phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	source_url stringclasses 1 value	commit stringclasses 1 value
exponent_cyclic X : cyclic X -> exponent X = #\|X\|.	Proof. by case/cyclicP=> x ->; apply: exponent_cycle. Qed.	Lemma	exponent_cyclic	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "cyclic", "cyclicP", "exponent", "exponent_cycle" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primes_exponent G : primes (exponent G) = primes (#\|G\|).	Proof. apply/eq_primes => p; rewrite !mem_primes exponent_gt0 cardG_gt0 /=. by apply: andb_id2l => p_pr; apply: negb_inj; rewrite -!p'natE // pnat_exponent. Qed.	Lemma	primes_exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "cardG_gt0", "eq_primes", "exponent", "exponent_gt0", "mem_primes", "p'natE", "p_pr", "pnat_exponent", "primes" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_of_exponent G : \pi(exponent G) = \pi(G).	Proof. by rewrite /pi_of primes_exponent. Qed.	Lemma	pi_of_exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "exponent", "pi", "pi_of", "primes_exponent" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
partn_exponentS pi H G : H \subset G -> #\|G\|`_pi %\| #\|H\| -> ((exponent H)`_pi = (exponent G)`_pi)%N.	Proof. move=> sHG Gpi_dvd_H; apply/eqP; rewrite eqn_dvd. rewrite partn_dvd ?exponentS ?exponent_gt0 //=; apply/dvdn_partP=> // p. rewrite pi_of_part ?exponent_gt0 // => /andP[_ /= pi_p]. have sppi: {subset (p : nat_pred) <= pi} by move=> q /eqnP->. have [P sylP] := Sylow_exists p H; have sPH := pHall_sub sylP. have{} s...	Lemma	partn_exponentS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Sylow", "Sylow_Jsub", "Sylow_exists", "apply", "bigD1", "big_ind", "cardG_gt0", "cardSg", "card_Hall", "cycleJ", "cycle_subG", "dvd1n", "dvdn", "dvdn_lcm", "dvdn_lcml", "dvdn_partP", "eqnP", "eqn_dvd", "exponent", "exponentS", "exponent_gt0", "groupX", "nat_pred", "ord...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_Hall pi G H : pi.-Hall(G) H -> exponent H = ((exponent G)`_pi)%N.	Proof. move=> hallH; have [sHG piH _] := and3P hallH. rewrite -(partn_exponentS sHG) -?(card_Hall hallH) ?part_pnat_id //. by apply: pnat_dvd piH; apply: exponent_dvdn. Qed.	Lemma	exponent_Hall	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Hall", "apply", "card_Hall", "exponent", "exponent_dvdn", "part_pnat_id", "partn_exponentS", "pi", "pnat_dvd", "sHG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_Zgroup G : Zgroup G -> exponent G = #\|G\|.	Proof. move/forall_inP=> ZgG; apply/eqP; rewrite eqn_dvd exponent_dvdn. apply/(dvdn_partP _ (cardG_gt0 _)) => p _. have [S sylS] := Sylow_exists p G; rewrite -(card_Hall sylS). have /cyclicP[x defS]: cyclic S by rewrite ZgG ?(p_Sylow sylS). by rewrite defS dvdn_exponent // -cycle_subG -defS (pHall_sub sylS). Qed.	Lemma	exponent_Zgroup	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Sylow_exists", "Zgroup", "apply", "cardG_gt0", "card_Hall", "cycle_subG", "cyclic", "cyclicP", "dvdn_exponent", "dvdn_partP", "eqn_dvd", "exponent", "exponent_dvdn", "forall_inP", "pHall_sub", "p_Sylow" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprod_exponent A B G : A \* B = G -> lcmn (exponent A) (exponent B) = (exponent G).	Proof. case/cprodP=> [[K H -> ->{A B}] <- cKH]. apply/eqP; rewrite eqn_dvd dvdn_lcm !exponentS ?mulG_subl ?mulG_subr //=. apply/exponentP=> _ /imset2P[x y Kx Hy ->]. rewrite -[1]mulg1 expgMn; first by red; rewrite -(centsP cKH). congr (_ * _); apply/eqP; rewrite -order_dvdn. by rewrite (dvdn_trans (dvdn_exponent Kx))...	Lemma	cprod_exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "cKH", "centsP", "cprodP", "dvdn_exponent", "dvdn_lcm", "dvdn_lcml", "dvdn_lcmr", "dvdn_trans", "eqn_dvd", "expgMn", "exponent", "exponentP", "exponentS", "imset2P", "lcmn", "mulG_subl", "mulG_subr", "mulg1", "order_dvdn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dprod_exponent A B G : A \x B = G -> lcmn (exponent A) (exponent B) = (exponent G).	Proof. case/dprodP=> [[K H -> ->{A B}] defG cKH _]. by apply: cprod_exponent; rewrite cprodE. Qed.	Lemma	dprod_exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "cKH", "cprodE", "cprod_exponent", "defG", "dprodP", "exponent", "lcmn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_LdivT A n : (A \subset 'Ldiv_n()) = (exponent A %\| n).	Proof. by apply/subsetP/exponentP=> eAn x /eAn /[1!inE] /eqP. Qed.	Lemma	sub_LdivT	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "exponent", "exponentP", "inE", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
