phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
properG_ltn_log p G H : p.-group G -> H \proper G -> logn p #\|H\| < logn p #\|G\|.	Proof. move=> pG; rewrite properEneq eqEcard andbC ltnNge => /andP[sHG]. rewrite sHG /= {1}(card_pgroup pG) {1}(card_pgroup (pgroupS sHG pG)). by apply: contra; case: p {pG} => [\|p] leHG; rewrite ?logn0 // leq_pexp2l. Qed.	Lemma	properG_ltn_log	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", "eqEcard", "group", "leq_pexp2l", "logn", "logn0", "ltnNge", "pG", "pgroupS", "proper", "properEneq", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroupM pi G H : pi.-group (G * H) = pi.-group G && pi.-group H.	Proof. have GH_gt0: 0 < #\|G :&: H\| := cardG_gt0 _. rewrite /pgroup -(mulnK #\|_\| GH_gt0) -mul_cardG -(LagrangeI G H) -mulnA. by rewrite mulKn // -(LagrangeI H G) setIC !pnatM andbCA; case: (pnat _). Qed.	Lemma	pgroupM	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "LagrangeI", "cardG_gt0", "group", "mulKn", "mul_cardG", "mulnA", "mulnK", "pgroup", "pi", "pnat", "pnatM", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroupJ pi G x : pi.-group (G :^ x) = pi.-group G.	Proof. by rewrite /pgroup cardJg. Qed.	Lemma	pgroupJ	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "cardJg", "group", "pgroup", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroup_p p P : p.-group P -> p_group P.	Proof. case: (leqP #\|P\| 1); first by move=> /card_le1_trivg-> _; apply: pgroup1. move/pdiv_prime=> pr_q pgP; have:= pgroupP pgP _ pr_q (pdiv_dvd _). by rewrite /p_group => /eqnP->. Qed.	Lemma	pgroup_p	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_le1_trivg", "eqnP", "group", "leqP", "p_group", "pdiv_dvd", "pdiv_prime", "pgroup1", "pgroupP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_groupP P : p_group P -> exists2 p, prime p & p.-group P.	Proof. case: (ltnP 1 #\|P\|); first by move/pdiv_prime; exists (pdiv #\|P\|). by move/card_le1_trivg=> -> _; exists 2 => //; apply: pgroup1. Qed.	Lemma	p_groupP	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_le1_trivg", "group", "ltnP", "p_group", "pdiv", "pdiv_prime", "pgroup1", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroup_pdiv p G : p.-group G -> G :!=: 1 -> [/\ prime p, p %\| #\|G\| & exists m, #\|G\| = p ^ m.+1]%N.	Proof. move=> pG; rewrite trivg_card1; case/p_groupP: (pgroup_p pG) => q q_pr qG. move/implyP: (pgroupP pG q q_pr); case/p_natP: qG => // [[\|m] ->] //. by rewrite dvdn_exp // => /eqnP <- _; split; rewrite ?dvdn_exp //; exists m. Qed.	Lemma	pgroup_pdiv	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "dvdn_exp", "eqnP", "group", "pG", "p_groupP", "p_natP", "pgroupP", "pgroup_p", "prime", "split", "trivg_card1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_p'group p K R : coprime #\|K\| #\|R\| -> p.-group R -> R :!=: 1 -> p^'.-group K.	Proof. move=> coKR pR ntR; have [p_pr _ [e oK]] := pgroup_pdiv pR ntR. by rewrite oK coprime_sym coprime_pexpl // prime_coprime // -p'natE in coKR. Qed.	Lemma	coprime_p'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" ]	[ "coprime", "coprime_pexpl", "coprime_sym", "group", "p'natE", "p_pr", "pgroup_pdiv", "prime_coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_Hall pi G H : pi.-Hall(G) H -> #\|H\| = (#\|G\|`_pi)%N.	Proof. case/and3P=> sHG piH pi'H; rewrite -(Lagrange sHG). by rewrite partnM ?Lagrange // part_pnat_id ?part_p'nat ?muln1. Qed.	Lemma	card_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", "Lagrange", "muln1", "part_p'nat", "part_pnat_id", "partnM", "pi", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHall_sub pi A B : pi.-Hall(A) B -> B \subset A.	Proof. by case/andP. Qed.	Lemma	pHall_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" ]	[ "Hall", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHall_pgroup pi A B : pi.-Hall(A) B -> pi.-group B.	Proof. by case/and3P. Qed.	Lemma	pHall_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" ]	[ "Hall", "group", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHallP pi G H : reflect (H \subset G /\ #\|H\| = #\|G\|`_pi)%N (pi.-Hall(G) H).	Proof. apply: (iffP idP) => [piH \| [sHG oH]]. by split; [apply: pHall_sub piH \| apply: card_Hall]. rewrite /pHall sHG -divgS // /pgroup oH. by rewrite -{2}(@partnC pi #\|G\|) ?mulKn ?part_pnat. Qed.	Lemma	pHallP	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", "card_Hall", "divgS", "mulKn", "pHall", "pHall_sub", "part_pnat", "partnC", "pgroup", "pi", "sHG", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHallE pi G H : pi.-Hall(G) H = (H \subset G) && (#\|H\| == #\|G\|`_pi)%N.	Proof. by apply/pHallP/andP=> [] [->] /eqP. Qed.	Lemma	pHallE	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", "pHallP", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_mulpG_Hall pi G K R : K * R = G -> pi.-group K -> pi^'.-group R -> pi.-Hall(G) K /\ pi^'.-Hall(G) R.	Proof. move=> defG piK pi'R; apply/andP. rewrite /pHall piK -!divgS /= -defG ?mulG_subl ?mulg_subr //= pnatNK. by rewrite coprime_cardMg ?(pnat_coprime piK) // mulKn ?mulnK //; apply/and3P. Qed.	Lemma	coprime_mulpG_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", "apply", "coprime_cardMg", "defG", "divgS", "group", "mulG_subl", "mulKn", "mulg_subr", "mulnK", "pHall", "pi", "piK", "pnatNK", "pnat_coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_mulGp_Hall pi G K R : K * R = G -> pi^'.-group K -> pi.-group R -> pi^'.-Hall(G) K /\ pi.-Hall(G) R.	Proof. move=> defG pi'K piR; apply/andP; rewrite andbC; apply/andP. by apply: coprime_mulpG_Hall => //; rewrite -(comm_group_setP _) defG ?groupP. Qed.	Lemma	coprime_mulGp_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", "apply", "comm_group_setP", "coprime_mulpG_Hall", "defG", "group", "groupP", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_pHall pi rho G H : {in \pi(G), pi =i rho} -> pi.-Hall(G) H = rho.-Hall(G) H.	Proof. move=> eq_pi_rho; apply: andb_id2l => sHG. congr (_ && _); apply: eq_in_pnat => p piHp. by apply: eq_pi_rho; apply: (piSg sHG). by congr (~~ _); apply: eq_pi_rho; apply: (pi_of_dvd (dvdn_indexg G H)). Qed.	Lemma	eq_in_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", "apply", "dvdn_indexg", "eq_in_pnat", "pi", "piSg", "pi_of_dvd", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_pHall pi rho G H : pi =i rho -> pi.-Hall(G) H = rho.-Hall(G) H.	Proof. by move=> eq_pi_rho; apply: eq_in_pHall (in1W eq_pi_rho). Qed.	Lemma	eq_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", "apply", "eq_in_pHall", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_p'Hall pi rho G H : pi =i rho -> pi^'.-Hall(G) H = rho^'.-Hall(G) H.	Proof. by move=> eq_pi_rho; apply: eq_pHall (eq_negn _). Qed.	Lemma	eq_p'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", "apply", "eq_negn", "eq_pHall", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHallNK pi G H : pi^'^'.-Hall(G) H = pi.-Hall(G) H.	Proof. exact: eq_pHall (negnK _). Qed.	Lemma	pHallNK	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", "eq_pHall", "negnK", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subHall_Hall pi rho G H K : rho.-Hall(G) H -> {subset pi <= rho} -> pi.-Hall(H) K -> pi.-Hall(G) K.	Proof. move=> hallH pi_sub_rho hallK. rewrite pHallE (subset_trans (pHall_sub hallK) (pHall_sub hallH)) /=. by rewrite (card_Hall hallK) (card_Hall hallH) partn_part. Qed.	Lemma	subHall_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", "card_Hall", "pHallE", "pHall_sub", "partn_part", "pi", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subHall_Sylow pi p G H P : pi.-Hall(G) H -> p \in pi -> p.-Sylow(H) P -> p.-Sylow(G) P.	Proof. move=> hallH pi_p sylP; have [sHG piH _] := and3P hallH. rewrite pHallE (subset_trans (pHall_sub sylP) sHG) /=. by rewrite (card_Hall sylP) (card_Hall hallH) partn_part // => q; move/eqnP->. Qed.	Lemma	subHall_Sylow	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", "Sylow", "card_Hall", "eqnP", "pHallE", "pHall_sub", "partn_part", "pi", "pi_p", "sHG", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHall_Hall pi A B : pi.-Hall(A) B -> Hall A B.	Proof. by case/and3P=> sBA piB pi'B; rewrite /Hall sBA (pnat_coprime piB). Qed.	Lemma	pHall_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", "pi", "pnat_coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Hall_pi G H : Hall G H -> \pi(H).-Hall(G) H.	Proof. by case/andP=> sHG coHG /=; rewrite /pHall sHG /pgroup pnat_pi -?coprime_pi'. Qed.	Lemma	Hall_pi	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", "coprime_pi'", "pHall", "pgroup", "pi", "pnat_pi", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
HallP G H : Hall G H -> exists pi, pi.-Hall(G) H.	Proof. by exists \pi(H); apply: Hall_pi. Qed.	Lemma	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", "Hall_pi", "apply", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_Hall G K H : K ><\| H = G -> Hall G K = Hall G H.	Proof. case/sdprod_context=> /andP[sKG _] sHG defG _ tiKH. by rewrite /Hall sKG sHG -!divgS // -defG TI_cardMg // coprime_sym mulKn ?mulnK. Qed.	Lemma	sdprod_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", "TI_cardMg", "coprime_sym", "defG", "divgS", "mulKn", "mulnK", "sHG", "sKG", "sdprod_context", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_sdprod_Hall_l G K H : K ><\| H = G -> coprime #\|K\| #\|H\| = Hall G K.	Proof. case/sdprod_context=> /andP[sKG _] _ defG _ tiKH. by rewrite /Hall sKG -divgS // -defG TI_cardMg ?mulKn. Qed.	Lemma	coprime_sdprod_Hall_l	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", "TI_cardMg", "coprime", "defG", "divgS", "mulKn", "sKG", "sdprod_context", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_sdprod_Hall_r G K H : K ><\| H = G -> coprime #\|K\| #\|H\| = Hall G H.	Proof. by move=> defG; rewrite (coprime_sdprod_Hall_l defG) (sdprod_Hall defG). Qed.	Lemma	coprime_sdprod_Hall_r	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", "coprime", "coprime_sdprod_Hall_l", "defG", "sdprod_Hall" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
