normal G). Qed.	Lemma	sdprod_Hall_pcoreP	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "compl_pHall", "complgC", "pcore_normal", "pi", "sdprod_normal_complP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_pcore_HallP H G : pi^'.-Hall(G) H -> reflect ('O_pi(G) ><\| H = G) (pi.-Hall(G) 'O_pi(G)).	Proof. exact: sdprod_normal_p'HallP (pcore_normal G). Qed.	Lemma	sdprod_pcore_HallP	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "pcore_normal", "pi", "sdprod_normal_p'HallP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcoreJ G x : 'O_pi(G :^ x) = 'O_pi(G) :^ x.	Proof. apply/eqP; rewrite eqEsubset -sub_conjgV. rewrite !pcore_max ?pgroupJ ?pcore_pgroup ?normalJ ?pcore_normal //. by rewrite -(normalJ _ _ x) conjsgKV pcore_normal. Qed.	Lemma	pcoreJ	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "conjsgKV", "eqEsubset", "normalJ", "pcore_max", "pcore_normal", "pcore_pgroup", "pgroupJ", "sub_conjgV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pcore pi : GFunctor.pcontinuous (@pcore pi).	Proof. move=> gT rT D G f; apply/bigcapsP=> M /normal_sub_max_pgroup; apply. by rewrite morphim_pgroup ?pcore_pgroup. by apply: morphim_normal; apply: pcore_normal. Qed.	Lemma	morphim_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "bigcapsP", "gT", "morphim_normal", "morphim_pgroup", "normal_sub_max_pgroup", "pcontinuous", "pcore", "pcore_normal", "pcore_pgroup", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcoreS pi gT (G H : {group gT}) : H \subset G -> H :&: 'O_pi(G) \subset 'O_pi(H).	Proof. move=> sHG; rewrite -{2}(setIidPl sHG). by do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_pcore. Qed.	Lemma	pcoreS	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "gT", "group", "morphimIdom", "morphim_idm", "morphim_pcore", "pi", "sHG", "setIidPl", "subsetIl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_igFun pi	:= [igFun by pcore_sub pi & morphim_pcore pi].	Canonical	pcore_igFun	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "morphim_pcore", "pcore_sub", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_gFun pi	:= [gFun by morphim_pcore pi].	Canonical	pcore_gFun	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "morphim_pcore", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_pgFun pi	:= [pgFun by morphim_pcore pi].	Canonical	pcore_pgFun	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "morphim_pcore", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_char pi gT (G : {group gT}) : 'O_pi(G) \char G.	Proof. exact: gFchar. Qed.	Lemma	pcore_char	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "char", "gFchar", "gT", "group", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_mod_sub pi gT (G : {group gT}) : pcore_mod G pi (F _ G) \subset G.	Proof. by rewrite sub_morphpre_im ?gFsub_trans ?morphimS ?gFnorm //= ker_coset gFsub. Qed.	Lemma	pcore_mod_sub	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gFnorm", "gFsub", "gFsub_trans", "gT", "group", "ker_coset", "morphimS", "pcore_mod", "pi", "sub_morphpre_im" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_pcore_mod pi gT (G : {group gT}) (B : {set gT}) : pcore_mod G pi B / B = 'O_pi(G / B).	Proof. exact/morphpreK/gFsub_trans/morphim_sub. Qed.	Lemma	quotient_pcore_mod	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gFsub_trans", "gT", "group", "morphim_sub", "morphpreK", "pcore_mod", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pcore_mod pi gT rT (D G : {group gT}) (f : {morphism D >-> rT}) : f @* pcore_mod G pi (F _ G) \subset pcore_mod (f @* G) pi (F _ (f @* G)).	Proof. have sDF: D :&: G \subset 'dom (coset (F _ G)). by rewrite setIC subIset ?gFnorm. have sDFf: D :&: G \subset 'dom (coset (F _ (f @* G)) \o f). by rewrite -sub_morphim_pre ?subsetIl // morphimIdom gFnorm. pose K := 'ker (restrm sDFf (coset (F _ (f @* G)) \o f)). have sFK: 'ker (restrm sDF (coset (F _ G))) \su...	Lemma	morphim_pcore_mod	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "coset", "dom", "factm", "gFnorm", "gFsub", "gT", "group", "ker", "ker_comp", "ker_coset", "ker_restrm", "morphimIG", "morphimIdom", "morphimS", "morphim_comp", "morphim_factm", "morphim_pcore", "morphim_restrm", "morphism", "morphpreS", "pcore_mod", "pcore_mod_s...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_mod_res pi gT rT (D : {group gT}) (f : {morphism D >-> rT}) : f @* pcore_mod D pi (F _ D) \subset pcore_mod (f @* D) pi (F _ (f @* D)).	Proof. exact: morphim_pcore_mod. Qed.	Lemma	pcore_mod_res	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "morphim_pcore_mod", "morphism", "pcore_mod", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_mod1 pi gT (G : {group gT}) : pcore_mod G pi 1 = 'O_pi(G).	Proof. rewrite /pcore_mod; have inj1 := coset1_injm gT; rewrite -injmF ?norms1 //. by rewrite -(morphim_invmE inj1) morphim_invm ?norms1. Qed.	Lemma	pcore_mod1	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "coset1_injm", "gT", "group", "injmF", "morphim_invm", "morphim_invmE", "norms1", "pcore_mod", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_rcons pi pis gT (A : {set gT}) : pseries (rcons pis pi) A = pcore_mod A pi (pseries pis A).	Proof. by rewrite /pseries rev_rcons. Qed.	Lemma	pseries_rcons	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "pcore_mod", "pi", "pseries", "rcons", "rev_rcons" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_subfun pis : GFunctor.closed (@pseries pis) /\ GFunctor.pcontinuous (@pseries pis).	Proof. elim/last_ind: pis => [\|pis pi [sFpi fFpi]]. by split=> [gT G \| gT rT D G f]; rewrite (sub1G, morphim1). pose fF := [gFun by fFpi : GFunctor.continuous [igFun by sFpi & fFpi]]. pose F := [pgFun by fFpi : GFunctor.hereditary fF]. split=> [gT G \| gT rT D G f]; rewrite !pseries_rcons ?(pcore_mod_sub F) //. exact:...	Lemma	pseries_subfun	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "closed", "continuous", "fF", "gT", "hereditary", "last_ind", "morphim1", "morphim_pcore_mod", "pcontinuous", "pcore_mod_sub", "pi", "pseries", "pseries_rcons", "split", "sub1G" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_sub pis : GFunctor.closed (@pseries pis).	Proof. by case: (pseries_subfun pis). Qed.	Lemma	pseries_sub	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "closed", "pseries", "pseries_subfun" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pseries pis : GFunctor.pcontinuous (@pseries pis).	Proof. by case: (pseries_subfun pis). Qed.	Lemma	morphim_pseries	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "pcontinuous", "pseries", "pseries_subfun" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseriesS pis : GFunctor.hereditary (@pseries pis).	Proof. exact: (morphim_pseries pis). Qed.	Lemma	pseriesS	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "hereditary", "morphim_pseries", "pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_igFun pis	:= [igFun by pseries_sub pis & morphim_pseries pis].	Canonical	pseries_igFun	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "morphim_pseries", "pseries_sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_gFun pis	:= [gFun by morphim_pseries pis].	Canonical	pseries_gFun	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "morphim_pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_pgFun pis	:= [pgFun by morphim_pseries pis].	Canonical	pseries_pgFun	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "morphim_pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_char pis gT (G : {group gT}) : pseries pis G \char G.	Proof. exact: gFchar. Qed.	Lemma	pseries_char	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "char", "gFchar", "gT", "group", "pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_normal pis gT (G : {group gT}) : pseries pis G <\| G.	Proof. exact: gFnormal. Qed.	Lemma	pseries_normal	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gFnormal", "gT", "group", "pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseriesJ pis gT (G : {group gT}) x : pseries pis (G :^ x) = pseries pis G :^ x.	Proof. rewrite -{1}(setIid G) -morphim_conj -(injmF _ (injm_conj G x)) //=. by rewrite morphim_conj (setIidPr (pseries_sub _ _)). Qed.	Lemma	pseriesJ	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "injmF", "injm_conj", "morphim_conj", "pseries", "pseries_sub", "setIid", "setIidPr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries1 pi gT (G : {group gT}) : 'O_{pi}(G) = 'O_pi(G).	Proof. exact: pcore_mod1. Qed.	Lemma	pseries1	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "pcore_mod1", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_pop pi pis gT (G : {group gT}) : 'O_pi(G) = 1 -> pseries (pi :: pis) G = pseries pis G.	Proof. by move=> OG1; rewrite /pseries rev_cons -cats1 foldr_cat /= pcore_mod1 OG1. Qed.	Lemma	pseries_pop	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "cats1", "foldr_cat", "gT", "group", "pcore_mod1", "pi", "pseries", "rev_cons" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_pop2 pi1 pi2 gT (G : {group gT}) : 'O_pi1(G) = 1 -> 'O_{pi1, pi2}(G) = 'O_pi2(G).	Proof. by move/pseries_pop->; apply: pseries1. Qed.	Lemma	pseries_pop2	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "gT", "group", "pseries1", "pseries_pop" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_sub_catl pi1s pi2s gT (G : {group gT}) : pseries pi1s G \subset pseries (pi1s ++ pi2s) G.	