LdivT_J n x : 'Ldiv_n() :^ x = 'Ldiv_n().	Proof. apply/setP=> y; rewrite !inE mem_conjg inE -conjXg. by rewrite (canF_eq (conjgKV x)) conj1g. Qed.	Lemma	LdivT_J	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "canF_eq", "conj1g", "conjXg", "conjgKV", "inE", "mem_conjg", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
LdivJ n A x : 'Ldiv_n(A :^ x) = 'Ldiv_n(A) :^ x.	Proof. by rewrite conjIg LdivT_J. Qed.	Lemma	LdivJ	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "LdivT_J", "conjIg" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_Ldiv A n : (A \subset 'Ldiv_n(A)) = (exponent A %\| n).	Proof. by rewrite subsetI subxx sub_LdivT. Qed.	Lemma	sub_Ldiv	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "exponent", "sub_LdivT", "subsetI", "subxx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_Ldiv G n : abelian G -> group_set 'Ldiv_n(G).	Proof. move=> cGG; apply/group_setP. split=> [\|x y]; rewrite !inE ?group1 ?expg1n //=. case/andP=> Gx /eqP xn /andP[Gy /eqP yn]. by rewrite groupM //= expgMn ?xn ?yn ?mulg1 //; apply: (centsP cGG). Qed.	Lemma	group_Ldiv	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelian", "apply", "cGG", "centsP", "expg1n", "expgMn", "group1", "groupM", "group_set", "group_setP", "inE", "mulg1", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_exponent_gen A : abelian A -> exponent <<A>> = exponent A.	Proof. rewrite -abelian_gen; set n := exponent A; set G := <<A>> => cGG. apply/eqP; rewrite eqn_dvd andbC exponentS ?subset_gen //= -sub_Ldiv. rewrite -(gen_set_id (group_Ldiv n cGG)) genS // subsetI subset_gen /=. by rewrite sub_LdivT. Qed.	Lemma	abelian_exponent_gen	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelian", "abelian_gen", "apply", "cGG", "eqn_dvd", "exponent", "exponentS", "genS", "gen_set_id", "group_Ldiv", "sub_Ldiv", "sub_LdivT", "subsetI", "subset_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_pgroup p A : p.-abelem A -> p.-group A.	Proof. by case/andP. Qed.	Lemma	abelem_pgroup	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_abelian p A : p.-abelem A -> abelian A.	Proof. by case/and3P. Qed.	Lemma	abelem_abelian	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelian" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem1 p : p.-abelem [1 gT].	Proof. by rewrite /abelem pgroup1 abelian1 exponent1 dvd1n. Qed.	Lemma	abelem1	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelian1", "dvd1n", "exponent1", "gT", "pgroup1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelemE p G : prime p -> p.-abelem G = abelian G && (exponent G %\| p).	Proof. move=> p_pr; rewrite /abelem -pnat_exponent andbA -!(andbC (_ %\| _)). by case: (dvdn_pfactor _ 1 p_pr) => // [[k _ ->]]; rewrite pnatX pnat_id. Qed.	Lemma	abelemE	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelian", "dvdn_pfactor", "exponent", "p_pr", "pnatX", "pnat_exponent", "pnat_id", "prime" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelemP p G : prime p -> reflect (abelian G /\ forall x, x \in G -> x ^+ p = 1) (p.-abelem G).	Proof. by move=> p_pr; rewrite abelemE //; apply: (iffP andP) => [] [-> /exponentP]. Qed.	Lemma	abelemP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelemE", "abelian", "apply", "exponentP", "p_pr", "prime" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_order_p p G x : p.-abelem G -> x \in G -> x != 1 -> #[x] = p.	Proof. case/and3P=> pG _ eG Gx; rewrite -cycle_eq1 => ntX. have{ntX} [p_pr p_x _] := pgroup_pdiv (mem_p_elt pG Gx) ntX. by apply/eqP; rewrite eqn_dvd p_x andbT order_dvdn (exponentP eG). Qed.	Lemma	abelem_order_p	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "apply", "cycle_eq1", "eqn_dvd", "exponentP", "mem_p_elt", "order_dvdn", "pG", "p_pr", "pgroup_pdiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cyclic_abelem_prime p X : p.-abelem X -> cyclic X -> X :!=: 1 -> #\|X\| = p.	Proof. move=> abelX cycX; case/cyclicP: cycX => x -> in abelX *. by rewrite cycle_eq1; apply: abelem_order_p abelX (cycle_id x). Qed.	Lemma	cyclic_abelem_prime	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelem_order_p", "apply", "cycle_eq1", "cycle_id", "cyclic", "cyclicP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle_abelem p x : p.-elt x \|\| prime p -> p.-abelem <[x]> = (#[x] %\| p).	Proof. move=> p_xVpr; rewrite /abelem cycle_abelian /=. apply/andP/idP=> [[_ xp1] \| x_dvd_p]. by rewrite order_dvdn (exponentP xp1) ?cycle_id. split; last exact: dvdn_trans (exponent_dvdn _) x_dvd_p. by case/orP: p_xVpr => // /pnat_id; apply: pnat_dvd. Qed.	Lemma	cycle_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "apply", "cycle_abelian", "cycle_id", "dvdn_trans", "exponentP", "exponent_dvdn", "last", "order_dvdn", "pnat_dvd", "pnat_id", "prime", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent2_abelem G : exponent G %\| 2 -> 2.