compl_pHall pi K H G : pi.-Hall(G) K -> (H \in [complements to K in G]) = pi^'.-Hall(G) H.	Proof. move=> hallK; apply/complP/idP=> [[tiKH mulKH] \| hallH]. have [_] := andP hallK; rewrite /pHall pnatNK -{3}(invGid G) -mulKH mulG_subr. rewrite invMG !indexMg -indexgI andbC. by rewrite -[#\|K : H\|]indexgI setIC tiKH !indexg1. have [[sKG piK _] [sHG pi'H _]] := (and3P hallK, and3P hallH). have tiKH: K :&: H...	Lemma	compl_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", "TI_cardMg", "apply", "card_Hall", "complP", "coprime_TIg", "eqEcard", "indexMg", "indexg1", "indexgI", "invGid", "invMG", "mulG_subr", "mul_subG", "pHall", "partnC", "pi", "piK", "pnatNK", "pnat_coprime", "sHG", "sKG", "setIC", "split", "tiKH", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
compl_p'Hall pi K H G : pi^'.-Hall(G) K -> (H \in [complements to K in G]) = pi.-Hall(G) H.	Proof. by move/compl_pHall->; apply: eq_pHall (negnK pi). Qed.	Lemma	compl_p'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", "apply", "compl_pHall", "eq_pHall", "negnK", "pi", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_normal_p'HallP pi K H G : K <\| G -> pi^'.-Hall(G) H -> reflect (K ><\| H = G) (pi.-Hall(G) K).	Proof. move=> nsKG hallH; rewrite -(compl_p'Hall K hallH). exact: sdprod_normal_complP. Qed.	Lemma	sdprod_normal_p'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", "compl_p'Hall", "nsKG", "pi", "sdprod_normal_complP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_normal_pHallP pi K H G : K <\| G -> pi.-Hall(G) H -> reflect (K ><\| H = G) (pi^'.-Hall(G) K).	Proof. by move=> nsKG hallH; apply: sdprod_normal_p'HallP; rewrite ?pHallNK. Qed.	Lemma	sdprod_normal_pHallP	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", "nsKG", "pHallNK", "pi", "sdprod_normal_p'HallP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHallJ2 pi G H x : pi.-Hall(G :^ x) (H :^ x) = pi.-Hall(G) H.	Proof. by rewrite !pHallE conjSg !cardJg. Qed.	Lemma	pHallJ2	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", "cardJg", "conjSg", "pHallE", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHallJnorm pi G H x : x \in 'N(G) -> pi.-Hall(G) (H :^ x) = pi.-Hall(G) H.	Proof. by move=> Nx; rewrite -{1}(normP Nx) pHallJ2. Qed.	Lemma	pHallJnorm	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", "normP", "pHallJ2", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHallJ pi G H x : x \in G -> pi.-Hall(G) (H :^ x) = pi.-Hall(G) H.	Proof. by move=> Gx; rewrite -{1}(conjGid Gx) pHallJ2. Qed.	Lemma	pHallJ	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", "conjGid", "pHallJ2", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
HallJ G H x : x \in G -> Hall G (H :^ x) = Hall G H.	Proof. by move=> Gx; rewrite /Hall -!divgI -{1 3}(conjGid Gx) conjSg -conjIg !cardJg. Qed.	Lemma	HallJ	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", "cardJg", "conjGid", "conjIg", "conjSg", "divgI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
psubgroupJ pi G H x : x \in G -> pi.-subgroup(G) (H :^ x) = pi.-subgroup(G) H.	Proof. by move=> Gx; rewrite /psubgroup pgroupJ -{1}(conjGid Gx) conjSg. Qed.	Lemma	psubgroupJ	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "conjGid", "conjSg", "pgroupJ", "pi", "psubgroup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_groupJ P x : p_group (P :^ x) = p_group P.	Proof. by rewrite /p_group cardJg pgroupJ. Qed.	Lemma	p_groupJ	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "cardJg", "p_group", "pgroupJ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
SylowJ G P x : x \in G -> Sylow G (P :^ x) = Sylow G P.	Proof. by move=> Gx; rewrite /Sylow p_groupJ HallJ. Qed.	Lemma	SylowJ	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "HallJ", "Sylow", "p_groupJ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_Sylow p G P : p.-Sylow(G) P -> Sylow G P.	Proof. by move=> pP; rewrite /Sylow (pgroup_p (pHall_pgroup pP)) (pHall_Hall pP). Qed.	Lemma	p_Sylow	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Sylow", "pHall_Hall", "pHall_pgroup", "pP", "pgroup_p" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHall_subl pi G K H : H \subset K -> K \subset G -> pi.-Hall(G) H -> pi.-Hall(K) H.	Proof. by move=> sHK sKG; rewrite /pHall sHK => /and3P[_ ->]; apply/pnat_dvd/indexSg. Qed.	Lemma	pHall_subl	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", "indexSg", "pHall", "pi", "pnat_dvd", "sHK", "sKG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Hall1 G : Hall G 1.	Proof. by rewrite /Hall sub1G cards1 coprime1n. Qed.	Lemma	Hall1	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", "cards1", "coprime1n", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_group1 : @p_group gT 1.	Proof. by rewrite (@pgroup_p 2) ?pgroup1. Qed.	Lemma	p_group1	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", "p_group", "pgroup1", "pgroup_p" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Sylow1 G : Sylow G 1.	Proof. by rewrite /Sylow p_group1 Hall1. Qed.	Lemma	Sylow1	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Hall1", "Sylow", "p_group1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
SylowP G P : reflect (exists2 p, prime p & p.-Sylow(G) P) (Sylow G P).	Proof. apply: (iffP idP) => [\| [p _]]; last exact: p_Sylow. case/andP=> /p_groupP[p p_pr] /p_natP[[P1 _ \| n oP /Hall_pi]]; last first. by rewrite /= oP pi_of_exp // (eq_pHall _ _ (pi_of_prime _)) //; exists p. have{p p_pr P1} ->: P :=: 1 by apply: card1_trivg; rewrite P1. pose p := pdiv #\|G\|.+1; have p_pr: prime p by...	Lemma	SylowP	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Euclid_dvd1", "Hall_pi", "P1", "Sylow", "addn1", "apply", "card1_trivg", "cards1", "def_q", "dvdn_addr", "eq_pHall", "eqnP", "last", "ltnS", "pHallE", "p_Sylow", "p_groupP", "p_natP", "p_pr", "part_p'nat", "pdiv", "pdiv_dvd", "pdiv_prime", "pgroupP", "pi_of_exp", "...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_elt_exp pi x m : pi.-elt (x ^+ m) = (#[x]`_pi^' %\| m).	Proof. apply/idP/idP=> [pi_xm \| /dvdnP[q ->{m}]]; last first. rewrite mulnC; apply: pnat_dvd (part_pnat pi #[x]). by rewrite order_dvdn -expgM mulnC mulnA partnC // -order_dvdn dvdn_mulr. rewrite -(@Gauss_dvdr _ #[x ^+ m]). by rewrite coprime_sym (pnat_coprime pi_xm) ?part_pnat. apply: (@dvdn_trans #[x]); first b...	Lemma	p_elt_exp	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Gauss_dvdr", "apply", "coprime_sym", "dvdnP", "dvdn_mull", "dvdn_mulr", "dvdn_trans", "expgM", "expg_order", "last", "mulnA", "mulnC", "order_dvdn", "part_pnat", "partnC", "pi", "pnat_coprime", "pnat_dvd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_p_elt pi x G : pi.-group G -> x \in G -> pi.-elt x.	Proof. by move=> piG Gx; apply: pgroupS piG; rewrite cycle_subG. Qed.	Lemma	mem_p_elt	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", "cycle_subG", "group", "pgroupS", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_eltM_norm pi x y : x \in 'N(<[y]>) -> pi.-elt x -> pi.-elt y -> pi.-elt (x * y).	Proof. move=> nyx pi_x pi_y; apply: (@mem_p_elt pi _ (<[x]> <*> <[y]>)%G). by rewrite /= norm_joinEl ?cycle_subG // pgroupM; apply/andP. by rewrite groupM // mem_gen // inE cycle_id ?orbT. Qed.	Lemma	p_eltM_norm	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", "cycle_id", "cycle_subG", "groupM", "inE", "mem_gen", "mem_p_elt", "norm_joinEl", "pgroupM", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_eltM pi x y : commute x y -> pi.-elt x -> pi.-elt y -> pi.-elt (x * y).	Proof. move=> cxy; apply: p_eltM_norm; apply: (subsetP (cent_sub _)). by rewrite cent_gen cent_set1; apply/cent1P. Qed.	Lemma	p_eltM	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", "cent1P", "cent_gen", "cent_set1", "cent_sub", "commute", "p_eltM_norm", "pi", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_elt1 pi : pi.-elt (1 : gT).	Proof. by rewrite /p_elt order1. Qed.	Lemma	p_elt1	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", "order1", "p_elt", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_eltV pi x : pi.-elt x^-1 = pi.-elt x.	Proof. by rewrite /p_elt orderV. Qed.	Lemma	p_eltV	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "orderV", "p_elt", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_eltX pi x n : pi.-elt x -> pi.-elt (x ^+ n).	Proof. by rewrite -{1}[x]expg1 !p_elt_exp dvdn1 => /eqnP->. Qed.	Lemma	p_eltX	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "dvdn1", "eqnP", "expg1", "p_elt_exp", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_eltJ pi x y : pi.-elt (x ^ y) = pi.-elt x.	Proof. by congr pnat; rewrite orderJ. Qed.	Lemma	p_eltJ	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "orderJ", "pi", "pnat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_p_elt pi1 pi2 x : {subset pi1 <= pi2} -> pi1.-elt x -> pi2.-elt x.	Proof. by move=> pi12; apply: sub_in_pnat => q _; apply: pi12. Qed.	Lemma	sub_p_elt	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", "sub_in_pnat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_p_elt pi1 pi2 x : pi1 =i pi2 -> pi1.-elt x = pi2.-elt x.	Proof. by move=> pi12; apply: eq_pnat. Qed.	Lemma	eq_p_elt	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_pnat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_eltNK pi x : pi^'^'.-elt x = pi.-elt x.	Proof. exact: pnatNK. Qed.	Lemma	p_eltNK	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "pi", "pnatNK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_constt pi1 pi2 x : pi1 =i pi2 -> x.`_pi1 = x.`_pi2.	Proof. move=> pi12; congr (x ^+ (chinese _ _ 1 0)); apply: eq_partn => // a. by congr (~~ _); apply: pi12. Qed.	Lemma	eq_constt	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", "chinese", "eq_partn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