Proof. elim/last_ind: pi2s => [\|pi pis IHpi]; rewrite ?cats0 // -rcons_cat. by rewrite pseries_rcons; apply: subset_trans IHpi _; rewrite sub_cosetpre. Qed.	Lemma	pseries_sub_catl	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "cats0", "gT", "group", "last_ind", "pi", "pseries", "pseries_rcons", "rcons_cat", "sub_cosetpre", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_pseries pis pi gT (G : {group gT}) : pseries (rcons pis pi) G / pseries pis G = 'O_pi(G / pseries pis G).	Proof. by rewrite pseries_rcons quotient_pcore_mod. Qed.	Lemma	quotient_pseries	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "pi", "pseries", "pseries_rcons", "quotient_pcore_mod", "rcons" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_norm2 pi1s pi2s gT (G : {group gT}) : pseries pi2s G \subset 'N(pseries pi1s G).	Proof. by rewrite gFsub_trans ?gFnorm. Qed.	Lemma	pseries_norm2	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gFnorm", "gFsub_trans", "gT", "group", "pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_sub_catr pi1s pi2s gT (G : {group gT}) : pseries pi2s G \subset pseries (pi1s ++ pi2s) G.	Proof. elim: pi1s => //= pi1 pi1s /subset_trans; apply. elim/last_ind: {pi1s pi2s}(_ ++ _) => [\|pis pi IHpi]; first exact: sub1G. rewrite -rcons_cons (pseries_rcons _ (pi1 :: pis)). rewrite -sub_morphim_pre ?pseries_norm2 //. apply: pcore_max; last by rewrite morphim_normal ?pseries_normal. have: pi.-group (pseries (rc...	Lemma	pseries_sub_catr	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "card_quotient", "gT", "group", "indexgS", "last", "last_ind", "morphim_normal", "pcore_max", "pcore_pgroup", "pi", "pnat_dvd", "pseries", "pseries_norm2", "pseries_normal", "pseries_rcons", "quotient_pseries", "rcons", "rcons_cons", "sub1G", "sub_morphim_pre", "su...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_pseries2 pi1 pi2 gT (G : {group gT}) : 'O_{pi1, pi2}(G) / 'O_pi1(G) = 'O_pi2(G / 'O_pi1(G)).	Proof. by rewrite -pseries1 -quotient_pseries. Qed.	Lemma	quotient_pseries2	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "pseries1", "quotient_pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_pseries_cat pi1s pi2s gT (G : {group gT}) : pseries (pi1s ++ pi2s) G / pseries pi1s G = pseries pi2s (G / pseries pi1s G).	Proof. elim/last_ind: pi2s => [\|pi2s pi IHpi]; first by rewrite cats0 trivg_quotient. have psN := pseries_normal _ G; set K := pseries _ G. case: (third_isom (pseries_sub_catl pi1s pi2s G) (psN _)) => //= f inj_f im_f. have nH2H: pseries pi2s (G / K) <\| pseries (pi1s ++ rcons pi2s pi) G / K. rewrite -IHpi morphim_nor...	Lemma	quotient_pseries_cat	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "catA", "cats0", "cats1", "gT", "group", "inj_f", "injmF", "injm_invm", "last_ind", "morphimS", "morphim_invm", "morphim_normal", "normal_sub", "pcore", "pi", "pseries", "pseries_norm2", "pseries_normal", "pseries_sub_catl", "quotient_inj", "quotient_pseries", "r...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_catl_id pi1s pi2s gT (G : {group gT}) : pseries pi1s (pseries (pi1s ++ pi2s) G) = pseries pi1s G.	Proof. elim/last_ind: pi1s => [//\|pi1s pi IHpi] in pi2s *. apply: (@quotient_inj _ (pseries_group pi1s G)). - rewrite /= -(IHpi (pi :: pi2s)) cat_rcons /(_ <\| _) pseries_norm2. by rewrite -cats1 pseries_sub_catl. - by rewrite /= /(_ <\| _) pseries_norm2 -cats1 pseries_sub_catl. rewrite /= cat_rcons -(IHpi (pi :: pi2s)...	Lemma	pseries_catl_id	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "cat_rcons", "cats1", "eqEsubset", "gFnormal", "gFnormal_trans", "gT", "group", "last_ind", "morphim_normal", "pcore_max", "pcore_pgroup", "pi", "pseries", "pseries_group", "pseries_norm2", "pseries_sub_catl", "quotient_inj", "quotient_normal", "quotient_pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_char_catl pi1s pi2s gT (G : {group gT}) : pseries pi1s G \char pseries (pi1s ++ pi2s) G.	Proof. by rewrite -(pseries_catl_id pi1s pi2s G) pseries_char. Qed.	Lemma	pseries_char_catl	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "char", "gT", "group", "pseries", "pseries_catl_id", "pseries_char" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_catr_id pi1s pi2s gT (G : {group gT}) : pseries pi2s (pseries (pi1s ++ pi2s) G) = pseries pi2s G.	Proof. elim/last_ind: pi2s => [//\|pi2s pi IHpi] in G *. have Epis: pseries pi2s (pseries (pi1s ++ rcons pi2s pi) G) = pseries pi2s G. by rewrite -cats1 catA -[RHS]IHpi -[LHS]IHpi /= [pseries (_ ++ _) _]pseries_catl_id. apply: (@quotient_inj _ (pseries_group pi2s G)). - by rewrite /= -Epis /(_ <\| _) pseries_norm2 -cat...	Lemma	pseries_catr_id	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "catA", "cats1", "eqEsubset", "gFnormal", "gFnormal_trans", "gT", "group", "last_ind", "morphim_normal", "pcore_max", "pcore_pgroup", "pi", "pseries", "pseries_catl_id", "pseries_group", "pseries_norm2", "pseries_sub_catl", "pseries_sub_catr", "quotient_inj", "quotie...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_char_catr pi1s pi2s gT (G : {group gT}) : pseries pi2s G \char pseries (pi1s ++ pi2s) G.	Proof. by rewrite -(pseries_catr_id pi1s pi2s G) pseries_char. Qed.	Lemma	pseries_char_catr	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "char", "gT", "group", "pseries", "pseries_catr_id", "pseries_char" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_modp pi gT (G H : {group gT}) : H <\| G -> pi.-group H -> pcore_mod G pi H = 'O_pi(G).	Proof. move=> nsHG piH; have nHG := normal_norm nsHG; apply/eqP. rewrite eqEsubset andbC -sub_morphim_pre ?(gFsub_trans, morphim_pcore) //=. rewrite -[G in 'O_pi(G)](quotientGK nsHG) pcore_max //. by rewrite -(pquotient_pgroup piH) ?subsetIl // cosetpreK pcore_pgroup. by rewrite morphpre_normal ?gFnormal ?gFsub_trans...	Lemma	pcore_modp	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "cosetpreK", "eqEsubset", "gFnormal", "gFsub_trans", "gT", "group", "morphim_pcore", "morphim_sub", "morphpre_normal", "nHG", "normal_norm", "nsHG", "pcore_max", "pcore_mod", "pcore_pgroup", "pi", "pquotient_pgroup", "quotientGK", "sub_morphim_pre", "subsetIl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pquotient_pcore pi gT (G H : {group gT}) : H <\| G -> pi.-group H -> 'O_pi(G / H) = 'O_pi(G) / H.	Proof. by move=> nsHG piH; rewrite -quotient_pcore_mod pcore_modp. Qed.	Lemma	pquotient_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gT", "group", "nsHG", "pcore_modp", "pi", "quotient_pcore_mod" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_pcore_quotient pi gT (G : {group gT}) : 'O_pi(G / 'O_pi(G)) = 1.	Proof. by rewrite pquotient_pcore ?gFnormal ?pcore_pgroup ?trivg_quotient. Qed.	Lemma	trivg_pcore_quotient	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gFnormal", "gT", "group", "pcore_pgroup", "pi", "pquotient_pcore", "trivg_quotient" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pseries_rcons_id pis pi gT (G : {group gT}) : pseries (rcons (rcons pis pi) pi) G = pseries (rcons pis pi) G.	Proof. apply/eqP; rewrite -!cats1 eqEsubset pseries_sub_catl andbT -catA. rewrite -(quotientSGK _ (pseries_sub_catl _ _ _)) ?pseries_norm2 //. rewrite !quotient_pseries_cat -quotient_sub1 ?pseries_norm2 //. by rewrite quotient_pseries_cat /= !pseries1 trivg_pcore_quotient. Qed.	Lemma	pseries_rcons_id	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "catA", "cats1", "eqEsubset", "gT", "group", "pi", "pseries", "pseries1", "pseries_norm2", "pseries_sub_catl", "quotientSGK", "quotient_pseries_cat", "quotient_sub1", "rcons", "trivg_pcore_quotient" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_in_pcore pi rho G : {in \pi(G), {subset pi <= rho}} -> 'O_pi(G) \subset 'O_rho(G).	Proof. move=> pi_sub_rho; rewrite pcore_max ?pcore_normal //. apply: sub_in_pnat (pcore_pgroup _ _) => p. by move/(piSg (pcore_sub _ _)); apply: pi_sub_rho. Qed.	Lemma	sub_in_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "pcore_max", "pcore_normal", "pcore_pgroup", "pcore_sub", "pi", "piSg", "sub_in_pnat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_pcore pi rho G : {subset pi <= rho} -> 'O_pi(G) \subset 'O_rho(G).	Proof. by move=> pi_sub_rho; apply: sub_in_pcore (in1W pi_sub_rho). Qed.	Lemma	sub_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "pi", "sub_in_pcore" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_pcore pi rho G : {in \pi(G), pi =i rho} -> 'O_pi(G) = 'O_rho(G).	Proof. move=> eq_pi_rho; apply/eqP; rewrite eqEsubset. by rewrite !sub_in_pcore // => p /eq_pi_rho->. Qed.	Lemma	eq_in_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "eqEsubset", "pi", "sub_in_pcore" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_pcore pi rho G : pi =i rho -> 'O_pi(G) = 'O_rho(G).	Proof. by move=> eq_pi_rho; apply: eq_in_pcore (in1W eq_pi_rho). Qed.	Lemma	eq_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "eq_in_pcore", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcoreNK pi G : 'O_pi^'^'(G) = 'O_pi(G).	Proof. by apply: eq_pcore; apply: negnK. Qed.	Lemma	pcoreNK	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "eq_pcore", "negnK", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_p'core pi rho G : pi =i rho -> 'O_pi^'(G) = 'O_rho^'(G).	