-abelem G.	Proof. move/exponentP=> expG; apply/abelemP=> //; split=> //. apply/centsP=> x Gx y Gy; apply: (mulIg x); apply: (mulgI y). by rewrite -!mulgA !(mulgA y) -!(expgS _ 1) !expG ?mulg1 ?groupM. Qed.	Lemma	exponent2_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelemP", "apply", "centsP", "expgS", "exponent", "exponentP", "groupM", "mulIg", "mulg1", "mulgA", "mulgI", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_abelem p G : prime p -> #\|G\| = p -> p.-abelem G.	Proof. move=> p_pr oG; rewrite /abelem -oG exponent_dvdn. by rewrite /pgroup cyclic_abelian ?prime_cyclic ?oG ?pnat_id. Qed.	Lemma	prime_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "cyclic_abelian", "exponent_dvdn", "p_pr", "pgroup", "pnat_id", "prime", "prime_cyclic" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_cyclic p G : p.-abelem G -> cyclic G = (logn p #\|G\| <= 1).	Proof. move=> abelG; have [pG _ expGp] := and3P abelG. case: (eqsVneq G 1) => [-> \| ntG]; first by rewrite cyclic1 cards1 logn1. have [p_pr _ [e oG]] := pgroup_pdiv pG ntG; apply/idP/idP. case/cyclicP=> x defG; rewrite -(pfactorK 1 p_pr) dvdn_leq_log ?prime_gt0 //. by rewrite defG order_dvdn (exponentP expGp) // de...	Lemma	abelem_cyclic	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "apply", "cards1", "cycle_id", "cyclic", "cyclic1", "cyclicP", "defG", "dvdn_leq_log", "e0", "eqsVneq", "exponentP", "leqn0", "logn", "logn1", "ltnS", "order_dvdn", "pG", "p_pr", "pfactorK", "pgroup_pdiv", "prime_cyclic", "prime_gt0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelemS p H G : H \subset G -> p.-abelem G -> p.-abelem H.	Proof. move=> sHG /and3P[cGG pG Gp1]; rewrite /abelem. by rewrite (pgroupS sHG) // (abelianS sHG) // (dvdn_trans (exponentS sHG)). Qed.	Lemma	abelemS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelianS", "cGG", "dvdn_trans", "exponentS", "pG", "pgroupS", "sHG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelemJ p G x : p.-abelem (G :^ x) = p.-abelem G.	Proof. by rewrite /abelem pgroupJ abelianJ exponentJ. Qed.	Lemma	abelemJ	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelianJ", "exponentJ", "pgroupJ" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprod_abelem p A B G : A \* B = G -> p.-abelem G = p.-abelem A && p.-abelem B.	Proof. case/cprodP=> [[H K -> ->{A B}] defG cHK]. apply/idP/andP=> [abelG \| []]. by rewrite !(abelemS _ abelG) // -defG (mulG_subl, mulG_subr). case/and3P=> pH cHH expHp; case/and3P=> pK cKK expKp. rewrite -defG /abelem pgroupM pH pK abelianM cHH cKK cHK /=. apply/exponentP=> _ /imset2P[x y Hx Ky ->]. rewrite expgMn;...	Lemma	cprod_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelemS", "abelianM", "apply", "centsP", "cprodP", "defG", "expgMn", "exponentP", "imset2P", "mul1g", "mulG_subl", "mulG_subr", "pgroupM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dprod_abelem p A B G : A \x B = G -> p.-abelem G = p.-abelem A && p.-abelem B.	Proof. move=> defG; case/dprodP: (defG) => _ _ _ tiHK. by apply: cprod_abelem; rewrite -dprodEcp. Qed.	Lemma	dprod_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "apply", "cprod_abelem", "defG", "dprodEcp", "dprodP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_abelem_pgroup p G : p.-group G -> is_abelem G = p.-abelem G.	Proof. rewrite /is_abelem => pG. case: (eqsVneq G 1) => [-> \| ntG]; first by rewrite !abelem1. by have [p_pr _ [k ->]] := pgroup_pdiv pG ntG; rewrite pdiv_pfactor. Qed.	Lemma	is_abelem_pgroup	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelem1", "eqsVneq", "group", "is_abelem", "pG", "p_pr", "pdiv_pfactor", "pgroup_pdiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_abelemP G : reflect (exists2 p, prime p & p.-abelem G) (is_abelem G).	Proof. apply: (iffP idP) => [abelG \| [p p_pr abelG]]. case: (eqsVneq G 1) => [-> \| ntG]; first by exists 2; rewrite ?abelem1. by exists (pdiv #\|G\|); rewrite ?pdiv_prime // ltnNge -trivg_card_le1. by rewrite (is_abelem_pgroup (abelem_pgroup abelG)). Qed.	Lemma	is_abelemP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelem1", "abelem_pgroup", "apply", "eqsVneq", "is_abelem", "is_abelem_pgroup", "ltnNge", "p_pr", "pdiv", "pdiv_prime", "prime", "trivg_card_le1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pElemP p A E : reflect (E \subset A /\ p.-abelem E) (E \in 'E_p(A)).	Proof. by rewrite inE; apply: andP. Qed.	Lemma	pElemP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "apply", "inE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pElemS p A B : A \subset B -> 'E_p(A) \subset 'E_p(B).	