consttNK pi x : x.`_pi^'^' = x.`_pi.	Proof. by rewrite /constt !partnNK. Qed.	Lemma	consttNK	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "constt", "partnNK", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle_constt pi x : x.`_pi \in <[x]>.	Proof. exact: mem_cycle. Qed.	Lemma	cycle_constt	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "mem_cycle", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
consttV pi x : (x^-1).`_pi = (x.`_pi)^-1.	Proof. by rewrite /constt expgVn orderV. Qed.	Lemma	consttV	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "constt", "expgVn", "orderV", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
constt1 pi : 1.`_pi = 1 :> gT.	Proof. exact: expg1n. Qed.	Lemma	constt1	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "expg1n", "gT", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
consttJ pi x y : (x ^ y).`_pi = x.`_pi ^ y.	Proof. by rewrite /constt orderJ conjXg. Qed.	Lemma	consttJ	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "conjXg", "constt", "orderJ", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_elt_constt pi x : pi.-elt x.`_pi.	Proof. by rewrite p_elt_exp /chinese addn0 mul1n dvdn_mulr. Qed.	Lemma	p_elt_constt	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "addn0", "chinese", "dvdn_mulr", "mul1n", "p_elt_exp", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
consttC pi x : x.`_pi * x.`_pi^' = x.	Proof. apply/eqP; rewrite -{3}[x]expg1 -expgD eq_expg_mod_order. rewrite partnNK -{5 6}(@partnC pi #[x]) // /chinese !addn0. by rewrite chinese_remainder ?chinese_modl ?chinese_modr ?coprime_partC ?eqxx. Qed.	Lemma	consttC	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "addn0", "apply", "chinese", "chinese_modl", "chinese_modr", "chinese_remainder", "coprime_partC", "eq_expg_mod_order", "eqxx", "expg1", "expgD", "partnC", "partnNK", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p'_elt_constt pi x : pi^'.-elt (x * (x.`_pi)^-1).	Proof. by rewrite -{1}(consttC pi^' x) consttNK mulgK p_elt_constt. Qed.	Lemma	p'_elt_constt	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "consttC", "consttNK", "mulgK", "p_elt_constt", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
order_constt pi (x : gT) : #[x.`_pi] = (#[x]`_pi)%N.	Proof. rewrite -{2}(consttC pi x) orderM; [exact: commuteX2\| \|]. by apply: (@pnat_coprime pi); apply: p_elt_constt. by rewrite partnM // part_pnat_id ?part_p'nat ?muln1 //; apply: p_elt_constt. Qed.	Lemma	order_constt	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", "commuteX2", "consttC", "gT", "muln1", "orderM", "p_elt_constt", "part_p'nat", "part_pnat_id", "partnM", "pi", "pnat_coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
consttM pi x y : commute x y -> (x * y).`_pi = x.`_pi * y.`_pi.	Proof. move=> cxy; pose m := #\|<<[set x; y]>>\|; have m_gt0: 0 < m := cardG_gt0 _. pose k := chinese m`_pi m`_pi^' 1 0. suffices kXpi z: z \in <<[set x; y]>> -> z.`_pi = z ^+ k. by rewrite !kXpi ?expgMn // ?groupM ?mem_gen // !inE eqxx ?orbT. move=> xyz; have{xyz} zm: #[z] %\| m by rewrite cardSg ?cycle_subG. apply/eqP...	Lemma	consttM	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", "cardG_gt0", "cardSg", "chinese", "chinese_modl", "chinese_modr", "chinese_remainder", "commute", "coprime_partC", "cycle_subG", "eq_expg_mod_order", "eqxx", "expgMn", "groupM", "inE", "mem_gen", "modn_dvdm", "partnC", "partn_dvd", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
consttX pi x n : (x ^+ n).`_pi = x.`_pi ^+ n.	Proof. elim: n => [\|n IHn]; first exact: constt1. by rewrite !expgS consttM ?IHn //; apply: commuteX. Qed.	Lemma	consttX	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", "commuteX", "constt1", "consttM", "expgS", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
constt1P pi x : reflect (x.`_pi = 1) (pi^'.-elt x).	Proof. rewrite -{2}[x]expg1 p_elt_exp -order_constt consttNK order_dvdn expg1. exact: eqP. Qed.	Lemma	constt1P	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "consttNK", "expg1", "order_constt", "order_dvdn", "p_elt_exp", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
constt_p_elt pi x : pi.-elt x -> x.`_pi = x.	Proof. by rewrite -p_eltNK -{3}(consttC pi x) => /constt1P->; rewrite mulg1. Qed.	Lemma	constt_p_elt	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "constt1P", "consttC", "mulg1", "p_eltNK", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_in_constt pi1 pi2 x : {in \pi(#[x]), {subset pi1 <= pi2}} -> x.`_pi2.`_pi1 = x.`_pi1.	Proof. move=> pi12; rewrite -{2}(consttC pi2 x) consttM; first exact: commuteX2. rewrite (constt1P _ x.`_pi2^' _) ?mulg1 //. apply: sub_in_pnat (p_elt_constt _ x) => p; rewrite order_constt => pi_p. by apply/contra/pi12; rewrite -[#[x]](partnC pi2^') // primesM // pi_p. Qed.	Lemma	sub_in_constt	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", "commuteX2", "constt1P", "consttC", "consttM", "mulg1", "order_constt", "p_elt_constt", "partnC", "pi", "pi_p", "primesM", "sub_in_pnat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_constt x : \prod_(0 <= p < #[x].+1) x.`_p = x.	Proof. pose lp n := [pred p \| p < n]. have: (lp #[x].+1).-elt x by apply/pnatP=> // p _; apply: dvdn_leq. move/constt_p_elt=> def_x; symmetry; rewrite -{1}def_x {def_x}. elim: _.+1 => [\|p IHp]. by rewrite big_nil; apply/constt1P; apply/pgroupP. rewrite big_nat_recr //= -{}IHp -(consttC (lp p) x.`__); congr (_ * _). ...	Lemma	prod_constt	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", "big_nat_recr", "big_nil", "constt1P", "consttC", "constt_p_elt", "dvdn_leq", "eqnP", "eqn_leq", "inE", "last", "leqNgt", "leqW", "ltnS", "ltnn", "mulg1", "order_constt", "p_elt", "part_pnat", "partnI", "pgroupP", "pnatNK", "pnatP", "sub_in_constt", "sub_in_p...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_pgroupJ pi M G x : x \in G -> [max M \| pi.-subgroup(G) M] -> [max M :^ x of M \| pi.-subgroup(G) M].	Proof. move=> Gx /maxgroupP[piM maxM]; apply/maxgroupP. split=> [\|H piH]; first by rewrite psubgroupJ. by rewrite -(conjsgKV x H) conjSg => /maxM/=-> //; rewrite psubgroupJ ?groupV. Qed.	Lemma	max_pgroupJ	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", "conjSg", "conjsgKV", "groupV", "max", "maxgroupP", "pi", "psubgroupJ", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_sub_max_pgroup pi H M G : [max M \| pi.-subgroup(G) M] -> pi.-group H -> H \subset G -> commute H M -> H \subset M.	Proof. case/maxgroupP=> /andP[sMG piM] maxM piH sHG cHM. rewrite -(maxM (H <*> M)%G) /= comm_joingE ?(mulG_subl, mulG_subr) //. by rewrite /psubgroup pgroupM piM piH mul_subG. Qed.	Lemma	comm_sub_max_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" ]	[ "comm_joingE", "commute", "group", "max", "maxgroupP", "mulG_subl", "mulG_subr", "mul_subG", "pgroupM", "pi", "psubgroup", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_sub_max_pgroup pi H M G : [max M \| pi.-subgroup(G) M] -> pi.-group H -> H <\| G -> H \subset M.	Proof. move=> maxM piH /andP[sHG nHG]. apply: comm_sub_max_pgroup piH sHG _ => //; apply: commute_sym; apply: normC. by apply: subset_trans nHG; case/andP: (maxgroupp maxM). Qed.	Lemma	normal_sub_max_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" ]	[ "apply", "comm_sub_max_pgroup", "commute_sym", "group", "max", "maxgroupp", "nHG", "normC", "pi", "sHG", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_sub_max_pgroup pi H M G : [max M \| pi.-subgroup(G) M] -> pi.-group H -> H \subset G -> H \subset 'N(M) -> H \subset M.	Proof. by move=> maxM piH sHG /normC; apply: comm_sub_max_pgroup piH sHG. Qed.	Lemma	norm_sub_max_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" ]	[ "apply", "comm_sub_max_pgroup", "group", "max", "normC", "pi", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_pHall pi H G K : pi.-Hall(G) H -> pi.-group K -> H \subset K -> K \subset G -> K :=: H.	Proof. move=> hallH piK sHK sKG; apply/eqP; rewrite eq_sym eqEcard sHK. by rewrite (card_Hall hallH) -(part_pnat_id piK) dvdn_leq ?partn_dvd ?cardSg. Qed.	Lemma	sub_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", "apply", "cardSg", "card_Hall", "dvdn_leq", "eqEcard", "eq_sym", "group", "part_pnat_id", "partn_dvd", "pi", "piK", "sHK", "sKG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Hall_max pi H G : pi.-Hall(G) H -> [max H \| pi.-subgroup(G) H].	Proof. move=> hallH; apply/maxgroupP; split=> [\|K /andP[sKG piK] sHK]. by rewrite /psubgroup; case/and3P: hallH => ->. exact: (sub_pHall hallH). Qed.	Lemma	Hall_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" ]	[ "Hall", "apply", "max", "maxgroupP", "pi", "piK", "psubgroup", "sHK", "sKG", "split", "sub_pHall" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHall_id pi H G : pi.-Hall(G) H -> pi.-group G -> H :=: G.	Proof. by move=> hallH piG; rewrite (sub_pHall hallH piG) ?(pHall_sub hallH). Qed.	Lemma	pHall_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" ]	[ "Hall", "group", "pHall_sub", "pi", "sub_pHall" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