Proof. by move/eq_negn; apply: eq_pcore. Qed.	Lemma	eq_p'core	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "eq_negn", "eq_pcore", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_Hall_p'coreP pi H G : pi^'.-Hall(G) 'O_pi^'(G) -> reflect ('O_pi^'(G) ><\| H = G) (pi.-Hall(G) H).	Proof. by rewrite -(pHallNK pi G H); apply: sdprod_Hall_pcoreP. Qed.	Lemma	sdprod_Hall_p'coreP	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "apply", "pHallNK", "pi", "sdprod_Hall_pcoreP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_p'core_HallP pi H G : pi.-Hall(G) H -> reflect ('O_pi^'(G) ><\| H = G) (pi^'.-Hall(G) 'O_pi^'(G)).	Proof. by rewrite -(pHallNK pi G H); apply: sdprod_pcore_HallP. Qed.	Lemma	sdprod_p'core_HallP	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "apply", "pHallNK", "pi", "sdprod_pcore_HallP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcoreI pi rho G : 'O_[predI pi & rho](G) = 'O_pi('O_rho(G)).	Proof. apply/eqP; rewrite eqEsubset !pcore_max //. - by apply: sub_pgroup (pcore_pgroup _ _) => p /andP[]. - apply/andP; split; first by apply: sub_pcore => p /andP[]. by rewrite gFnorm_trans ?normsG ?gFsub. - rewrite /pgroup pnatI -!pgroupE. by rewrite pcore_pgroup (pgroupS (pcore_sub pi _))// pcore_pgroup. - by r...	Lemma	pcoreI	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "eqEsubset", "gFnorm_trans", "gFnormal_trans", "gFsub", "normsG", "pcore_max", "pcore_pgroup", "pcore_sub", "pgroup", "pgroupE", "pgroupS", "pi", "pnatI", "split", "sub_pcore", "sub_pgroup" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigcap_p'core pi G : G :&: \bigcap_(p < #\|G\|.+1 \| (p : nat) \in pi) 'O_p^'(G) = 'O_pi^'(G).	Proof. apply/eqP; rewrite eqEsubset subsetI pcore_sub pcore_max /=. - apply/pgroupP=> q q_pr qGpi'; apply: contraL (eqxx q) => /= pi_q. apply: (pgroupP (pcore_pgroup q^' G)) => //. have qG: q %\| #\|G\| by rewrite (dvdn_trans qGpi') // cardSg ?subsetIl. have ltqG: q < #\|G\|.+1 by rewrite ltnS dvdn_leq. rewrite (dvd...	Lemma	bigcap_p'core	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "bigcap_inf", "bigcapsP", "cardSg", "contraNneq", "dvdn_leq", "dvdn_trans", "eqEsubset", "eqxx", "gFnorm", "ltnS", "nat", "normG", "normal", "normsI", "norms_bigcap", "pcore_max", "pcore_pgroup", "pcore_sub", "pgroupP", "pi", "pi_p", "subIset", "sub_pcore", "...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_pcoreC (rT : finGroupType) pi G (R : {group rT}) : coprime #\|'O_pi(G)\| #\|'O_pi^'(R)\|.	Proof. exact: pnat_coprime (pcore_pgroup _ _) (pcore_pgroup _ _). Qed.	Lemma	coprime_pcoreC	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "coprime", "group", "pcore_pgroup", "pi", "pnat_coprime" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
TI_pcoreC pi G H : 'O_pi(G) :&: 'O_pi^'(H) = 1.	Proof. by rewrite coprime_TIg ?coprime_pcoreC. Qed.	Lemma	TI_pcoreC	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "coprime_TIg", "coprime_pcoreC", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcore_setI_normal pi G H : H <\| G -> 'O_pi(G) :&: H = 'O_pi(H).	Proof. move=> nsHG; apply/eqP; rewrite eqEsubset subsetI pcore_sub setIC. rewrite !pcore_max ?(pgroupS (subsetIr H _)) ?pcore_pgroup ?gFnormal_trans //=. by rewrite norm_normalI ?gFnorm_trans ?normsG ?normal_sub. Qed.	Lemma	pcore_setI_normal	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "apply", "eqEsubset", "gFnorm_trans", "gFnormal_trans", "norm_normalI", "normal_sub", "normsG", "nsHG", "pcore_max", "pcore_pgroup", "pcore_sub", "pgroupS", "pi", "setIC", "subsetI", "subsetIr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_pgroup pi A : A \subset D -> pi.-group (f @* A) = pi.-group A.	Proof. by move=> sAD; rewrite /pgroup card_injm. Qed.	Lemma	injm_pgroup	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "card_injm", "group", "pgroup", "pi", "sAD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_pelt pi x : x \in D -> pi.-elt (f x) = pi.-elt x.	Proof. by move=> Dx; rewrite /p_elt order_injm. Qed.	Lemma	injm_pelt	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Dx", "order_injm", "p_elt", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_pHall pi G H : G \subset D -> H \subset D -> pi.-Hall(f @* G) (f @* H) = pi.-Hall(G) H.	Proof. by move=> sGD sGH; rewrite !pHallE injmSK ?card_injm. Qed.	Lemma	injm_pHall	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall", "card_injm", "injmSK", "pHallE", "pi", "sGD", "sGH" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_pcore pi G : G \subset D -> f @* 'O_pi(G) = 'O_pi(f @* G).	Proof. exact: injmF. Qed.	Lemma	injm_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "injmF", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_pseries pis G : G \subset D -> f @* pseries pis G = pseries pis (f @* G).	Proof. exact: injmF. Qed.	Lemma	injm_pseries	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "injmF", "pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_pgroup pi : G \isog H -> pi.-group G = pi.-group H.	Proof. by move=> isoGH; rewrite /pgroup (card_isog isoGH). Qed.	Lemma	isog_pgroup	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "card_isog", "group", "isoGH", "isog", "pgroup", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_pcore pi : G \isog H -> 'O_pi(G) \isog 'O_pi(H).	Proof. exact: gFisog. Qed.	Lemma	isog_pcore	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gFisog", "isog", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_pseries pis : G \isog H -> pseries pis G \isog pseries pis H.	Proof. exact: gFisog. Qed.	Lemma	isog_pseries	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "gFisog", "isog", "pseries" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
imprimitivity_system Q	:= [&& partition Q S, [acts A, on Q \| to^*] & 1 < #\|Q\| < #\|S\|].	Definition	imprimitivity_system	solvable	solvable/primitive_action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]	[ "on", "partition", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primitive	:= [transitive A, on S \| to] && ~~ [exists Q, imprimitivity_system Q].	Definition	primitive	solvable	solvable/primitive_action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]	[ "imprimitivity_system", "on", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'primitive' A , 'on' S \| to ]"	:= (primitive A S to) (format "[ 'primitive' A , 'on' S \| to ]") : form_scope.	Notation	[ 'primitive' A , 'on' S \| to ]	solvable	solvable/primitive_action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]	[ "primitive", "to" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trans_prim_astab x : x \in S -> [transitive G, on S \| to] -> [primitive G, on S \| to] = maximal_eq 'C_G[x \| to] G.	Proof. move=> Sx trG; rewrite /primitive trG negb_exists. apply/forallP/maximal_eqP=> /= [primG \| [_ maxCx] Q]. split=> [\|H sCH sHG]; first exact: subsetIl. pose X := orbit to H x; pose Q := orbit (to^*)%act G X. have Xx: x \in X by apply: orbit_refl. have defH: 'N_(G)(X \| to) = H. have trH: [transitive H, ...	Lemma	trans_prim_astab	solvable	solvable/primitive_action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]	[ "Lagrange", "a1", "act", "actK", "actKV", "actM", "actsP", "acts_sub_orbit", "apply", "astab1P", "astab1_set", "atransP", "atransP2", "atrans_acts", "bigcupP", "canF_eq", "card_orbit", "card_orbit_stab", "contraNeq", "contraNneq", "defQ", "eqEcard", "eqEsubset", "eqVpro...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_trans_norm (H : {group aT}) : [primitive G, on S \| to] -> H <\| G -> H \subset 'C_G(S \| to) \/ [transitive H, on S \| to].	Proof. move=> primG /andP[sHG nHG]; rewrite subsetI sHG. have [trG _] := andP primG; have [x Sx defS] := imsetP trG. move: primG; rewrite (trans_prim_astab Sx) // => /maximal_eqP[_]. case/(_ ('C_G[x \| to] <*> H)%G) => /= [\|\|cxH\|]; first exact: joing_subl. - by rewrite join_subG subsetIl. - have{} cxH: H \subset 'C_G[x ...	Lemma	prim_trans_norm	solvable	solvable/primitive_action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]	[ "aT", "actCJV", "apply", "astab1P", "astabP", "atransP2", "group", "imsetP", "join_subG", "joing_subl", "joing_subr", "maximal_eqP", "mem_conjg", "nHG", "norm_joinEl", "normsP", "on", "primitive", "sHG", "subIset", "subgroup_transitiveP", "subsetI", "subsetIl", "subsetP...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
n_act (t : n.-tuple sT) a	:= [tuple of map (to^~ a) t].	Definition	n_act	solvable	solvable/primitive_action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]	[ "map", "sT", "to", "tuple" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
n_act_is_action : is_action setT n_act.	Proof. by apply: is_total_action => [t\|t a b]; apply: eq_from_tnth => i; rewrite !tnth_map ?act1 ?actM. Qed.	Fact	n_act_is_action	solvable	solvable/primitive_action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]	[ "act1", "actM", "apply", "eq_from_tnth", "is_action", "is_total_action", "n_act", "setT", "tnth_map" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
n_act_action	:= Action n_act_is_action.	Canonical	n_act_action	solvable	solvable/primitive_action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]	[ "n_act_is_action" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