Proof. by move=> sAB; apply/subsetP=> E /[!inE] /andP[/subset_trans->]. Qed.	Lemma	pElemS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "inE", "subsetP", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pElemI p A B : 'E_p(A :&: B) = 'E_p(A) :&: subgroups B.	Proof. by apply/setP=> E; rewrite !inE subsetI andbAC. Qed.	Lemma	pElemI	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "inE", "setP", "subgroups", "subsetI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pElemJ x p A E : ((E :^ x)%G \in 'E_p(A :^ x)) = (E \in 'E_p(A)).	Proof. by rewrite !inE conjSg abelemJ. Qed.	Lemma	pElemJ	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelemJ", "conjSg", "inE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElemP p n A E : reflect [/\ E \subset A, p.-abelem E & logn p #\|E\| = n] (E \in 'E_p^n(A)).	Proof. by rewrite !inE -andbA; apply: (iffP and3P) => [] [-> -> /eqP]. Qed.	Lemma	pnElemP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "apply", "inE", "logn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElemPcard p n A E : E \in 'E_p^n(A) -> [/\ E \subset A, p.-abelem E & #\|E\| = p ^ n]%N.	Proof. by case/pnElemP=> -> abelE <-; rewrite -card_pgroup // abelem_pgroup. Qed.	Lemma	pnElemPcard	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelE", "abelem", "abelem_pgroup", "card_pgroup", "pnElemP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_pnElem p n A E : E \in 'E_p^n(A) -> #\|E\| = (p ^ n)%N.	Proof. by case/pnElemPcard. Qed.	Lemma	card_pnElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "pnElemPcard" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElem0 p G : 'E_p^0(G) = [set 1%G].	Proof. apply/setP=> E; rewrite !inE -andbA; apply/and3P/idP=> [[_ pE] \| /eqP->]. apply: contraLR; case/(pgroup_pdiv (abelem_pgroup pE)) => p_pr _ [k ->]. by rewrite pfactorK. by rewrite sub1G abelem1 cards1 logn1. Qed.	Lemma	pnElem0	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem1", "abelem_pgroup", "apply", "cards1", "inE", "logn1", "pE", "p_pr", "pfactorK", "pgroup_pdiv", "setP", "sub1G" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElem_prime p n A E : E \in 'E_p^n.+1(A) -> prime p.	Proof. by case/pnElemP=> _ _; rewrite lognE; case: prime. Qed.	Lemma	pnElem_prime	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "lognE", "pnElemP", "prime" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElemE p n A : prime p -> 'E_p^n(A) = [set E in 'E_p(A) \| #\|E\| == (p ^ n)%N].	Proof. move/pfactorK=> pnK; apply/setP=> E; rewrite 3!inE. case: (@andP (E \subset A)) => //= [[_]] /andP[/p_natP[k ->] _]. by rewrite pnK (can_eq pnK). Qed.	Lemma	pnElemE	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "can_eq", "inE", "p_natP", "pfactorK", "prime", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElemS p n A B : A \subset B -> 'E_p^n(A) \subset 'E_p^n(B).	Proof. move=> sAB; apply/subsetP=> E. by rewrite !inE -!andbA => /andP[/subset_trans->]. Qed.	Lemma	pnElemS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "inE", "subsetP", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElemI p n A B : 'E_p^n(A :&: B) = 'E_p^n(A) :&: subgroups B.	Proof. by apply/setP=> E; rewrite !inE subsetI -!andbA; do !bool_congr. Qed.	Lemma	pnElemI	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "inE", "setP", "subgroups", "subsetI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElemJ x p n A E : ((E :^ x)%G \in 'E_p^n(A :^ x)) = (E \in 'E_p^n(A)).	Proof. by rewrite inE pElemJ cardJg !inE. Qed.	Lemma	pnElemJ	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "cardJg", "inE", "pElemJ" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_pnElem p n G : p.-abelem G -> n <= logn p #\|G\| -> exists E, E \in 'E_p^n(G).	Proof. case: n => [\|n] abelG lt_nG; first by exists 1%G; rewrite pnElem0 set11. have p_pr: prime p by move: lt_nG; rewrite lognE; case: prime. case/(normal_pgroup (abelem_pgroup abelG)): lt_nG => // E [sEG _ oE]. by exists E; rewrite pnElemE // !inE oE sEG (abelemS sEG) /=. Qed.	Lemma	abelem_pnElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelemS", "abelem_pgroup", "inE", "logn", "lognE", "normal_pgroup", "p_pr", "pnElem0", "pnElemE", "prime", "set11" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_p1Elem p A X : X \in 'E_p^1(A) -> #\|X\| = p.	Proof. exact: card_pnElem. Qed.	Lemma	card_p1Elem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "card_pnElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p1ElemE p A : prime p -> 'E_p^1(A) = [set X in subgroups A \| #\|X\| == p].	Proof. move=> p_pr; apply/setP=> X; rewrite pnElemE // !inE -andbA; congr (_ && _). by apply: andb_idl => /eqP oX; rewrite prime_abelem ?oX. Qed.	Lemma	p1ElemE	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "inE", "p_pr", "pnElemE", "prime", "prime_abelem", "setP", "subgroups" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