psubgroup1 pi G : pi.-subgroup(G) 1.	Proof. by rewrite /psubgroup sub1G pgroup1. Qed.	Lemma	psubgroup1	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "pgroup1", "pi", "psubgroup", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Cauchy p G : prime p -> p %\| #\|G\| -> {x \| x \in G & #[x] = p}.	Proof. move=> p_pr; have [n] := ubnP #\|G\|; elim: n G => // n IHn G /ltnSE-leGn pG. pose xpG := [pred x in G \| #[x] == p]. have [x /andP[Gx /eqP] \| no_x] := pickP xpG; first by exists x. have{pG n leGn IHn} pZ: p %\| #\|'C_G(G)\|. suffices /dvdn_addl <-: p %\| #\|G :\: 'C(G)\| by rewrite cardsID. have /acts_sum_card_orbi...	Lemma	Cauchy	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Euclid_dvdM", "LagrangeI", "actsP", "acts_sum_card_orbit", "apply", "big_rec", "cardSg", "cardsID", "centJ", "centsC", "cents_norm", "conjGid", "cycle_subG", "dvdnP", "dvdn_addl", "dvdn_gt0", "dvdn_mulr", "dvdn_trans", "gcdnC", "gcdnMr", "gen_subG", "groupV", "groupX", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_normal_Hall pi G H K : pi.-Hall(G) H -> H <\| G -> K \subset G -> (K \subset H) = pi.-group K.	Proof. move=> hallH nsHG sKG; apply/idP/idP=> [sKH \| piK]. by rewrite (pgroupS sKH) ?(pHall_pgroup hallH). apply: norm_sub_max_pgroup (Hall_max hallH) piK _ _ => //. exact: subset_trans sKG (normal_norm nsHG). Qed.	Lemma	sub_normal_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", "group", "norm_sub_max_pgroup", "normal_norm", "nsHG", "pHall_pgroup", "pgroupS", "pi", "piK", "sKG", "subset_trans" ]	derive from the Cauchy lemma that a normal max pi-group is Hall.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_normal_Hall pi H G x : pi.-Hall(G) H -> H <\| G -> x \in G -> (x \in H) = pi.-elt x.	Proof. by rewrite -!cycle_subG; apply: sub_normal_Hall. Qed.	Lemma	mem_normal_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", "apply", "cycle_subG", "pi", "sub_normal_Hall" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
uniq_normal_Hall pi H G K : pi.-Hall(G) H -> H <\| G -> [max K \| pi.-subgroup(G) K] -> K :=: H.	Proof. move=> hallH nHG /maxgroupP[/andP[sKG piK] /(_ H) -> //]. exact: (maxgroupp (Hall_max hallH)). by rewrite (sub_normal_Hall hallH). Qed.	Lemma	uniq_normal_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", "max", "maxgroupP", "maxgroupp", "nHG", "pi", "piK", "sKG", "sub_normal_Hall" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_max_pgroup_Hall G H : [max H \| pi.-subgroup(G) H] -> H <\| G -> pi.-Hall(G) H.	Proof. case/maxgroupP=> /andP[sHG piH] maxH nsHG; have [_ nHG] := andP nsHG. rewrite /pHall sHG piH; apply/pnatP=> // p p_pr. rewrite inE /= -pnatE // -card_quotient //. case/Cauchy=> //= Hx; rewrite -sub1set -gen_subG -/<[Hx]> /order. case/inv_quotientS=> //= K -> sHK sKG {Hx}. rewrite card_quotient ?(subset_trans sKG...	Lemma	normal_max_pgroup_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" ]	[ "Cauchy", "Hall", "Lagrange", "apply", "cardG_gt0", "card_quotient", "divgS", "divnn", "gen_subG", "inE", "inv_quotientS", "max", "maxgroupP", "mulnC", "nHG", "nsHG", "order", "pHall", "p_pr", "pgroup", "pi", "pi_p", "pnatE", "pnatM", "pnatP", "psubgroup", "sHG", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setI_normal_Hall G H K : H <\| G -> pi.-Hall(G) H -> K \subset G -> pi.-Hall(K) (H :&: K).	Proof. move=> nsHG hallH sKG; apply: normal_max_pgroup_Hall; last first. by rewrite /= setIC (normalGI sKG nsHG). apply/maxgroupP; split=> [\|M /andP[sMK piM] sHK_M]. by rewrite /psubgroup subsetIr (pgroupS (subsetIl _ _) (pHall_pgroup hallH)). apply/eqP; rewrite eqEsubset sHK_M subsetI sMK !andbT. by rewrite (sub_n...	Lemma	setI_normal_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", "apply", "eqEsubset", "last", "maxgroupP", "normalGI", "normal_max_pgroup_Hall", "nsHG", "pHall_pgroup", "pgroupS", "pi", "psubgroup", "sKG", "setIC", "split", "sub_normal_Hall", "subsetI", "subsetIl", "subsetIr", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pgroup pi G : pi.-group G -> pi.-group (f @* G).	Proof. by apply: pnat_dvd; apply: dvdn_morphim. Qed.	Lemma	morphim_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" ]	[ "apply", "dvdn_morphim", "group", "pi", "pnat_dvd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_odd G : odd #\|G\| -> odd #\|f @* G\|.	Proof. by rewrite !odd_2'nat; apply: morphim_pgroup. Qed.	Lemma	morphim_odd	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", "morphim_pgroup", "odd", "odd_2'nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmorphim_pgroup pi G : pi.-group ('ker f) -> G \subset D -> pi.-group (f @* G) = pi.-group G.	Proof. move=> piker sGD; apply/idP/idP=> [pifG\|]; last exact: morphim_pgroup. apply: (@pgroupS _ _ (f @^-1 (f @ G))); first by rewrite -sub_morphim_pre. by rewrite /pgroup card_morphpre ?morphimS // pnatM; apply/andP. Qed.	Lemma	pmorphim_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" ]	[ "apply", "card_morphpre", "group", "ker", "last", "morphimS", "morphim_pgroup", "pgroup", "pgroupS", "pi", "pnatM", "sGD", "sub_morphim_pre" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_p_index pi G H : H \subset D -> pi.-nat #\|G : H\| -> pi.-nat #\|f @* G : f @* H\|.	Proof. by move=> sHD; apply: pnat_dvd; rewrite index_morphim ?subIset // sHD orbT. Qed.	Lemma	morphim_p_index	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", "index_morphim", "nat", "pi", "pnat_dvd", "sHD", "subIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pHall pi G H : H \subset D -> pi.-Hall(G) H -> pi.-Hall(f @* G) (f @* H).	Proof. move=> sHD /and3P[sHG piH pi'GH]. by rewrite /pHall morphimS // morphim_pgroup // morphim_p_index. Qed.	Lemma	morphim_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", "morphimS", "morphim_p_index", "morphim_pgroup", "pHall", "pi", "sHD", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmorphim_pHall pi G H : G \subset D -> H \subset D -> pi.-subgroup(H :&: G) ('ker f) -> pi.-Hall(f @* G) (f @* H) = pi.-Hall(G) H.	Proof. move=> sGD sHD /andP[/subsetIP[sKH sKG] piK]; rewrite !pHallE morphimSGK //. apply: andb_id2l => sHG; rewrite -(Lagrange sKH) -(Lagrange sKG) partnM //. by rewrite (part_pnat_id piK) !card_morphim !(setIidPr _) // eqn_pmul2l. Qed.	Lemma	pmorphim_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", "Lagrange", "apply", "card_morphim", "eqn_pmul2l", "ker", "morphimSGK", "pHallE", "part_pnat_id", "partnM", "pi", "piK", "sGD", "sHD", "sHG", "sKG", "setIidPr", "subsetIP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_Hall G H : H \subset D -> Hall G H -> Hall (f @* G) (f @* H).	Proof. by move=> sHD /HallP[pi piH]; apply: (@pHall_Hall _ pi); apply: morphim_pHall. Qed.	Lemma	morphim_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", "HallP", "apply", "morphim_pHall", "pHall_Hall", "pi", "sHD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pSylow p G P : P \subset D -> p.-Sylow(G) P -> p.-Sylow(f @* G) (f @* P).	Proof. exact: morphim_pHall. Qed.	Lemma	morphim_pSylow	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Sylow", "morphim_pHall" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_p_group P : p_group P -> p_group (f @* P).	Proof. by move/morphim_pgroup; apply: pgroup_p. Qed.	Lemma	morphim_p_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" ]	[ "apply", "morphim_pgroup", "p_group", "pgroup_p" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_Sylow G P : P \subset D -> Sylow G P -> Sylow (f @* G) (f @* P).	Proof. by move=> sPD /andP[pP hallP]; rewrite /Sylow morphim_p_group // morphim_Hall. Qed.	Lemma	morphim_Sylow	solvable	solvable/pgroup.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]	[ "Sylow", "morphim_Hall", "morphim_p_group", "pP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_p_elt pi x : x \in D -> pi.-elt x -> pi.-elt (f x).	Proof. by move=> Dx; apply: pnat_dvd; apply: morph_order. Qed.	Lemma	morph_p_elt	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", "apply", "morph_order", "pi", "pnat_dvd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_constt pi x : x \in D -> f x.`_pi = (f x).`_pi.	Proof. move=> Dx; rewrite -{2}(consttC pi x) morphM ?groupX //. rewrite consttM; first by rewrite !morphX //; apply: commuteX2. have: pi.-elt (f x.`_pi) by rewrite morph_p_elt ?groupX ?p_elt_constt //. have: pi^'.-elt (f x.`_pi^') by rewrite morph_p_elt ?groupX ?p_elt_constt //. by move/constt1P->; move/constt_p_elt->;...	Lemma	morph_constt	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", "apply", "commuteX2", "constt1P", "consttC", "consttM", "constt_p_elt", "groupX", "morphM", "morphX", "morph_p_elt", "mulg1", "p_elt_constt", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
piK : pi.-group K.		Hypothesis	piK	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", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_pgroup : pi.-group (K / H).	Proof. exact: morphim_pgroup. Qed.	Lemma	quotient_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", "morphim_pgroup", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_pHall : K \subset 'N(H) -> pi.-Hall(G) K -> pi.-Hall(G / H) (K / H).	Proof. exact: morphim_pHall. Qed.	Lemma	quotient_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", "morphim_pHall", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_odd : odd #\|K\| -> odd #\|K / H\|.	Proof. exact: morphim_odd. Qed.	Lemma	quotient_odd	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_odd", "odd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