pquotient_pgroup : G \subset 'N(K) -> pi.-group (G / K) = pi.-group G.

Proof. by move=> nKG; rewrite pmorphim_pgroup ?ker_coset. Qed.

Lemma

pquotient_pgroup

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "group", "ker_coset", "nKG", "pi", "pmorphim_pgroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pquotient_pHall : K <| G -> K <| H -> pi.-Hall(G / K) (H / K) = pi.-Hall(G) H.

Proof. case/andP=> sKG nKG; case/andP=> sKH nKH. by rewrite pmorphim_pHall // ker_coset /psubgroup subsetI sKH sKG. Qed.

Lemma

pquotient_pHall

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "ker_coset", "nKG", "nKH", "pi", "pmorphim_pHall", "psubgroup", "sKG", "subsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltn_log_quotient : p.-group G -> H :!=: 1 -> H \subset G -> logn p #|G / H| < logn p #|G|.

Proof. move=> pG ntH sHG; apply: contraLR (ltn_quotient ntH sHG); rewrite -!leqNgt. rewrite {2}(card_pgroup pG) {2}(card_pgroup (morphim_pgroup _ pG)). by case: (posnP p) => [-> //|]; apply: leq_pexp2l. Qed.

Lemma

ltn_log_quotient

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "card_pgroup", "group", "leqNgt", "leq_pexp2l", "logn", "ltn_quotient", "morphim_pgroup", "pG", "posnP", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nCG : G \subset 'N(C).

Hypothesis

nCG

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn_quotient_cent_cyclic_pgroup : p.-group C -> cyclic C -> logn p #|G / 'C_G(C)| <= (logn p #|C|).-1.

Proof. move=> pC cycC; have [-> | ntC] := eqsVneq C 1. by rewrite cent1T setIT trivg_quotient cards1 logn1. have [p_pr _ [e oC]] := pgroup_pdiv pC ntC. rewrite -ker_conj_aut (card_isog (first_isog_loc _ _)) //. apply: leq_trans (dvdn_leq_log _ _ (cardSg (Aut_conj_aut _ _))) _ => //. rewrite card_Aut_cyclic // oC toti...

Lemma

logn_quotient_cent_cyclic_pgroup

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Aut_conj_aut", "apply", "cardSg", "card_Aut_cyclic", "card_isog", "cards1", "cent1T", "cyclic", "dvdn_leq_log", "eqsVneq", "first_isog_loc", "group", "gtnNdvd", "ker_conj_aut", "leq_trans", "logn", "logn1", "logn_Gauss", "p_pr", "pfactorK", "pgroup_pdiv", "prime_coprime", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p'group_quotient_cent_prime : prime p -> #|C| %| p -> p^'.-group (G / 'C_G(C)).

Proof. move=> p_pr pC; have pgC: p.-group C := pnat_dvd pC (pnat_id p_pr). have [_ dv_p] := primeP p_pr; case/pred2P: {dv_p pC}(dv_p _ pC) => [|pC]. by move/card1_trivg->; rewrite cent1T setIT trivg_quotient pgroup1. have le_oGC := logn_quotient_cent_cyclic_pgroup pgC. rewrite /pgroup -partn_eq1 ?cardG_gt0 // -dvdn1 ...

Lemma

p'group_quotient_cent_prime

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "card1_trivg", "cardG_gt0", "cent1T", "dvdn1", "group", "leq_trans", "logn1", "logn_quotient_cent_cyclic_pgroup", "p_part", "p_pr", "partn_eq1", "pfactorK", "pfactor_dvdn", "pgroup", "pgroup1", "pnat_dvd", "pnat_id", "pred2P", "prime", "primeP", "prime_cyclic", "setIT", "...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore

:= \bigcap_(G | [max G | pi.-subgroup(A) G]) G.

Definition

pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "max", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_group : {group gT}

:= Eval hnf in [group of pcore].

Canonical

pcore_group

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "pcore" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''O_' pi ( G )"

:= (pcore pi G) (pi at level 2, format "''O_' pi ( G )") : group_scope.

Notation

''O_' pi ( G )

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "pcore", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''O_' pi ( G )"

:= (pcore_group pi G) : Group_scope.

Notation

''O_' pi ( G )

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "pcore_group", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_mod pi B

:= coset B @*^-1 'O_pi(A / B).

Definition

pcore_mod

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "coset", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_mod_group pi B : {group gT}

:= Eval hnf in [group of pcore_mod pi B].

Canonical

pcore_mod_group

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "pcore_mod", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries

:= foldr pcore_mod 1 (rev pis).

Definition

pseries

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "foldr", "pcore_mod", "rev" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_group_set : group_set pseries.

Proof. by rewrite /pseries; case: rev => [|pi1 pi1']; apply: groupP. Qed.