TIp1ElemP p A X Y : X \in 'E_p^1(A) -> Y \in 'E_p^1(A) -> reflect (X :&: Y = 1) (X :!=: Y).	Proof. move=> EpX EpY; have p_pr := pnElem_prime EpX. have [oX oY] := (card_p1Elem EpX, card_p1Elem EpY). have [<-\|] := eqVneq. by right=> X1; rewrite -oX -(setIid X) X1 cards1 in p_pr. by rewrite eqEcard oX oY leqnn andbT; left; rewrite prime_TIg ?oX. Qed.	Lemma	TIp1ElemP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "card_p1Elem", "cards1", "eqEcard", "eqVneq", "leqnn", "p_pr", "pnElem_prime", "prime_TIg", "setIid" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_p1Elem_pnElem p n A E : E \in 'E_p^n(A) -> #\|'E_p^1(E)\| = (\sum_(i < n) p ^ i)%N.	Proof. case/pnElemP=> _ {A} abelE dimE; have [pE cEE _] := and3P abelE. have [E1 \| ntE] := eqsVneq E 1. rewrite -dimE E1 cards1 logn1 big_ord0 eq_card0 // => X. by rewrite !inE subG1 trivg_card1; case: eqP => // ->; rewrite logn1 andbF. have [p_pr _ _] := pgroup_pdiv pE ntE; have p_gt1 := prime_gt1 p_pr. apply/eqP;...	Lemma	card_p1Elem_pnElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "TIp1ElemP", "X'", "abelE", "abelem_order_p", "apply", "big_ord0", "bigcupP", "bigcupsP", "card_imset", "card_p1Elem", "card_pgroup", "card_uniform_partition", "cards0", "cards1", "cardsD1", "cycle_id", "cycle_subG", "eqEsubset", "eq_card0", "eq_sym", "eqn_pmul2l", "eqsVneq...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_p1Elem_p2Elem p A E : E \in 'E_p^2(A) -> #\|'E_p^1(E)\| = p.+1.	Proof. by move/card_p1Elem_pnElem->; rewrite big_ord_recl big_ord1. Qed.	Lemma	card_p1Elem_p2Elem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "big_ord1", "big_ord_recl", "card_p1Elem_pnElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p2Elem_dprodP p A E X Y : E \in 'E_p^2(A) -> X \in 'E_p^1(E) -> Y \in 'E_p^1(E) -> reflect (X \x Y = E) (X :!=: Y).	Proof. move=> Ep2E EpX EpY; have [_ abelE oE] := pnElemPcard Ep2E. apply: (iffP (TIp1ElemP EpX EpY)) => [tiXY\|]; last by case/dprodP. have [[sXE _ oX] [sYE _ oY]] := (pnElemPcard EpX, pnElemPcard EpY). rewrite dprodE ?(sub_abelian_cent2 (abelem_abelian abelE)) //. by apply/eqP; rewrite eqEcard mul_subG //= TI_cardMg //...	Lemma	p2Elem_dprodP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "TI_cardMg", "TIp1ElemP", "abelE", "abelem_abelian", "apply", "dprodE", "dprodP", "eqEcard", "last", "mul_subG", "pnElemPcard", "sub_abelian_cent2" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nElemP n G E : reflect (exists p, E \in 'E_p^n(G)) (E \in 'E^n(G)).	Proof. rewrite ['E^n(G)]big_mkord. apply: (iffP bigcupP) => [[[p /= _] _] \| [p]]; first by exists p. case: n => [\|n EpnE]; first by rewrite pnElem0; exists ord0; rewrite ?pnElem0. suffices lepG: p < #\|G\|.+1 by exists (Ordinal lepG). have:= EpnE; rewrite pnElemE ?(pnElem_prime EpnE) // !inE -andbA ltnS. case/and3P=> sE...	Lemma	nElemP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "big_mkord", "bigcupP", "cardSg", "dvdn_exp", "dvdn_leq", "dvdn_trans", "inE", "ltnS", "ord0", "pnElem0", "pnElemE", "pnElem_prime" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nElem0 G : 'E^0(G) = [set 1%G].	Proof. apply/setP=> E; apply/nElemP/idP=> [[p] \|]; first by rewrite pnElem0. by exists 2; rewrite pnElem0. Qed.	Lemma	nElem0	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "nElemP", "pnElem0", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nElem1P G E : reflect (E \subset G /\ exists2 p, prime p & #\|E\| = p) (E \in 'E^1(G)).	Proof. apply: (iffP nElemP) => [[p pE] \| [sEG [p p_pr oE]]]. have p_pr := pnElem_prime pE; rewrite pnElemE // !inE -andbA in pE. by case/and3P: pE => -> _ /eqP; split; last exists p. exists p; rewrite pnElemE // !inE sEG oE eqxx abelemE // -oE exponent_dvdn. by rewrite cyclic_abelian // prime_cyclic // oE. Qed.	Lemma	nElem1P	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelemE", "apply", "cyclic_abelian", "eqxx", "exponent_dvdn", "inE", "last", "nElemP", "pE", "p_pr", "pnElemE", "pnElem_prime", "prime", "prime_cyclic", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nElemS n G H : G \subset H -> 'E^n(G) \subset 'E^n(H).	Proof. move=> sGH; apply/subsetP=> E /nElemP[p EpnG_E]. by apply/nElemP; exists p; rewrite // (subsetP (pnElemS _ _ sGH)). Qed.	Lemma	nElemS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "nElemP", "pnElemS", "sGH", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nElemI n G H : 'E^n(G :&: H) = 'E^n(G) :&: subgroups H.	Proof. apply/setP=> E; apply/nElemP/setIP=> [[p] \| []]. by rewrite pnElemI; case/setIP; split=> //; apply/nElemP; exists p. by case/nElemP=> p EpnG_E sHE; exists p; rewrite pnElemI inE EpnG_E. Qed.	Lemma	nElemI	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "inE", "nElemP", "pnElemI", "setIP", "setP", "split", "subgroups" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