properG_ltn_log p G H : p.-group G -> H \proper G -> logn p #|H| < logn p #|G|.

Proof. move=> pG; rewrite properEneq eqEcard andbC ltnNge => /andP[sHG]. rewrite sHG /= {1}(card_pgroup pG) {1}(card_pgroup (pgroupS sHG pG)). by apply: contra; case: p {pG} => [|p] leHG; rewrite ?logn0 // leq_pexp2l. Qed.

Lemma

properG_ltn_log

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", "eqEcard", "group", "leq_pexp2l", "logn", "logn0", "ltnNge", "pG", "pgroupS", "proper", "properEneq", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pgroupM pi G H : pi.-group (G * H) = pi.-group G && pi.-group H.

Proof. have GH_gt0: 0 < #|G :&: H| := cardG_gt0 _. rewrite /pgroup -(mulnK #|_| GH_gt0) -mul_cardG -(LagrangeI G H) -mulnA. by rewrite mulKn // -(LagrangeI H G) setIC !pnatM andbCA; case: (pnat _). Qed.

Lemma

pgroupM

solvable

solvable/pgroup.v

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

[ "LagrangeI", "cardG_gt0", "group", "mulKn", "mul_cardG", "mulnA", "mulnK", "pgroup", "pi", "pnat", "pnatM", "setIC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pgroupJ pi G x : pi.-group (G :^ x) = pi.-group G.

Proof. by rewrite /pgroup cardJg. Qed.

Lemma

pgroupJ

solvable

solvable/pgroup.v

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

[ "cardJg", "group", "pgroup", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pgroup_p p P : p.-group P -> p_group P.

Proof. case: (leqP #|P| 1); first by move=> /card_le1_trivg-> _; apply: pgroup1. move/pdiv_prime=> pr_q pgP; have:= pgroupP pgP _ pr_q (pdiv_dvd _). by rewrite /p_group => /eqnP->. Qed.

Lemma

pgroup_p

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_le1_trivg", "eqnP", "group", "leqP", "p_group", "pdiv_dvd", "pdiv_prime", "pgroup1", "pgroupP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_groupP P : p_group P -> exists2 p, prime p & p.-group P.

Proof. case: (ltnP 1 #|P|); first by move/pdiv_prime; exists (pdiv #|P|). by move/card_le1_trivg=> -> _; exists 2 => //; apply: pgroup1. Qed.

Lemma

p_groupP

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_le1_trivg", "group", "ltnP", "p_group", "pdiv", "pdiv_prime", "pgroup1", "prime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pgroup_pdiv p G : p.-group G -> G :!=: 1 -> [/\ prime p, p %| #|G| & exists m, #|G| = p ^ m.+1]%N.

Proof. move=> pG; rewrite trivg_card1; case/p_groupP: (pgroup_p pG) => q q_pr qG. move/implyP: (pgroupP pG q q_pr); case/p_natP: qG => // [[|m] ->] //. by rewrite dvdn_exp // => /eqnP <- _; split; rewrite ?dvdn_exp //; exists m. Qed.

Lemma

pgroup_pdiv

solvable

solvable/pgroup.v

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

[ "dvdn_exp", "eqnP", "group", "pG", "p_groupP", "p_natP", "pgroupP", "pgroup_p", "prime", "split", "trivg_card1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_p'group p K R : coprime #|K| #|R| -> p.-group R -> R :!=: 1 -> p^'.-group K.

Proof. move=> coKR pR ntR; have [p_pr _ [e oK]] := pgroup_pdiv pR ntR. by rewrite oK coprime_sym coprime_pexpl // prime_coprime // -p'natE in coKR. Qed.

Lemma

coprime_p'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" ]

[ "coprime", "coprime_pexpl", "coprime_sym", "group", "p'natE", "p_pr", "pgroup_pdiv", "prime_coprime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_Hall pi G H : pi.-Hall(G) H -> #|H| = (#|G|`_pi)%N.

Proof. case/and3P=> sHG piH pi'H; rewrite -(Lagrange sHG). by rewrite partnM ?Lagrange // part_pnat_id ?part_p'nat ?muln1. Qed.

Lemma

card_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", "Lagrange", "muln1", "part_p'nat", "part_pnat_id", "partnM", "pi", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHall_sub pi A B : pi.-Hall(A) B -> B \subset A.

Proof. by case/andP. Qed.

Lemma

pHall_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" ]

[ "Hall", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHall_pgroup pi A B : pi.-Hall(A) B -> pi.-group B.

Proof. by case/and3P. Qed.

Lemma

pHall_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" ]

[ "Hall", "group", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHallP pi G H : reflect (H \subset G /\ #|H| = #|G|`_pi)%N (pi.-Hall(G) H).

Proof. apply: (iffP idP) => [piH | [sHG oH]]. by split; [apply: pHall_sub piH | apply: card_Hall]. rewrite /pHall sHG -divgS // /pgroup oH. by rewrite -{2}(@partnC pi #|G|) ?mulKn ?part_pnat. Qed.

Lemma

pHallP

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", "card_Hall", "divgS", "mulKn", "pHall", "pHall_sub", "part_pnat", "partnC", "pgroup", "pi", "sHG", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHallE pi G H : pi.-Hall(G) H = (H \subset G) && (#|H| == #|G|`_pi)%N.

Proof. by apply/pHallP/andP=> [] [->] /eqP. Qed.

Lemma

pHallE

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", "pHallP", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_mulpG_Hall pi G K R : K * R = G -> pi.-group K -> pi^'.-group R -> pi.-Hall(G) K /\ pi^'.-Hall(G) R.

Proof. move=> defG piK pi'R; apply/andP. rewrite /pHall piK -!divgS /= -defG ?mulG_subl ?mulg_subr //= pnatNK. by rewrite coprime_cardMg ?(pnat_coprime piK) // mulKn ?mulnK //; apply/and3P. Qed.

Lemma

coprime_mulpG_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", "apply", "coprime_cardMg", "defG", "divgS", "group", "mulG_subl", "mulKn", "mulg_subr", "mulnK", "pHall", "pi", "piK", "pnatNK", "pnat_coprime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_mulGp_Hall pi G K R : K * R = G -> pi^'.-group K -> pi.-group R -> pi^'.-Hall(G) K /\ pi.-Hall(G) R.

Proof. move=> defG pi'K piR; apply/andP; rewrite andbC; apply/andP. by apply: coprime_mulpG_Hall => //; rewrite -(comm_group_setP _) defG ?groupP. Qed.

Lemma

coprime_mulGp_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", "apply", "comm_group_setP", "coprime_mulpG_Hall", "defG", "group", "groupP", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_pHall pi rho G H : {in \pi(G), pi =i rho} -> pi.-Hall(G) H = rho.-Hall(G) H.

Proof. move=> eq_pi_rho; apply: andb_id2l => sHG. congr (_ && _); apply: eq_in_pnat => p piHp. by apply: eq_pi_rho; apply: (piSg sHG). by congr (~~ _); apply: eq_pi_rho; apply: (pi_of_dvd (dvdn_indexg G H)). Qed.

Lemma

eq_in_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", "apply", "dvdn_indexg", "eq_in_pnat", "pi", "piSg", "pi_of_dvd", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_pHall pi rho G H : pi =i rho -> pi.-Hall(G) H = rho.-Hall(G) H.

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

Lemma

eq_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", "apply", "eq_in_pHall", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_p'Hall pi rho G H : pi =i rho -> pi^'.-Hall(G) H = rho^'.-Hall(G) H.

Proof. by move=> eq_pi_rho; apply: eq_pHall (eq_negn _). Qed.

Lemma

eq_p'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", "apply", "eq_negn", "eq_pHall", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHallNK pi G H : pi^'^'.-Hall(G) H = pi.-Hall(G) H.

Proof. exact: eq_pHall (negnK _). Qed.

Lemma

pHallNK

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", "eq_pHall", "negnK", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subHall_Hall pi rho G H K : rho.-Hall(G) H -> {subset pi <= rho} -> pi.-Hall(H) K -> pi.-Hall(G) K.

Proof. move=> hallH pi_sub_rho hallK. rewrite pHallE (subset_trans (pHall_sub hallK) (pHall_sub hallH)) /=. by rewrite (card_Hall hallK) (card_Hall hallH) partn_part. Qed.

Lemma

subHall_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", "card_Hall", "pHallE", "pHall_sub", "partn_part", "pi", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subHall_Sylow pi p G H P : pi.-Hall(G) H -> p \in pi -> p.-Sylow(H) P -> p.-Sylow(G) P.

Proof. move=> hallH pi_p sylP; have [sHG piH _] := and3P hallH. rewrite pHallE (subset_trans (pHall_sub sylP) sHG) /=. by rewrite (card_Hall sylP) (card_Hall hallH) partn_part // => q; move/eqnP->. Qed.

Lemma

subHall_Sylow

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", "Sylow", "card_Hall", "eqnP", "pHallE", "pHall_sub", "partn_part", "pi", "pi_p", "sHG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHall_Hall pi A B : pi.-Hall(A) B -> Hall A B.

Proof. by case/and3P=> sBA piB pi'B; rewrite /Hall sBA (pnat_coprime piB). Qed.

Lemma

pHall_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", "pi", "pnat_coprime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Hall_pi G H : Hall G H -> \pi(H).-Hall(G) H.

Proof. by case/andP=> sHG coHG /=; rewrite /pHall sHG /pgroup pnat_pi -?coprime_pi'. Qed.

Lemma

Hall_pi

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", "coprime_pi'", "pHall", "pgroup", "pi", "pnat_pi", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

HallP G H : Hall G H -> exists pi, pi.-Hall(G) H.

Proof. by exists \pi(H); apply: Hall_pi. Qed.

Lemma

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", "Hall_pi", "apply", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_Hall G K H : K ><| H = G -> Hall G K = Hall G H.

Proof. case/sdprod_context=> /andP[sKG _] sHG defG _ tiKH. by rewrite /Hall sKG sHG -!divgS // -defG TI_cardMg // coprime_sym mulKn ?mulnK. Qed.

Lemma

sdprod_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", "TI_cardMg", "coprime_sym", "defG", "divgS", "mulKn", "mulnK", "sHG", "sKG", "sdprod_context", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_sdprod_Hall_l G K H : K ><| H = G -> coprime #|K| #|H| = Hall G K.

Proof. case/sdprod_context=> /andP[sKG _] _ defG _ tiKH. by rewrite /Hall sKG -divgS // -defG TI_cardMg ?mulKn. Qed.

Lemma

coprime_sdprod_Hall_l

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", "TI_cardMg", "coprime", "defG", "divgS", "mulKn", "sKG", "sdprod_context", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_sdprod_Hall_r G K H : K ><| H = G -> coprime #|K| #|H| = Hall G H.