Lemma

pseries_group_set

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "groupP", "group_set", "pseries", "rev" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_group : {group gT}

:= group pseries_group_set.

Canonical

pseries_group

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "pseries_group_set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ConsPred p

:= (@Cons nat_pred p%N) (only parsing).

Notation

ConsPred

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Cons", "nat_pred" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''O_{' p1 , .. , pn } ( A )"

:= (pseries (ConsPred p1 .. (ConsPred pn [::]) ..) A) (format "''O_{' p1 , .. , pn } ( A )") : group_scope.

Notation

''O_{' p1 , .. , pn } ( A )

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "ConsPred", "pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''O_{' p1 , .. , pn } ( A )"

:= (pseries_group (ConsPred p1 .. (ConsPred pn [::]) ..) A) : Group_scope.

Notation

''O_{' p1 , .. , pn } ( A )

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "ConsPred", "pseries_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_psubgroup G : pi.-subgroup(G) 'O_pi(G).

Proof. have [M maxM _]: {M | [max M | pi.-subgroup(G) M] & 1%G \subset M}. by apply: maxgroup_exists; rewrite /psubgroup sub1G pgroup1. have sOM: 'O_pi(G) \subset M by apply: bigcap_inf. have /andP[piM sMG] := maxgroupp maxM. by rewrite /psubgroup (pgroupS sOM) // (subset_trans sOM). Qed.

Lemma

pcore_psubgroup

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "bigcap_inf", "max", "maxgroup_exists", "maxgroupp", "pgroup1", "pgroupS", "pi", "psubgroup", "sub1G", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_pgroup G : pi.-group 'O_pi(G).

Proof. by case/andP: (pcore_psubgroup G). Qed.

Lemma

pcore_pgroup

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "group", "pcore_psubgroup", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_sub G : 'O_pi(G) \subset G.

Proof. by case/andP: (pcore_psubgroup G). Qed.

Lemma

pcore_sub

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "pcore_psubgroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_sub_Hall G H : pi.-Hall(G) H -> 'O_pi(G) \subset H.

Proof. by move/Hall_max=> maxH; apply: bigcap_inf. Qed.

Lemma

pcore_sub_Hall

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "Hall_max", "apply", "bigcap_inf", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_max G H : pi.-group H -> H <| G -> H \subset 'O_pi(G).

Proof. move=> piH nHG; apply/bigcapsP=> M maxM. exact: normal_sub_max_pgroup piH nHG. Qed.

Lemma

pcore_max

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "bigcapsP", "group", "nHG", "normal_sub_max_pgroup", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_pgroup_id G : pi.-group G -> 'O_pi(G) = G.

Proof. by move=> piG; apply/eqP; rewrite eqEsubset pcore_sub pcore_max. Qed.

Lemma

pcore_pgroup_id

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "eqEsubset", "group", "pcore_max", "pcore_sub", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_normal G : 'O_pi(G) <| G.

Proof. rewrite /(_ <| G) pcore_sub; apply/subsetP=> x Gx. rewrite inE; apply/bigcapsP=> M maxM; rewrite sub_conjg. by apply: bigcap_inf; apply: max_pgroupJ; rewrite ?groupV. Qed.

Lemma

pcore_normal

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "bigcap_inf", "bigcapsP", "groupV", "inE", "max_pgroupJ", "pcore_sub", "sub_conjg", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_Hall_pcore H G : pi.-Hall(G) H -> H <| G -> 'O_pi(G) = H.

Proof. move=> hallH nHG; apply/eqP. rewrite eqEsubset (sub_normal_Hall hallH) ?pcore_sub ?pcore_pgroup //=. by rewrite pcore_max //= (pHall_pgroup hallH). Qed.

Lemma

normal_Hall_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "apply", "eqEsubset", "nHG", "pHall_pgroup", "pcore_max", "pcore_pgroup", "pcore_sub", "pi", "sub_normal_Hall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_Hall_pcore G H : pi.-Hall(G) 'O_pi(G) -> pi.-Hall(G) H -> H :=: 'O_pi(G).

Proof. move=> hallGpi hallH. exact: uniq_normal_Hall (pcore_normal G) (Hall_max hallH). Qed.

Lemma

eq_Hall_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "Hall_max", "pcore_normal", "pi", "uniq_normal_Hall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_Hall_pcore G K : pi.-Hall(G) 'O_pi(G) -> K \subset G -> (K \subset 'O_pi(G)) = pi.-group K.

Proof. by move=> hallGpi; apply: sub_normal_Hall (pcore_normal G). Qed.

Lemma

sub_Hall_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "apply", "group", "pcore_normal", "pi", "sub_normal_Hall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_Hall_pcore G x : pi.-Hall(G) 'O_pi(G) -> x \in G -> (x \in 'O_pi(G)) = pi.-elt x.

Proof. by move=> hallGpi; apply: mem_normal_Hall (pcore_normal G). Qed.

Lemma

mem_Hall_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "apply", "mem_normal_Hall", "pcore_normal", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_Hall_pcoreP H G : pi.-Hall(G) 'O_pi(G) -> reflect ('O_pi(G) ><| H = G) (pi^'.-Hall(G) H).

Proof. move=> hallGpi; rewrite -(compl_pHall H hallGpi) complgC. exact: sdprod_normal_complP (pcore_normal G). Qed.

Lemma

sdprod_Hall_pcoreP

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "compl_pHall", "complgC", "pcore_normal", "pi", "sdprod_normal_complP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_pcore_HallP H G : pi^'.-Hall(G) H -> reflect ('O_pi(G) ><| H = G) (pi.-Hall(G) 'O_pi(G)).

Proof. exact: sdprod_normal_p'HallP (pcore_normal G). Qed.

Lemma

sdprod_pcore_HallP

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "pcore_normal", "pi", "sdprod_normal_p'HallP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcoreJ G x : 'O_pi(G :^ x) = 'O_pi(G) :^ x.

Proof. apply/eqP; rewrite eqEsubset -sub_conjgV. rewrite !pcore_max ?pgroupJ ?pcore_pgroup ?normalJ ?pcore_normal //. by rewrite -(normalJ _ _ x) conjsgKV pcore_normal. Qed.

Lemma

pcoreJ

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "conjsgKV", "eqEsubset", "normalJ", "pcore_max", "pcore_normal", "pcore_pgroup", "pgroupJ", "sub_conjgV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pcore pi : GFunctor.pcontinuous (@pcore pi).

Proof. move=> gT rT D G f; apply/bigcapsP=> M /normal_sub_max_pgroup; apply. by rewrite morphim_pgroup ?pcore_pgroup. by apply: morphim_normal; apply: pcore_normal. Qed.

Lemma

morphim_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "bigcapsP", "gT", "morphim_normal", "morphim_pgroup", "normal_sub_max_pgroup", "pcontinuous", "pcore", "pcore_normal", "pcore_pgroup", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcoreS pi gT (G H : {group gT}) : H \subset G -> H :&: 'O_pi(G) \subset 'O_pi(H).

Proof. move=> sHG; rewrite -{2}(setIidPl sHG). by do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_pcore. Qed.

Lemma

pcoreS

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "gT", "group", "morphimIdom", "morphim_idm", "morphim_pcore", "pi", "sHG", "setIidPl", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_igFun pi

:= [igFun by pcore_sub pi & morphim_pcore pi].

Canonical

pcore_igFun

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "morphim_pcore", "pcore_sub", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_gFun pi

:= [gFun by morphim_pcore pi].

Canonical

pcore_gFun

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "morphim_pcore", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_pgFun pi

:= [pgFun by morphim_pcore pi].

Canonical

pcore_pgFun

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "morphim_pcore", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_char pi gT (G : {group gT}) : 'O_pi(G) \char G.

Proof. exact: gFchar. Qed.

Lemma

pcore_char

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_mod_sub pi gT (G : {group gT}) : pcore_mod G pi (F _ G) \subset G.

Proof. by rewrite sub_morphpre_im ?gFsub_trans ?morphimS ?gFnorm //= ker_coset gFsub. Qed.

Lemma

pcore_mod_sub

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gFnorm", "gFsub", "gFsub_trans", "gT", "group", "ker_coset", "morphimS", "pcore_mod", "pi", "sub_morphpre_im" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_pcore_mod pi gT (G : {group gT}) (B : {set gT}) : pcore_mod G pi B / B = 'O_pi(G / B).

Proof. exact/morphpreK/gFsub_trans/morphim_sub. Qed.