def_pnElem p n G : 'E_p^n(G) = 'E_p(G) :&: 'E^n(G).	Proof. apply/setP=> E; rewrite inE in_setI; apply: andb_id2l => /pElemP[sEG abelE]. apply/idP/nElemP=> [\|[q]]; first by exists p; rewrite !inE sEG abelE. rewrite !inE -2!andbA => /and4P[_ /pgroupP qE _]. have [->\|] := eqVneq E 1%G; first by rewrite cards1 !logn1. case/(pgroup_pdiv (abelem_pgroup abelE)) => p_pr pE _. b...	Lemma	def_pnElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelE", "abelem_pgroup", "apply", "cards1", "eqVneq", "eqnP", "inE", "in_setI", "logn1", "nElemP", "pE", "pElemP", "p_pr", "pgroupP", "pgroup_pdiv", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmaxElemP p A E : reflect (E \in 'E_p(A) /\ forall H, H \in 'E_p(A) -> E \subset H -> H :=: E) (E \in 'E*_p(A)).	Proof. by rewrite [E \in 'E*_p(A)]inE; apply: (iffP maxgroupP). Qed.	Lemma	pmaxElemP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "inE", "maxgroupP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmaxElem_exists p A D : D \in 'E_p(A) -> {E \| E \in 'E*_p(A) & D \subset E}.	Proof. move=> EpD; have [E maxE sDE] := maxgroup_exists (EpD : mem 'E_p(A) D). by exists E; rewrite // inE. Qed.	Lemma	pmaxElem_exists	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "inE", "maxgroup_exists" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmaxElem_LdivP p G E : prime p -> reflect ('Ldiv_p('C_G(E)) = E) (E \in 'E*_p(G)).	Proof. move=> p_pr; apply: (iffP (pmaxElemP p G E)) => [[] \| defE]. case/pElemP=> sEG abelE maxE; have [_ cEE eE] := and3P abelE. apply/setP=> x; rewrite !inE -andbA; apply/and3P/idP=> [[Gx cEx xp] \| Ex]. rewrite -(maxE (<[x]> <*> E)%G) ?joing_subr //; last first. by rewrite -cycle_subG joing_subl. re...	Lemma	pmaxElem_LdivP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelE", "abelemP", "abelian", "apply", "centsC", "centsS", "cprodEY", "cprod_abelem", "cycle_abelem", "cycle_subG", "eqEsubset", "exponentP", "inE", "join_subG", "joing_subl", "joing_subr", "last", "order_dvdn", "pElemP", "p_pr", "pmaxElemP", "prime", "sHG", "setIA", ...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmaxElemS p A B : A \subset B -> 'E_p(B) :&: subgroups A \subset 'E_p(A).	Proof. move=> sAB; apply/subsetP=> E /[!inE]. case/andP=> /maxgroupP[/pElemP[_ abelE] maxE] sEA. apply/maxgroupP; rewrite inE sEA; split=> // D EpD. by apply: maxE; apply: subsetP EpD; apply: pElemS. Qed.	Lemma	pmaxElemS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelE", "apply", "inE", "maxgroupP", "pElemP", "pElemS", "split", "subgroups", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmaxElemJ p A E x : ((E :^ x)%G \in 'E_p(A :^ x)) = (E \in 'E_p(A)).	Proof. apply/pmaxElemP/pmaxElemP=> [] [EpE maxE]. rewrite pElemJ in EpE; split=> //= H EpH sEH; apply: (act_inj 'Js x). by apply: maxE; rewrite ?conjSg ?pElemJ. rewrite pElemJ; split=> // H; rewrite -(actKV 'JG x H) pElemJ conjSg => EpHx'. by move/maxE=> /= ->. Qed.	Lemma	pmaxElemJ	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "actKV", "act_inj", "apply", "conjSg", "pElemJ", "pmaxElemP", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
grank_min B : 'm(<<B>>) <= #\|B\|.	Proof. by rewrite /gen_rank; case: arg_minnP => [\|_ _ -> //]; rewrite genGid. Qed.	Lemma	grank_min	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "arg_minnP", "genGid", "gen_rank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
grank_witness G : {B \| <<B>> = G & #\|B\| = 'm(G)}.	Proof. rewrite /gen_rank; case: arg_minnP => [\|B defG _]; first by rewrite genGid. by exists B; first apply/eqP. Qed.	Lemma	grank_witness	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "arg_minnP", "defG", "genGid", "gen_rank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_witness p G : {E \| E \in 'E_p^('r_p(G))(G)}.	Proof. have [E EG_E mE]: {E \| E \in 'E_p(G) & 'r_p(G) = logn p #\|E\| }. by apply: eq_bigmax_cond; rewrite (cardD1 1%G) inE sub1G abelem1. by exists E; rewrite inE EG_E -mE /=. Qed.	Lemma	p_rank_witness	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem1", "apply", "cardD1", "eq_bigmax_cond", "inE", "logn", "sub1G" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_geP p n G : reflect (exists E, E \in 'E_p^n(G)) (n <= 'r_p(G)).	