Proof. by move=> defG; rewrite (coprime_sdprod_Hall_l defG) (sdprod_Hall defG). Qed.

Lemma

coprime_sdprod_Hall_r

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", "coprime", "coprime_sdprod_Hall_l", "defG", "sdprod_Hall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

compl_pHall pi K H G : pi.-Hall(G) K -> (H \in [complements to K in G]) = pi^'.-Hall(G) H.

Proof. move=> hallK; apply/complP/idP=> [[tiKH mulKH] | hallH]. have [_] := andP hallK; rewrite /pHall pnatNK -{3}(invGid G) -mulKH mulG_subr. rewrite invMG !indexMg -indexgI andbC. by rewrite -[#|K : H|]indexgI setIC tiKH !indexg1. have [[sKG piK _] [sHG pi'H _]] := (and3P hallK, and3P hallH). have tiKH: K :&: H...

Lemma

compl_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", "TI_cardMg", "apply", "card_Hall", "complP", "coprime_TIg", "eqEcard", "indexMg", "indexg1", "indexgI", "invGid", "invMG", "mulG_subr", "mul_subG", "pHall", "partnC", "pi", "piK", "pnatNK", "pnat_coprime", "sHG", "sKG", "setIC", "split", "tiKH", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

compl_p'Hall pi K H G : pi^'.-Hall(G) K -> (H \in [complements to K in G]) = pi.-Hall(G) H.

Proof. by move/compl_pHall->; apply: eq_pHall (negnK pi). Qed.

Lemma

compl_p'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", "apply", "compl_pHall", "eq_pHall", "negnK", "pi", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_normal_p'HallP pi K H G : K <| G -> pi^'.-Hall(G) H -> reflect (K ><| H = G) (pi.-Hall(G) K).

Proof. move=> nsKG hallH; rewrite -(compl_p'Hall K hallH). exact: sdprod_normal_complP. Qed.

Lemma

sdprod_normal_p'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", "compl_p'Hall", "nsKG", "pi", "sdprod_normal_complP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> nsKG hallH; apply: sdprod_normal_p'HallP; rewrite ?pHallNK. Qed.

Lemma

sdprod_normal_pHallP

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", "nsKG", "pHallNK", "pi", "sdprod_normal_p'HallP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHallJ2 pi G H x : pi.-Hall(G :^ x) (H :^ x) = pi.-Hall(G) H.

Proof. by rewrite !pHallE conjSg !cardJg. Qed.

Lemma

pHallJ2

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", "cardJg", "conjSg", "pHallE", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHallJnorm pi G H x : x \in 'N(G) -> pi.-Hall(G) (H :^ x) = pi.-Hall(G) H.

Proof. by move=> Nx; rewrite -{1}(normP Nx) pHallJ2. Qed.

Lemma

pHallJnorm

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", "normP", "pHallJ2", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHallJ pi G H x : x \in G -> pi.-Hall(G) (H :^ x) = pi.-Hall(G) H.

Proof. by move=> Gx; rewrite -{1}(conjGid Gx) pHallJ2. Qed.

Lemma

pHallJ

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", "conjGid", "pHallJ2", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

HallJ G H x : x \in G -> Hall G (H :^ x) = Hall G H.

Proof. by move=> Gx; rewrite /Hall -!divgI -{1 3}(conjGid Gx) conjSg -conjIg !cardJg. Qed.

Lemma

HallJ

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", "cardJg", "conjGid", "conjIg", "conjSg", "divgI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

psubgroupJ pi G H x : x \in G -> pi.-subgroup(G) (H :^ x) = pi.-subgroup(G) H.

Proof. by move=> Gx; rewrite /psubgroup pgroupJ -{1}(conjGid Gx) conjSg. Qed.

Lemma

psubgroupJ

solvable

solvable/pgroup.v

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

[ "conjGid", "conjSg", "pgroupJ", "pi", "psubgroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_groupJ P x : p_group (P :^ x) = p_group P.

Proof. by rewrite /p_group cardJg pgroupJ. Qed.

Lemma

p_groupJ

solvable

solvable/pgroup.v

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

[ "cardJg", "p_group", "pgroupJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

SylowJ G P x : x \in G -> Sylow G (P :^ x) = Sylow G P.

Proof. by move=> Gx; rewrite /Sylow p_groupJ HallJ. Qed.

Lemma

SylowJ

solvable

solvable/pgroup.v

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

[ "HallJ", "Sylow", "p_groupJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_Sylow p G P : p.-Sylow(G) P -> Sylow G P.

Proof. by move=> pP; rewrite /Sylow (pgroup_p (pHall_pgroup pP)) (pHall_Hall pP). Qed.

Lemma

p_Sylow

solvable

solvable/pgroup.v

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

[ "Sylow", "pHall_Hall", "pHall_pgroup", "pP", "pgroup_p" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHall_subl pi G K H : H \subset K -> K \subset G -> pi.-Hall(G) H -> pi.-Hall(K) H.

Proof. by move=> sHK sKG; rewrite /pHall sHK => /and3P[_ ->]; apply/pnat_dvd/indexSg. Qed.

Lemma

pHall_subl

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", "indexSg", "pHall", "pi", "pnat_dvd", "sHK", "sKG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Hall1 G : Hall G 1.

Proof. by rewrite /Hall sub1G cards1 coprime1n. Qed.

Lemma

Hall1

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", "cards1", "coprime1n", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_group1 : @p_group gT 1.

Proof. by rewrite (@pgroup_p 2) ?pgroup1. Qed.

Lemma

p_group1

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", "p_group", "pgroup1", "pgroup_p" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Sylow1 G : Sylow G 1.

Proof. by rewrite /Sylow p_group1 Hall1. Qed.

Lemma

Sylow1

solvable

solvable/pgroup.v

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

[ "Hall1", "Sylow", "p_group1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

SylowP G P : reflect (exists2 p, prime p & p.-Sylow(G) P) (Sylow G P).

Proof. apply: (iffP idP) => [| [p _]]; last exact: p_Sylow. case/andP=> /p_groupP[p p_pr] /p_natP[[P1 _ | n oP /Hall_pi]]; last first. by rewrite /= oP pi_of_exp // (eq_pHall _ _ (pi_of_prime _)) //; exists p. have{p p_pr P1} ->: P :=: 1 by apply: card1_trivg; rewrite P1. pose p := pdiv #|G|.+1; have p_pr: prime p by...

Lemma

SylowP

solvable

solvable/pgroup.v

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

[ "Euclid_dvd1", "Hall_pi", "P1", "Sylow", "addn1", "apply", "card1_trivg", "cards1", "def_q", "dvdn_addr", "eq_pHall", "eqnP", "last", "ltnS", "pHallE", "p_Sylow", "p_groupP", "p_natP", "p_pr", "part_p'nat", "pdiv", "pdiv_dvd", "pdiv_prime", "pgroupP", "pi_of_exp", "...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_elt_exp pi x m : pi.-elt (x ^+ m) = (#[x]`_pi^' %| m).

Proof. apply/idP/idP=> [pi_xm | /dvdnP[q ->{m}]]; last first. rewrite mulnC; apply: pnat_dvd (part_pnat pi #[x]). by rewrite order_dvdn -expgM mulnC mulnA partnC // -order_dvdn dvdn_mulr. rewrite -(@Gauss_dvdr _ #[x ^+ m]). by rewrite coprime_sym (pnat_coprime pi_xm) ?part_pnat. apply: (@dvdn_trans #[x]); first b...

Lemma

p_elt_exp

solvable

solvable/pgroup.v

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

[ "Gauss_dvdr", "apply", "coprime_sym", "dvdnP", "dvdn_mull", "dvdn_mulr", "dvdn_trans", "expgM", "expg_order", "last", "mulnA", "mulnC", "order_dvdn", "part_pnat", "partnC", "pi", "pnat_coprime", "pnat_dvd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_p_elt pi x G : pi.-group G -> x \in G -> pi.-elt x.

Proof. by move=> piG Gx; apply: pgroupS piG; rewrite cycle_subG. Qed.

Lemma

mem_p_elt

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", "cycle_subG", "group", "pgroupS", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_eltM_norm pi x y : x \in 'N(<[y]>) -> pi.-elt x -> pi.-elt y -> pi.-elt (x * y).

Proof. move=> nyx pi_x pi_y; apply: (@mem_p_elt pi _ (<[x]> <*> <[y]>)%G). by rewrite /= norm_joinEl ?cycle_subG // pgroupM; apply/andP. by rewrite groupM // mem_gen // inE cycle_id ?orbT. Qed.

Lemma

p_eltM_norm

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", "cycle_id", "cycle_subG", "groupM", "inE", "mem_gen", "mem_p_elt", "norm_joinEl", "pgroupM", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_eltM pi x y : commute x y -> pi.-elt x -> pi.-elt y -> pi.-elt (x * y).

Proof. move=> cxy; apply: p_eltM_norm; apply: (subsetP (cent_sub _)). by rewrite cent_gen cent_set1; apply/cent1P. Qed.

Lemma

p_eltM

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", "cent1P", "cent_gen", "cent_set1", "cent_sub", "commute", "p_eltM_norm", "pi", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_elt1 pi : pi.-elt (1 : gT).

Proof. by rewrite /p_elt order1. Qed.

Lemma

p_elt1

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", "order1", "p_elt", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_eltV pi x : pi.-elt x^-1 = pi.-elt x.

Proof. by rewrite /p_elt orderV. Qed.

Lemma

p_eltV

solvable

solvable/pgroup.v

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

[ "orderV", "p_elt", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_eltX pi x n : pi.-elt x -> pi.-elt (x ^+ n).

Proof. by rewrite -{1}[x]expg1 !p_elt_exp dvdn1 => /eqnP->. Qed.

Lemma

p_eltX

solvable

solvable/pgroup.v

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

[ "dvdn1", "eqnP", "expg1", "p_elt_exp", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_eltJ pi x y : pi.-elt (x ^ y) = pi.-elt x.

Proof. by congr pnat; rewrite orderJ. Qed.

Lemma

p_eltJ

solvable

solvable/pgroup.v

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

[ "orderJ", "pi", "pnat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_p_elt pi1 pi2 x : {subset pi1 <= pi2} -> pi1.-elt x -> pi2.-elt x.

Proof. by move=> pi12; apply: sub_in_pnat => q _; apply: pi12. Qed.

Lemma

sub_p_elt

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", "sub_in_pnat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_p_elt pi1 pi2 x : pi1 =i pi2 -> pi1.-elt x = pi2.-elt x.

Proof. by move=> pi12; apply: eq_pnat. Qed.

Lemma

eq_p_elt

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_pnat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_eltNK pi x : pi^'^'.-elt x = pi.-elt x.