Lemma

quotient_pcore_mod

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gFsub_trans", "gT", "group", "morphim_sub", "morphpreK", "pcore_mod", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. have sDF: D :&: G \subset 'dom (coset (F _ G)). by rewrite setIC subIset ?gFnorm. have sDFf: D :&: G \subset 'dom (coset (F _ (f @* G)) \o f). by rewrite -sub_morphim_pre ?subsetIl // morphimIdom gFnorm. pose K := 'ker (restrm sDFf (coset (F _ (f @* G)) \o f)). have sFK: 'ker (restrm sDF (coset (F _ G))) \su...

Lemma

morphim_pcore_mod

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "coset", "dom", "factm", "gFnorm", "gFsub", "gT", "group", "ker", "ker_comp", "ker_coset", "ker_restrm", "morphimIG", "morphimIdom", "morphimS", "morphim_comp", "morphim_factm", "morphim_pcore", "morphim_restrm", "morphism", "morphpreS", "pcore_mod", "pcore_mod_s...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_mod_res pi gT rT (D : {group gT}) (f : {morphism D >-> rT}) : f @* pcore_mod D pi (F _ D) \subset pcore_mod (f @* D) pi (F _ (f @* D)).

Proof. exact: morphim_pcore_mod. Qed.

Lemma

pcore_mod_res

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "morphim_pcore_mod", "morphism", "pcore_mod", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_mod1 pi gT (G : {group gT}) : pcore_mod G pi 1 = 'O_pi(G).

Proof. rewrite /pcore_mod; have inj1 := coset1_injm gT; rewrite -injmF ?norms1 //. by rewrite -(morphim_invmE inj1) morphim_invm ?norms1. Qed.

Lemma

pcore_mod1

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "coset1_injm", "gT", "group", "injmF", "morphim_invm", "morphim_invmE", "norms1", "pcore_mod", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_rcons pi pis gT (A : {set gT}) : pseries (rcons pis pi) A = pcore_mod A pi (pseries pis A).

Proof. by rewrite /pseries rev_rcons. Qed.

Lemma

pseries_rcons

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "pcore_mod", "pi", "pseries", "rcons", "rev_rcons" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_subfun pis : GFunctor.closed (@pseries pis) /\ GFunctor.pcontinuous (@pseries pis).

Proof. elim/last_ind: pis => [|pis pi [sFpi fFpi]]. by split=> [gT G | gT rT D G f]; rewrite (sub1G, morphim1). pose fF := [gFun by fFpi : GFunctor.continuous [igFun by sFpi & fFpi]]. pose F := [pgFun by fFpi : GFunctor.hereditary fF]. split=> [gT G | gT rT D G f]; rewrite !pseries_rcons ?(pcore_mod_sub F) //. exact:...

Lemma

pseries_subfun

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "closed", "continuous", "fF", "gT", "hereditary", "last_ind", "morphim1", "morphim_pcore_mod", "pcontinuous", "pcore_mod_sub", "pi", "pseries", "pseries_rcons", "split", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_sub pis : GFunctor.closed (@pseries pis).

Proof. by case: (pseries_subfun pis). Qed.

Lemma

pseries_sub

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "closed", "pseries", "pseries_subfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pseries pis : GFunctor.pcontinuous (@pseries pis).

Proof. by case: (pseries_subfun pis). Qed.

Lemma

morphim_pseries

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "pcontinuous", "pseries", "pseries_subfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseriesS pis : GFunctor.hereditary (@pseries pis).

Proof. exact: (morphim_pseries pis). Qed.

Lemma

pseriesS

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "hereditary", "morphim_pseries", "pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_igFun pis

:= [igFun by pseries_sub pis & morphim_pseries pis].

Canonical

pseries_igFun

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "morphim_pseries", "pseries_sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_gFun pis

:= [gFun by morphim_pseries pis].

Canonical

pseries_gFun

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "morphim_pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_pgFun pis

:= [pgFun by morphim_pseries pis].

Canonical

pseries_pgFun

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "morphim_pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_char pis gT (G : {group gT}) : pseries pis G \char G.

Proof. exact: gFchar. Qed.

Lemma

pseries_char

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_normal pis gT (G : {group gT}) : pseries pis G <| G.

Proof. exact: gFnormal. Qed.

Lemma

pseries_normal

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gFnormal", "gT", "group", "pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseriesJ pis gT (G : {group gT}) x : pseries pis (G :^ x) = pseries pis G :^ x.

Proof. rewrite -{1}(setIid G) -morphim_conj -(injmF _ (injm_conj G x)) //=. by rewrite morphim_conj (setIidPr (pseries_sub _ _)). Qed.

Lemma

pseriesJ

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "injmF", "injm_conj", "morphim_conj", "pseries", "pseries_sub", "setIid", "setIidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries1 pi gT (G : {group gT}) : 'O_{pi}(G) = 'O_pi(G).

Proof. exact: pcore_mod1. Qed.

Lemma

pseries1

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "pcore_mod1", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_pop pi pis gT (G : {group gT}) : 'O_pi(G) = 1 -> pseries (pi :: pis) G = pseries pis G.

Proof. by move=> OG1; rewrite /pseries rev_cons -cats1 foldr_cat /= pcore_mod1 OG1. Qed.

Lemma

pseries_pop

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "cats1", "foldr_cat", "gT", "group", "pcore_mod1", "pi", "pseries", "rev_cons" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_pop2 pi1 pi2 gT (G : {group gT}) : 'O_pi1(G) = 1 -> 'O_{pi1, pi2}(G) = 'O_pi2(G).

Proof. by move/pseries_pop->; apply: pseries1. Qed.

Lemma

pseries_pop2

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "gT", "group", "pseries1", "pseries_pop" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_sub_catl pi1s pi2s gT (G : {group gT}) : pseries pi1s G \subset pseries (pi1s ++ pi2s) G.

Proof. elim/last_ind: pi2s => [|pi pis IHpi]; rewrite ?cats0 // -rcons_cat. by rewrite pseries_rcons; apply: subset_trans IHpi _; rewrite sub_cosetpre. Qed.

Lemma

pseries_sub_catl

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "cats0", "gT", "group", "last_ind", "pi", "pseries", "pseries_rcons", "rcons_cat", "sub_cosetpre", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_pseries pis pi gT (G : {group gT}) : pseries (rcons pis pi) G / pseries pis G = 'O_pi(G / pseries pis G).

Proof. by rewrite pseries_rcons quotient_pcore_mod. Qed.

Lemma

quotient_pseries

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "pi", "pseries", "pseries_rcons", "quotient_pcore_mod", "rcons" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_norm2 pi1s pi2s gT (G : {group gT}) : pseries pi2s G \subset 'N(pseries pi1s G).

Proof. by rewrite gFsub_trans ?gFnorm. Qed.

Lemma

pseries_norm2

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gFnorm", "gFsub_trans", "gT", "group", "pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_sub_catr pi1s pi2s gT (G : {group gT}) : pseries pi2s G \subset pseries (pi1s ++ pi2s) G.

Proof. elim: pi1s => //= pi1 pi1s /subset_trans; apply. elim/last_ind: {pi1s pi2s}(_ ++ _) => [|pis pi IHpi]; first exact: sub1G. rewrite -rcons_cons (pseries_rcons _ (pi1 :: pis)). rewrite -sub_morphim_pre ?pseries_norm2 //. apply: pcore_max; last by rewrite morphim_normal ?pseries_normal. have: pi.-group (pseries (rc...

Lemma

pseries_sub_catr

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "card_quotient", "gT", "group", "indexgS", "last", "last_ind", "morphim_normal", "pcore_max", "pcore_pgroup", "pi", "pnat_dvd", "pseries", "pseries_norm2", "pseries_normal", "pseries_rcons", "quotient_pseries", "rcons", "rcons_cons", "sub1G", "sub_morphim_pre", "su...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_pseries2 pi1 pi2 gT (G : {group gT}) : 'O_{pi1, pi2}(G) / 'O_pi1(G) = 'O_pi2(G / 'O_pi1(G)).

Proof. by rewrite -pseries1 -quotient_pseries. Qed.

Lemma

quotient_pseries2

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "pseries1", "quotient_pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_pseries_cat pi1s pi2s gT (G : {group gT}) : pseries (pi1s ++ pi2s) G / pseries pi1s G = pseries pi2s (G / pseries pi1s G).

Proof. elim/last_ind: pi2s => [|pi2s pi IHpi]; first by rewrite cats0 trivg_quotient. have psN := pseries_normal _ G; set K := pseries _ G. case: (third_isom (pseries_sub_catl pi1s pi2s G) (psN _)) => //= f inj_f im_f. have nH2H: pseries pi2s (G / K) <| pseries (pi1s ++ rcons pi2s pi) G / K. rewrite -IHpi morphim_nor...