Proof. apply: (iffP idP) => [\|[E]]; last first. by rewrite inE => /andP[Ep_E /eqP <-]; rewrite (bigmax_sup E). have [D /pnElemP[sDG abelD <-]] := p_rank_witness p G. by case/abelem_pnElem=> // E; exists E; apply: (subsetP (pnElemS _ _ sDG)). Qed.	Lemma	p_rank_geP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem_pnElem", "apply", "bigmax_sup", "inE", "last", "p_rank_witness", "pnElemP", "pnElemS", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_gt0 p H : ('r_p(H) > 0) = (p \in \pi(H)).	Proof. rewrite mem_primes cardG_gt0 /=; apply/p_rank_geP/andP=> [[E] \| [p_pr]]. case/pnElemP=> sEG _; rewrite lognE; case: and3P => // [[-> _ pE] _]. by rewrite (dvdn_trans _ (cardSg sEG)). case/Cauchy=> // x Hx ox; exists <[x]>%G; rewrite 2!inE [#\|_\|]ox cycle_subG. by rewrite Hx (pfactorK 1) ?abelemE // cycle_abel...	Lemma	p_rank_gt0	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Cauchy", "abelemE", "apply", "cardG_gt0", "cardSg", "cycle_abelian", "cycle_subG", "dvdn_trans", "exponent_dvdn", "inE", "lognE", "mem_primes", "pE", "p_pr", "p_rank_geP", "pfactorK", "pi", "pnElemP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank1 p : 'r_p([1 gT]) = 0.	Proof. by apply/eqP; rewrite eqn0Ngt p_rank_gt0 /= cards1. Qed.	Lemma	p_rank1	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "cards1", "eqn0Ngt", "gT", "p_rank_gt0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn_le_p_rank p A E : E \in 'E_p(A) -> logn p #\|E\| <= 'r_p(A).	Proof. by move=> EpA_E; rewrite (bigmax_sup E). Qed.	Lemma	logn_le_p_rank	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "bigmax_sup", "logn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_le_logn p G : 'r_p(G) <= logn p #\|G\|.	Proof. have [E EpE] := p_rank_witness p G. by have [sEG _ <-] := pnElemP EpE; apply: lognSg. Qed.	Lemma	p_rank_le_logn	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "logn", "lognSg", "p_rank_witness", "pnElemP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_abelem p G : p.-abelem G -> 'r_p(G) = logn p #\|G\|.	Proof. move=> abelG; apply/eqP; rewrite eqn_leq andbC (bigmax_sup G)//. by rewrite inE subxx. by apply/bigmax_leqP=> E /[1!inE] /andP[/lognSg->]. Qed.	Lemma	p_rank_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "apply", "bigmax_leqP", "bigmax_sup", "eqn_leq", "inE", "logn", "lognSg", "subxx" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rankS p A B : A \subset B -> 'r_p(A) <= 'r_p(B).	Proof. move=> sAB; apply/bigmax_leqP=> E /(subsetP (pElemS p sAB)) EpB_E. by rewrite (bigmax_sup E). Qed.	Lemma	p_rankS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "bigmax_leqP", "bigmax_sup", "pElemS", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rankElem_max p A : 'E_p^('r_p(A))(A) \subset 'E*_p(A).	Proof. apply/subsetP=> E /setIdP[EpE dimE]. apply/pmaxElemP; split=> // F EpF sEF; apply/eqP. have pF: p.-group F by case/pElemP: EpF => _ /and3P[]. have pE: p.-group E by case/pElemP: EpE => _ /and3P[]. rewrite eq_sym eqEcard sEF dvdn_leq // (card_pgroup pE) (card_pgroup pF). by rewrite (eqP dimE) dvdn_exp2l // logn_l...	Lemma	p_rankElem_max	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "card_pgroup", "dvdn_exp2l", "dvdn_leq", "eqEcard", "eq_sym", "group", "logn_le_p_rank", "pE", "pElemP", "pmaxElemP", "setIdP", "split", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rankJ p A x : 'r_p(A :^ x) = 'r_p(A).	Proof. rewrite /p_rank (reindex_inj (act_inj 'JG x)). by apply: eq_big => [E \| E _]; rewrite ?cardJg ?pElemJ. Qed.	Lemma	p_rankJ	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "act_inj", "apply", "cardJg", "eq_big", "pElemJ", "p_rank", "reindex_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_Sylow p G H : p.-Sylow(G) H -> 'r_p(H) = 'r_p(G).	Proof. move=> sylH; apply/eqP; rewrite eqn_leq (p_rankS _ (pHall_sub sylH)) /=. apply/bigmax_leqP=> E /[1!inE] /andP[sEG abelE]. have [P sylP sEP] := Sylow_superset sEG (abelem_pgroup abelE). have [x _ ->] := Sylow_trans sylP sylH. by rewrite p_rankJ -(p_rank_abelem abelE) (p_rankS _ sEP). Qed.	Lemma	p_rank_Sylow	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Sylow", "Sylow_superset", "Sylow_trans", "abelE", "abelem_pgroup", "apply", "bigmax_leqP", "eqn_leq", "inE", "pHall_sub", "p_rankJ", "p_rankS", "p_rank_abelem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_Hall pi p G H : pi.-Hall(G) H -> p \in pi -> 'r_p(H) = 'r_p(G).	Proof. move=> hallH pi_p; have [P sylP] := Sylow_exists p H. by rewrite -(p_rank_Sylow sylP) (p_rank_Sylow (subHall_Sylow hallH pi_p sylP)). Qed.	