Proof. exact: pnatNK. Qed.

Lemma

p_eltNK

solvable

solvable/pgroup.v

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

[ "pi", "pnatNK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_constt pi1 pi2 x : pi1 =i pi2 -> x.`_pi1 = x.`_pi2.

Proof. move=> pi12; congr (x ^+ (chinese _ _ 1 0)); apply: eq_partn => // a. by congr (~~ _); apply: pi12. Qed.

Lemma

eq_constt

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", "chinese", "eq_partn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

consttNK pi x : x.`_pi^'^' = x.`_pi.

Proof. by rewrite /constt !partnNK. Qed.

Lemma

consttNK

solvable

solvable/pgroup.v

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

[ "constt", "partnNK", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle_constt pi x : x.`_pi \in <[x]>.

Proof. exact: mem_cycle. Qed.

Lemma

cycle_constt

solvable

solvable/pgroup.v

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

[ "mem_cycle", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

consttV pi x : (x^-1).`_pi = (x.`_pi)^-1.

Proof. by rewrite /constt expgVn orderV. Qed.

Lemma

consttV

solvable

solvable/pgroup.v

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

[ "constt", "expgVn", "orderV", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

constt1 pi : 1.`_pi = 1 :> gT.

Proof. exact: expg1n. Qed.

Lemma

constt1

solvable

solvable/pgroup.v

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

[ "expg1n", "gT", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

consttJ pi x y : (x ^ y).`_pi = x.`_pi ^ y.

Proof. by rewrite /constt orderJ conjXg. Qed.

Lemma

consttJ

solvable

solvable/pgroup.v

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

[ "conjXg", "constt", "orderJ", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_elt_constt pi x : pi.-elt x.`_pi.

Proof. by rewrite p_elt_exp /chinese addn0 mul1n dvdn_mulr. Qed.

Lemma

p_elt_constt

solvable

solvable/pgroup.v

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

[ "addn0", "chinese", "dvdn_mulr", "mul1n", "p_elt_exp", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

consttC pi x : x.`_pi * x.`_pi^' = x.

Proof. apply/eqP; rewrite -{3}[x]expg1 -expgD eq_expg_mod_order. rewrite partnNK -{5 6}(@partnC pi #[x]) // /chinese !addn0. by rewrite chinese_remainder ?chinese_modl ?chinese_modr ?coprime_partC ?eqxx. Qed.

Lemma

consttC

solvable

solvable/pgroup.v

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

[ "addn0", "apply", "chinese", "chinese_modl", "chinese_modr", "chinese_remainder", "coprime_partC", "eq_expg_mod_order", "eqxx", "expg1", "expgD", "partnC", "partnNK", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p'_elt_constt pi x : pi^'.-elt (x * (x.`_pi)^-1).

Proof. by rewrite -{1}(consttC pi^' x) consttNK mulgK p_elt_constt. Qed.

Lemma

p'_elt_constt

solvable

solvable/pgroup.v

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

[ "consttC", "consttNK", "mulgK", "p_elt_constt", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

order_constt pi (x : gT) : #[x.`_pi] = (#[x]`_pi)%N.

Proof. rewrite -{2}(consttC pi x) orderM; [exact: commuteX2| |]. by apply: (@pnat_coprime pi); apply: p_elt_constt. by rewrite partnM // part_pnat_id ?part_p'nat ?muln1 //; apply: p_elt_constt. Qed.

Lemma

order_constt

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", "commuteX2", "consttC", "gT", "muln1", "orderM", "p_elt_constt", "part_p'nat", "part_pnat_id", "partnM", "pi", "pnat_coprime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

consttM pi x y : commute x y -> (x * y).`_pi = x.`_pi * y.`_pi.

Proof. move=> cxy; pose m := #|<<[set x; y]>>|; have m_gt0: 0 < m := cardG_gt0 _. pose k := chinese m`_pi m`_pi^' 1 0. suffices kXpi z: z \in <<[set x; y]>> -> z.`_pi = z ^+ k. by rewrite !kXpi ?expgMn // ?groupM ?mem_gen // !inE eqxx ?orbT. move=> xyz; have{xyz} zm: #[z] %| m by rewrite cardSg ?cycle_subG. apply/eqP...

Lemma

consttM

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", "cardG_gt0", "cardSg", "chinese", "chinese_modl", "chinese_modr", "chinese_remainder", "commute", "coprime_partC", "cycle_subG", "eq_expg_mod_order", "eqxx", "expgMn", "groupM", "inE", "mem_gen", "modn_dvdm", "partnC", "partn_dvd", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

consttX pi x n : (x ^+ n).`_pi = x.`_pi ^+ n.

Proof. elim: n => [|n IHn]; first exact: constt1. by rewrite !expgS consttM ?IHn //; apply: commuteX. Qed.

Lemma

consttX

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", "commuteX", "constt1", "consttM", "expgS", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

constt1P pi x : reflect (x.`_pi = 1) (pi^'.-elt x).

Proof. rewrite -{2}[x]expg1 p_elt_exp -order_constt consttNK order_dvdn expg1. exact: eqP. Qed.

Lemma

constt1P

solvable

solvable/pgroup.v

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

[ "consttNK", "expg1", "order_constt", "order_dvdn", "p_elt_exp", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

constt_p_elt pi x : pi.-elt x -> x.`_pi = x.

Proof. by rewrite -p_eltNK -{3}(consttC pi x) => /constt1P->; rewrite mulg1. Qed.

Lemma

constt_p_elt

solvable

solvable/pgroup.v

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

[ "constt1P", "consttC", "mulg1", "p_eltNK", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_in_constt pi1 pi2 x : {in \pi(#[x]), {subset pi1 <= pi2}} -> x.`_pi2.`_pi1 = x.`_pi1.

Proof. move=> pi12; rewrite -{2}(consttC pi2 x) consttM; first exact: commuteX2. rewrite (constt1P _ x.`_pi2^' _) ?mulg1 //. apply: sub_in_pnat (p_elt_constt _ x) => p; rewrite order_constt => pi_p. by apply/contra/pi12; rewrite -[#[x]](partnC pi2^') // primesM // pi_p. Qed.

Lemma

sub_in_constt

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", "commuteX2", "constt1P", "consttC", "consttM", "mulg1", "order_constt", "p_elt_constt", "partnC", "pi", "pi_p", "primesM", "sub_in_pnat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_constt x : \prod_(0 <= p < #[x].+1) x.`_p = x.

Proof. pose lp n := [pred p | p < n]. have: (lp #[x].+1).-elt x by apply/pnatP=> // p _; apply: dvdn_leq. move/constt_p_elt=> def_x; symmetry; rewrite -{1}def_x {def_x}. elim: _.+1 => [|p IHp]. by rewrite big_nil; apply/constt1P; apply/pgroupP. rewrite big_nat_recr //= -{}IHp -(consttC (lp p) x.`__); congr (_ * _). ...

Lemma

prod_constt

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", "big_nat_recr", "big_nil", "constt1P", "consttC", "constt_p_elt", "dvdn_leq", "eqnP", "eqn_leq", "inE", "last", "leqNgt", "leqW", "ltnS", "ltnn", "mulg1", "order_constt", "p_elt", "part_pnat", "partnI", "pgroupP", "pnatNK", "pnatP", "sub_in_constt", "sub_in_p...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_pgroupJ pi M G x : x \in G -> [max M | pi.-subgroup(G) M] -> [max M :^ x of M | pi.-subgroup(G) M].

Proof. move=> Gx /maxgroupP[piM maxM]; apply/maxgroupP. split=> [|H piH]; first by rewrite psubgroupJ. by rewrite -(conjsgKV x H) conjSg => /maxM/=-> //; rewrite psubgroupJ ?groupV. Qed.

Lemma

max_pgroupJ

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", "conjSg", "conjsgKV", "groupV", "max", "maxgroupP", "pi", "psubgroupJ", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_sub_max_pgroup pi H M G : [max M | pi.-subgroup(G) M] -> pi.-group H -> H \subset G -> commute H M -> H \subset M.

Proof. case/maxgroupP=> /andP[sMG piM] maxM piH sHG cHM. rewrite -(maxM (H <*> M)%G) /= comm_joingE ?(mulG_subl, mulG_subr) //. by rewrite /psubgroup pgroupM piM piH mul_subG. Qed.

Lemma

comm_sub_max_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" ]

[ "comm_joingE", "commute", "group", "max", "maxgroupP", "mulG_subl", "mulG_subr", "mul_subG", "pgroupM", "pi", "psubgroup", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_sub_max_pgroup pi H M G : [max M | pi.-subgroup(G) M] -> pi.-group H -> H <| G -> H \subset M.

Proof. move=> maxM piH /andP[sHG nHG]. apply: comm_sub_max_pgroup piH sHG _ => //; apply: commute_sym; apply: normC. by apply: subset_trans nHG; case/andP: (maxgroupp maxM). Qed.

Lemma

normal_sub_max_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" ]

[ "apply", "comm_sub_max_pgroup", "commute_sym", "group", "max", "maxgroupp", "nHG", "normC", "pi", "sHG", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_sub_max_pgroup pi H M G : [max M | pi.-subgroup(G) M] -> pi.-group H -> H \subset G -> H \subset 'N(M) -> H \subset M.

Proof. by move=> maxM piH sHG /normC; apply: comm_sub_max_pgroup piH sHG. Qed.

Lemma

norm_sub_max_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" ]

[ "apply", "comm_sub_max_pgroup", "group", "max", "normC", "pi", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_pHall pi H G K : pi.-Hall(G) H -> pi.-group K -> H \subset K -> K \subset G -> K :=: H.

Proof. move=> hallH piK sHK sKG; apply/eqP; rewrite eq_sym eqEcard sHK. by rewrite (card_Hall hallH) -(part_pnat_id piK) dvdn_leq ?partn_dvd ?cardSg. Qed.

Lemma

sub_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", "apply", "cardSg", "card_Hall", "dvdn_leq", "eqEcard", "eq_sym", "group", "part_pnat_id", "partn_dvd", "pi", "piK", "sHK", "sKG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Hall_max pi H G : pi.-Hall(G) H -> [max H | pi.-subgroup(G) H].

Proof. move=> hallH; apply/maxgroupP; split=> [|K /andP[sKG piK] sHK]. by rewrite /psubgroup; case/and3P: hallH => ->. exact: (sub_pHall hallH). Qed.

Lemma

Hall_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" ]

[ "Hall", "apply", "max", "maxgroupP", "pi", "piK", "psubgroup", "sHK", "sKG", "split", "sub_pHall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pHall_id pi H G : pi.-Hall(G) H -> pi.-group G -> H :=: G.