Lemma

quotient_pseries_cat

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "catA", "cats0", "cats1", "gT", "group", "inj_f", "injmF", "injm_invm", "last_ind", "morphimS", "morphim_invm", "morphim_normal", "normal_sub", "pcore", "pi", "pseries", "pseries_norm2", "pseries_normal", "pseries_sub_catl", "quotient_inj", "quotient_pseries", "r...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_catl_id pi1s pi2s gT (G : {group gT}) : pseries pi1s (pseries (pi1s ++ pi2s) G) = pseries pi1s G.

Proof. elim/last_ind: pi1s => [//|pi1s pi IHpi] in pi2s *. apply: (@quotient_inj _ (pseries_group pi1s G)). - rewrite /= -(IHpi (pi :: pi2s)) cat_rcons /(_ <| _) pseries_norm2. by rewrite -cats1 pseries_sub_catl. - by rewrite /= /(_ <| _) pseries_norm2 -cats1 pseries_sub_catl. rewrite /= cat_rcons -(IHpi (pi :: pi2s)...

Lemma

pseries_catl_id

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "cat_rcons", "cats1", "eqEsubset", "gFnormal", "gFnormal_trans", "gT", "group", "last_ind", "morphim_normal", "pcore_max", "pcore_pgroup", "pi", "pseries", "pseries_group", "pseries_norm2", "pseries_sub_catl", "quotient_inj", "quotient_normal", "quotient_pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_char_catl pi1s pi2s gT (G : {group gT}) : pseries pi1s G \char pseries (pi1s ++ pi2s) G.

Proof. by rewrite -(pseries_catl_id pi1s pi2s G) pseries_char. Qed.

Lemma

pseries_char_catl

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "char", "gT", "group", "pseries", "pseries_catl_id", "pseries_char" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_catr_id pi1s pi2s gT (G : {group gT}) : pseries pi2s (pseries (pi1s ++ pi2s) G) = pseries pi2s G.

Proof. elim/last_ind: pi2s => [//|pi2s pi IHpi] in G *. have Epis: pseries pi2s (pseries (pi1s ++ rcons pi2s pi) G) = pseries pi2s G. by rewrite -cats1 catA -[RHS]IHpi -[LHS]IHpi /= [pseries (_ ++ _) _]pseries_catl_id. apply: (@quotient_inj _ (pseries_group pi2s G)). - by rewrite /= -Epis /(_ <| _) pseries_norm2 -cat...

Lemma

pseries_catr_id

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "catA", "cats1", "eqEsubset", "gFnormal", "gFnormal_trans", "gT", "group", "last_ind", "morphim_normal", "pcore_max", "pcore_pgroup", "pi", "pseries", "pseries_catl_id", "pseries_group", "pseries_norm2", "pseries_sub_catl", "pseries_sub_catr", "quotient_inj", "quotie...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_char_catr pi1s pi2s gT (G : {group gT}) : pseries pi2s G \char pseries (pi1s ++ pi2s) G.

Proof. by rewrite -(pseries_catr_id pi1s pi2s G) pseries_char. Qed.

Lemma

pseries_char_catr

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "char", "gT", "group", "pseries", "pseries_catr_id", "pseries_char" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_modp pi gT (G H : {group gT}) : H <| G -> pi.-group H -> pcore_mod G pi H = 'O_pi(G).

Proof. move=> nsHG piH; have nHG := normal_norm nsHG; apply/eqP. rewrite eqEsubset andbC -sub_morphim_pre ?(gFsub_trans, morphim_pcore) //=. rewrite -[G in 'O_pi(G)](quotientGK nsHG) pcore_max //. by rewrite -(pquotient_pgroup piH) ?subsetIl // cosetpreK pcore_pgroup. by rewrite morphpre_normal ?gFnormal ?gFsub_trans...

Lemma

pcore_modp

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "cosetpreK", "eqEsubset", "gFnormal", "gFsub_trans", "gT", "group", "morphim_pcore", "morphim_sub", "morphpre_normal", "nHG", "normal_norm", "nsHG", "pcore_max", "pcore_mod", "pcore_pgroup", "pi", "pquotient_pgroup", "quotientGK", "sub_morphim_pre", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pquotient_pcore pi gT (G H : {group gT}) : H <| G -> pi.-group H -> 'O_pi(G / H) = 'O_pi(G) / H.

Proof. by move=> nsHG piH; rewrite -quotient_pcore_mod pcore_modp. Qed.

Lemma

pquotient_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gT", "group", "nsHG", "pcore_modp", "pi", "quotient_pcore_mod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_pcore_quotient pi gT (G : {group gT}) : 'O_pi(G / 'O_pi(G)) = 1.

Proof. by rewrite pquotient_pcore ?gFnormal ?pcore_pgroup ?trivg_quotient. Qed.

Lemma

trivg_pcore_quotient

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gFnormal", "gT", "group", "pcore_pgroup", "pi", "pquotient_pcore", "trivg_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pseries_rcons_id pis pi gT (G : {group gT}) : pseries (rcons (rcons pis pi) pi) G = pseries (rcons pis pi) G.

Proof. apply/eqP; rewrite -!cats1 eqEsubset pseries_sub_catl andbT -catA. rewrite -(quotientSGK _ (pseries_sub_catl _ _ _)) ?pseries_norm2 //. rewrite !quotient_pseries_cat -quotient_sub1 ?pseries_norm2 //. by rewrite quotient_pseries_cat /= !pseries1 trivg_pcore_quotient. Qed.

Lemma

pseries_rcons_id

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "catA", "cats1", "eqEsubset", "gT", "group", "pi", "pseries", "pseries1", "pseries_norm2", "pseries_sub_catl", "quotientSGK", "quotient_pseries_cat", "quotient_sub1", "rcons", "trivg_pcore_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_in_pcore pi rho G : {in \pi(G), {subset pi <= rho}} -> 'O_pi(G) \subset 'O_rho(G).

Proof. move=> pi_sub_rho; rewrite pcore_max ?pcore_normal //. apply: sub_in_pnat (pcore_pgroup _ _) => p. by move/(piSg (pcore_sub _ _)); apply: pi_sub_rho. Qed.

Lemma

sub_in_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "pcore_max", "pcore_normal", "pcore_pgroup", "pcore_sub", "pi", "piSg", "sub_in_pnat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_pcore pi rho G : {subset pi <= rho} -> 'O_pi(G) \subset 'O_rho(G).

Proof. by move=> pi_sub_rho; apply: sub_in_pcore (in1W pi_sub_rho). Qed.

Lemma

sub_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "pi", "sub_in_pcore" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_pcore pi rho G : {in \pi(G), pi =i rho} -> 'O_pi(G) = 'O_rho(G).

Proof. move=> eq_pi_rho; apply/eqP; rewrite eqEsubset. by rewrite !sub_in_pcore // => p /eq_pi_rho->. Qed.

Lemma

eq_in_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "eqEsubset", "pi", "sub_in_pcore" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_pcore pi rho G : pi =i rho -> 'O_pi(G) = 'O_rho(G).

Proof. by move=> eq_pi_rho; apply: eq_in_pcore (in1W eq_pi_rho). Qed.

Lemma

eq_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "eq_in_pcore", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcoreNK pi G : 'O_pi^'^'(G) = 'O_pi(G).

Proof. by apply: eq_pcore; apply: negnK. Qed.

Lemma

pcoreNK

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "eq_pcore", "negnK", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_p'core pi rho G : pi =i rho -> 'O_pi^'(G) = 'O_rho^'(G).

Proof. by move/eq_negn; apply: eq_pcore. Qed.

Lemma

eq_p'core

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "eq_negn", "eq_pcore", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_Hall_p'coreP pi H G : pi^'.-Hall(G) 'O_pi^'(G) -> reflect ('O_pi^'(G) ><| H = G) (pi.-Hall(G) H).

Proof. by rewrite -(pHallNK pi G H); apply: sdprod_Hall_pcoreP. Qed.

Lemma

sdprod_Hall_p'coreP

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "apply", "pHallNK", "pi", "sdprod_Hall_pcoreP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_p'core_HallP pi H G : pi.-Hall(G) H -> reflect ('O_pi^'(G) ><| H = G) (pi^'.-Hall(G) 'O_pi^'(G)).

Proof. by rewrite -(pHallNK pi G H); apply: sdprod_pcore_HallP. Qed.

Lemma

sdprod_p'core_HallP

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "apply", "pHallNK", "pi", "sdprod_pcore_HallP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcoreI pi rho G : 'O_[predI pi & rho](G) = 'O_pi('O_rho(G)).