Proof. by move=> hallH piG; rewrite (sub_pHall hallH piG) ?(pHall_sub hallH). Qed.

Lemma

pHall_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" ]

[ "Hall", "group", "pHall_sub", "pi", "sub_pHall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

psubgroup1 pi G : pi.-subgroup(G) 1.

Proof. by rewrite /psubgroup sub1G pgroup1. Qed.

Lemma

psubgroup1

solvable

solvable/pgroup.v

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

[ "pgroup1", "pi", "psubgroup", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Cauchy p G : prime p -> p %| #|G| -> {x | x \in G & #[x] = p}.

Proof. move=> p_pr; have [n] := ubnP #|G|; elim: n G => // n IHn G /ltnSE-leGn pG. pose xpG := [pred x in G | #[x] == p]. have [x /andP[Gx /eqP] | no_x] := pickP xpG; first by exists x. have{pG n leGn IHn} pZ: p %| #|'C_G(G)|. suffices /dvdn_addl <-: p %| #|G :\: 'C(G)| by rewrite cardsID. have /acts_sum_card_orbi...

Lemma

Cauchy

solvable

solvable/pgroup.v

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

[ "Euclid_dvdM", "LagrangeI", "actsP", "acts_sum_card_orbit", "apply", "big_rec", "cardSg", "cardsID", "centJ", "centsC", "cents_norm", "conjGid", "cycle_subG", "dvdnP", "dvdn_addl", "dvdn_gt0", "dvdn_mulr", "dvdn_trans", "gcdnC", "gcdnMr", "gen_subG", "groupV", "groupX", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_normal_Hall pi G H K : pi.-Hall(G) H -> H <| G -> K \subset G -> (K \subset H) = pi.-group K.

Proof. move=> hallH nsHG sKG; apply/idP/idP=> [sKH | piK]. by rewrite (pgroupS sKH) ?(pHall_pgroup hallH). apply: norm_sub_max_pgroup (Hall_max hallH) piK _ _ => //. exact: subset_trans sKG (normal_norm nsHG). Qed.

Lemma

sub_normal_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", "group", "norm_sub_max_pgroup", "normal_norm", "nsHG", "pHall_pgroup", "pgroupS", "pi", "piK", "sKG", "subset_trans" ]

derive from the Cauchy lemma that a normal max pi-group is Hall.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_normal_Hall pi H G x : pi.-Hall(G) H -> H <| G -> x \in G -> (x \in H) = pi.-elt x.

Proof. by rewrite -!cycle_subG; apply: sub_normal_Hall. Qed.

Lemma

mem_normal_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", "apply", "cycle_subG", "pi", "sub_normal_Hall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

uniq_normal_Hall pi H G K : pi.-Hall(G) H -> H <| G -> [max K | pi.-subgroup(G) K] -> K :=: H.

Proof. move=> hallH nHG /maxgroupP[/andP[sKG piK] /(_ H) -> //]. exact: (maxgroupp (Hall_max hallH)). by rewrite (sub_normal_Hall hallH). Qed.

Lemma

uniq_normal_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", "max", "maxgroupP", "maxgroupp", "nHG", "pi", "piK", "sKG", "sub_normal_Hall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_max_pgroup_Hall G H : [max H | pi.-subgroup(G) H] -> H <| G -> pi.-Hall(G) H.

Proof. case/maxgroupP=> /andP[sHG piH] maxH nsHG; have [_ nHG] := andP nsHG. rewrite /pHall sHG piH; apply/pnatP=> // p p_pr. rewrite inE /= -pnatE // -card_quotient //. case/Cauchy=> //= Hx; rewrite -sub1set -gen_subG -/<[Hx]> /order. case/inv_quotientS=> //= K -> sHK sKG {Hx}. rewrite card_quotient ?(subset_trans sKG...

Lemma

normal_max_pgroup_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" ]

[ "Cauchy", "Hall", "Lagrange", "apply", "cardG_gt0", "card_quotient", "divgS", "divnn", "gen_subG", "inE", "inv_quotientS", "max", "maxgroupP", "mulnC", "nHG", "nsHG", "order", "pHall", "p_pr", "pgroup", "pi", "pi_p", "pnatE", "pnatM", "pnatP", "psubgroup", "sHG", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setI_normal_Hall G H K : H <| G -> pi.-Hall(G) H -> K \subset G -> pi.-Hall(K) (H :&: K).

Proof. move=> nsHG hallH sKG; apply: normal_max_pgroup_Hall; last first. by rewrite /= setIC (normalGI sKG nsHG). apply/maxgroupP; split=> [|M /andP[sMK piM] sHK_M]. by rewrite /psubgroup subsetIr (pgroupS (subsetIl _ _) (pHall_pgroup hallH)). apply/eqP; rewrite eqEsubset sHK_M subsetI sMK !andbT. by rewrite (sub_n...

Lemma

setI_normal_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", "apply", "eqEsubset", "last", "maxgroupP", "normalGI", "normal_max_pgroup_Hall", "nsHG", "pHall_pgroup", "pgroupS", "pi", "psubgroup", "sKG", "setIC", "split", "sub_normal_Hall", "subsetI", "subsetIl", "subsetIr", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pgroup pi G : pi.-group G -> pi.-group (f @* G).

Proof. by apply: pnat_dvd; apply: dvdn_morphim. Qed.

Lemma

morphim_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" ]

[ "apply", "dvdn_morphim", "group", "pi", "pnat_dvd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_odd G : odd #|G| -> odd #|f @* G|.

Proof. by rewrite !odd_2'nat; apply: morphim_pgroup. Qed.

Lemma

morphim_odd

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", "morphim_pgroup", "odd", "odd_2'nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmorphim_pgroup pi G : pi.-group ('ker f) -> G \subset D -> pi.-group (f @* G) = pi.-group G.

Proof. move=> piker sGD; apply/idP/idP=> [pifG|]; last exact: morphim_pgroup. apply: (@pgroupS _ _ (f @*^-1 (f @* G))); first by rewrite -sub_morphim_pre. by rewrite /pgroup card_morphpre ?morphimS // pnatM; apply/andP. Qed.

Lemma

pmorphim_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" ]

[ "apply", "card_morphpre", "group", "ker", "last", "morphimS", "morphim_pgroup", "pgroup", "pgroupS", "pi", "pnatM", "sGD", "sub_morphim_pre" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_p_index pi G H : H \subset D -> pi.-nat #|G : H| -> pi.-nat #|f @* G : f @* H|.

Proof. by move=> sHD; apply: pnat_dvd; rewrite index_morphim ?subIset // sHD orbT. Qed.

Lemma

morphim_p_index

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", "index_morphim", "nat", "pi", "pnat_dvd", "sHD", "subIset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. move=> sHD /and3P[sHG piH pi'GH]. by rewrite /pHall morphimS // morphim_pgroup // morphim_p_index. Qed.

Lemma

morphim_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", "morphimS", "morphim_p_index", "morphim_pgroup", "pHall", "pi", "sHD", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmorphim_pHall pi G H : G \subset D -> H \subset D -> pi.-subgroup(H :&: G) ('ker f) -> pi.-Hall(f @* G) (f @* H) = pi.-Hall(G) H.

Proof. move=> sGD sHD /andP[/subsetIP[sKH sKG] piK]; rewrite !pHallE morphimSGK //. apply: andb_id2l => sHG; rewrite -(Lagrange sKH) -(Lagrange sKG) partnM //. by rewrite (part_pnat_id piK) !card_morphim !(setIidPr _) // eqn_pmul2l. Qed.

Lemma

pmorphim_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", "Lagrange", "apply", "card_morphim", "eqn_pmul2l", "ker", "morphimSGK", "pHallE", "part_pnat_id", "partnM", "pi", "piK", "sGD", "sHD", "sHG", "sKG", "setIidPr", "subsetIP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_Hall G H : H \subset D -> Hall G H -> Hall (f @* G) (f @* H).

Proof. by move=> sHD /HallP[pi piH]; apply: (@pHall_Hall _ pi); apply: morphim_pHall. Qed.

Lemma

morphim_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", "HallP", "apply", "morphim_pHall", "pHall_Hall", "pi", "sHD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pSylow p G P : P \subset D -> p.-Sylow(G) P -> p.-Sylow(f @* G) (f @* P).

Proof. exact: morphim_pHall. Qed.

Lemma

morphim_pSylow

solvable

solvable/pgroup.v

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

[ "Sylow", "morphim_pHall" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_p_group P : p_group P -> p_group (f @* P).

Proof. by move/morphim_pgroup; apply: pgroup_p. Qed.

Lemma

morphim_p_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" ]

[ "apply", "morphim_pgroup", "p_group", "pgroup_p" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_Sylow G P : P \subset D -> Sylow G P -> Sylow (f @* G) (f @* P).

Proof. by move=> sPD /andP[pP hallP]; rewrite /Sylow morphim_p_group // morphim_Hall. Qed.

Lemma

morphim_Sylow

solvable

solvable/pgroup.v

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

[ "Sylow", "morphim_Hall", "morphim_p_group", "pP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> Dx; apply: pnat_dvd; apply: morph_order. Qed.

Lemma

morph_p_elt

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", "apply", "morph_order", "pi", "pnat_dvd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_constt pi x : x \in D -> f x.`_pi = (f x).`_pi.

Proof. move=> Dx; rewrite -{2}(consttC pi x) morphM ?groupX //. rewrite consttM; first by rewrite !morphX //; apply: commuteX2. have: pi.-elt (f x.`_pi) by rewrite morph_p_elt ?groupX ?p_elt_constt //. have: pi^'.-elt (f x.`_pi^') by rewrite morph_p_elt ?groupX ?p_elt_constt //. by move/constt1P->; move/constt_p_elt->;...

Lemma

morph_constt

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", "apply", "commuteX2", "constt1P", "consttC", "consttM", "constt_p_elt", "groupX", "morphM", "morphX", "morph_p_elt", "mulg1", "p_elt_constt", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

piK : pi.-group K.

Hypothesis

piK

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", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_pgroup : pi.-group (K / H).

Proof. exact: morphim_pgroup. Qed.

Lemma

quotient_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", "morphim_pgroup", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_pHall : K \subset 'N(H) -> pi.-Hall(G) K -> pi.-Hall(G / H) (K / H).

Proof. exact: morphim_pHall. Qed.

Lemma

quotient_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", "morphim_pHall", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_odd : odd #|K| -> odd #|K / H|.

Proof. exact: morphim_odd. Qed.

Lemma

quotient_odd

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_odd", "odd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d