Proof. apply/eqP; rewrite eqEsubset !pcore_max //. - by apply: sub_pgroup (pcore_pgroup _ _) => p /andP[]. - apply/andP; split; first by apply: sub_pcore => p /andP[]. by rewrite gFnorm_trans ?normsG ?gFsub. - rewrite /pgroup pnatI -!pgroupE. by rewrite pcore_pgroup (pgroupS (pcore_sub pi _))// pcore_pgroup. - by r...

Lemma

pcoreI

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "eqEsubset", "gFnorm_trans", "gFnormal_trans", "gFsub", "normsG", "pcore_max", "pcore_pgroup", "pcore_sub", "pgroup", "pgroupE", "pgroupS", "pi", "pnatI", "split", "sub_pcore", "sub_pgroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigcap_p'core pi G : G :&: \bigcap_(p < #|G|.+1 | (p : nat) \in pi) 'O_p^'(G) = 'O_pi^'(G).

Proof. apply/eqP; rewrite eqEsubset subsetI pcore_sub pcore_max /=. - apply/pgroupP=> q q_pr qGpi'; apply: contraL (eqxx q) => /= pi_q. apply: (pgroupP (pcore_pgroup q^' G)) => //. have qG: q %| #|G| by rewrite (dvdn_trans qGpi') // cardSg ?subsetIl. have ltqG: q < #|G|.+1 by rewrite ltnS dvdn_leq. rewrite (dvd...

Lemma

bigcap_p'core

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "bigcap_inf", "bigcapsP", "cardSg", "contraNneq", "dvdn_leq", "dvdn_trans", "eqEsubset", "eqxx", "gFnorm", "ltnS", "nat", "normG", "normal", "normsI", "norms_bigcap", "pcore_max", "pcore_pgroup", "pcore_sub", "pgroupP", "pi", "pi_p", "subIset", "sub_pcore", "...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_pcoreC (rT : finGroupType) pi G (R : {group rT}) : coprime #|'O_pi(G)| #|'O_pi^'(R)|.

Proof. exact: pnat_coprime (pcore_pgroup _ _) (pcore_pgroup _ _). Qed.

Lemma

coprime_pcoreC

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "coprime", "group", "pcore_pgroup", "pi", "pnat_coprime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

TI_pcoreC pi G H : 'O_pi(G) :&: 'O_pi^'(H) = 1.

Proof. by rewrite coprime_TIg ?coprime_pcoreC. Qed.

Lemma

TI_pcoreC

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "coprime_TIg", "coprime_pcoreC", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcore_setI_normal pi G H : H <| G -> 'O_pi(G) :&: H = 'O_pi(H).

Proof. move=> nsHG; apply/eqP; rewrite eqEsubset subsetI pcore_sub setIC. rewrite !pcore_max ?(pgroupS (subsetIr H _)) ?pcore_pgroup ?gFnormal_trans //=. by rewrite norm_normalI ?gFnorm_trans ?normsG ?normal_sub. Qed.

Lemma

pcore_setI_normal

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "apply", "eqEsubset", "gFnorm_trans", "gFnormal_trans", "norm_normalI", "normal_sub", "normsG", "nsHG", "pcore_max", "pcore_pgroup", "pcore_sub", "pgroupS", "pi", "setIC", "subsetI", "subsetIr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_pgroup pi A : A \subset D -> pi.-group (f @* A) = pi.-group A.

Proof. by move=> sAD; rewrite /pgroup card_injm. Qed.

Lemma

injm_pgroup

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "card_injm", "group", "pgroup", "pi", "sAD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_pelt pi x : x \in D -> pi.-elt (f x) = pi.-elt x.

Proof. by move=> Dx; rewrite /p_elt order_injm. Qed.

Lemma

injm_pelt

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Dx", "order_injm", "p_elt", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_pHall pi G H : G \subset D -> H \subset D -> pi.-Hall(f @* G) (f @* H) = pi.-Hall(G) H.

Proof. by move=> sGD sGH; rewrite !pHallE injmSK ?card_injm. Qed.

Lemma

injm_pHall

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "Hall", "card_injm", "injmSK", "pHallE", "pi", "sGD", "sGH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_pcore pi G : G \subset D -> f @* 'O_pi(G) = 'O_pi(f @* G).

Proof. exact: injmF. Qed.

Lemma

injm_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "injmF", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_pseries pis G : G \subset D -> f @* pseries pis G = pseries pis (f @* G).

Proof. exact: injmF. Qed.

Lemma

injm_pseries

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "injmF", "pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_pgroup pi : G \isog H -> pi.-group G = pi.-group H.

Proof. by move=> isoGH; rewrite /pgroup (card_isog isoGH). Qed.

Lemma

isog_pgroup

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "card_isog", "group", "isoGH", "isog", "pgroup", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_pcore pi : G \isog H -> 'O_pi(G) \isog 'O_pi(H).

Proof. exact: gFisog. Qed.

Lemma

isog_pcore

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gFisog", "isog", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_pseries pis : G \isog H -> pseries pis G \isog pseries pis H.

Proof. exact: gFisog. Qed.

Lemma

isog_pseries

solvable

solvable/pgroup.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]

[ "gFisog", "isog", "pseries" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

imprimitivity_system Q

:= [&& partition Q S, [acts A, on Q | to^*] & 1 < #|Q| < #|S|].

Definition

imprimitivity_system

solvable

solvable/primitive_action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]

[ "on", "partition", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primitive

:= [transitive A, on S | to] && ~~ [exists Q, imprimitivity_system Q].

Definition

primitive

solvable

solvable/primitive_action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]

[ "imprimitivity_system", "on", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'primitive' A , 'on' S | to ]"

:= (primitive A S to) (format "[ 'primitive' A , 'on' S | to ]") : form_scope.

Notation

[ 'primitive' A , 'on' S | to ]

solvable

solvable/primitive_action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]

[ "primitive", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trans_prim_astab x : x \in S -> [transitive G, on S | to] -> [primitive G, on S | to] = maximal_eq 'C_G[x | to] G.

Proof. move=> Sx trG; rewrite /primitive trG negb_exists. apply/forallP/maximal_eqP=> /= [primG | [_ maxCx] Q]. split=> [|H sCH sHG]; first exact: subsetIl. pose X := orbit to H x; pose Q := orbit (to^*)%act G X. have Xx: x \in X by apply: orbit_refl. have defH: 'N_(G)(X | to) = H. have trH: [transitive H, ...

Lemma

trans_prim_astab

solvable

solvable/primitive_action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]

[ "Lagrange", "a1", "act", "actK", "actKV", "actM", "actsP", "acts_sub_orbit", "apply", "astab1P", "astab1_set", "atransP", "atransP2", "atrans_acts", "bigcupP", "canF_eq", "card_orbit", "card_orbit_stab", "contraNeq", "contraNneq", "defQ", "eqEcard", "eqEsubset", "eqVpro...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prim_trans_norm (H : {group aT}) : [primitive G, on S | to] -> H <| G -> H \subset 'C_G(S | to) \/ [transitive H, on S | to].

Proof. move=> primG /andP[sHG nHG]; rewrite subsetI sHG. have [trG _] := andP primG; have [x Sx defS] := imsetP trG. move: primG; rewrite (trans_prim_astab Sx) // => /maximal_eqP[_]. case/(_ ('C_G[x | to] <*> H)%G) => /= [||cxH|]; first exact: joing_subl. - by rewrite join_subG subsetIl. - have{} cxH: H \subset 'C_G[x ...

Lemma

prim_trans_norm

solvable

solvable/primitive_action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]

[ "aT", "actCJV", "apply", "astab1P", "astabP", "atransP2", "group", "imsetP", "join_subG", "joing_subl", "joing_subr", "maximal_eqP", "mem_conjg", "nHG", "norm_joinEl", "normsP", "on", "primitive", "sHG", "subIset", "subgroup_transitiveP", "subsetI", "subsetIl", "subsetP...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

n_act (t : n.-tuple sT) a

:= [tuple of map (to^~ a) t].

Definition

n_act

solvable

solvable/primitive_action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]

[ "map", "sT", "to", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

n_act_is_action : is_action setT n_act.

Proof. by apply: is_total_action => [t|t a b]; apply: eq_from_tnth => i; rewrite !tnth_map ?act1 ?actM. Qed.

Fact

n_act_is_action

solvable

solvable/primitive_action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]

[ "act1", "actM", "apply", "eq_from_tnth", "is_action", "is_total_action", "n_act", "setT", "tnth_map" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

n_act_action

:= Action n_act_is_action.

Canonical

n_act_action

solvable

solvable/primitive_action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "fintype", "tuple", "finset", "fingroup", "action", "gseries" ]

[ "n_act_is_action" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d