Lemma	p_rank_Hall	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Hall", "Sylow_exists", "p_rank_Sylow", "pi", "pi_p", "subHall_Sylow" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_pmaxElem_exists p r G : 'r_p(G) >= r -> exists2 E, E \in 'E*_p(G) & 'r_p(E) >= r.	Proof. case/p_rank_geP=> D /setIdP[EpD /eqP <- {r}]. have [E EpE sDE] := pmaxElem_exists EpD; exists E => //. case/pmaxElemP: EpE => /setIdP[_ abelE] _. by rewrite (p_rank_abelem abelE) lognSg. Qed.	Lemma	p_rank_pmaxElem_exists	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelE", "lognSg", "p_rank_abelem", "p_rank_geP", "pmaxElemP", "pmaxElem_exists", "setIdP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank1 : 'r([1 gT]) = 0.	Proof. by rewrite ['r(1)]big1_seq // => p _; rewrite p_rank1. Qed.	Lemma	rank1	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "big1_seq", "gT", "p_rank1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_le_rank p G : 'r_p(G) <= 'r(G).	Proof. case: (posnP 'r_p(G)) => [-> //\|]; rewrite p_rank_gt0 mem_primes. case/and3P=> p_pr _ pG; have lepg: p < #\|G\|.+1 by rewrite ltnS dvdn_leq. by rewrite ['r(G)]big_mkord (bigmax_sup (Ordinal lepg)). Qed.	Lemma	p_rank_le_rank	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "big_mkord", "bigmax_sup", "dvdn_leq", "ltnS", "mem_primes", "pG", "p_pr", "p_rank_gt0", "posnP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_gt0 G : ('r(G) > 0) = (G :!=: 1).	Proof. case: (eqsVneq G 1) => [-> \|]; first by rewrite rank1. case: (trivgVpdiv G) => [/eqP->// \| [p p_pr]]. case/Cauchy=> // x Gx oxp _; apply: leq_trans (p_rank_le_rank p G). have EpGx: <[x]>%G \in 'E_p(G). by rewrite inE cycle_subG Gx abelemE // cycle_abelian -oxp exponent_dvdn. by apply: leq_trans (logn_le_p_rank...	Lemma	rank_gt0	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Cauchy", "abelemE", "apply", "cycle_abelian", "cycle_subG", "eqsVneq", "eqxx", "exponent_dvdn", "inE", "leq_trans", "logn_le_p_rank", "logn_prime", "orderE", "p_pr", "p_rank_le_rank", "rank1", "trivgVpdiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_witness G : {p \| prime p & 'r(G) = 'r_p(G)}.	Proof. have [p _ defmG]: {p : 'I_(#\|G\|.+1) \| true & 'r(G) = 'r_p(G)}. by rewrite ['r(G)]big_mkord; apply: eq_bigmax_cond; rewrite card_ord. case: (eqsVneq G 1) => [-> \| ]; first by exists 2; rewrite // rank1 p_rank1. by rewrite -rank_gt0 defmG p_rank_gt0 mem_primes; case/andP; exists p. Qed.	Lemma	rank_witness	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "big_mkord", "card_ord", "eq_bigmax_cond", "eqsVneq", "mem_primes", "p_rank1", "p_rank_gt0", "prime", "rank1", "rank_gt0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_pgroup p G : p.-group G -> 'r(G) = 'r_p(G).	Proof. move=> pG; apply/eqP; rewrite eqn_leq p_rank_le_rank andbT. rewrite ['r(G)]big_mkord; apply/bigmax_leqP=> [[q /= _] _]. case: (posnP 'r_q(G)) => [-> // \|]; rewrite p_rank_gt0 mem_primes. by case/and3P=> q_pr _ qG; rewrite (eqnP (pgroupP pG q q_pr qG)). Qed.	Lemma	rank_pgroup	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "big_mkord", "bigmax_leqP", "eqnP", "eqn_leq", "group", "mem_primes", "pG", "p_rank_gt0", "p_rank_le_rank", "pgroupP", "posnP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_Sylow p G P : p.-Sylow(G) P -> 'r(P) = 'r_p(G).	Proof. move=> sylP; have pP := pHall_pgroup sylP. by rewrite -(p_rank_Sylow sylP) -(rank_pgroup pP). Qed.	Lemma	rank_Sylow	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Sylow", "pHall_pgroup", "pP", "p_rank_Sylow", "rank_pgroup" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_abelem p G : p.-abelem G -> 'r(G) = logn p #\|G\|.	Proof. by move=> abelG; rewrite (rank_pgroup (abelem_pgroup abelG)) p_rank_abelem. Qed.	Lemma	rank_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "abelem_pgroup", "logn", "p_rank_abelem", "rank_pgroup" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nt_pnElem p n E A : E \in 'E_p^n(A) -> n > 0 -> E :!=: 1.	Proof. by case/pnElemP=> _ /rank_abelem <- <-; rewrite rank_gt0. Qed.	Lemma	nt_pnElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "pnElemP", "rank_abelem", "rank_gt0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rankJ A x : 'r(A :^ x) = 'r(A).	Proof. by rewrite /rank cardJg; apply: eq_bigr => p _; rewrite p_rankJ. Qed.	Lemma	rankJ	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "cardJg", "eq_bigr", "p_rankJ", "rank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rankS A B : A \subset B -> 'r(A) <= 'r(B).	Proof. move=> sAB; rewrite /rank !big_mkord; apply/bigmax_leqP=> p _. have leAB: #\|A\| < #\|B\|.+1 by rewrite ltnS subset_leq_card. by rewrite (bigmax_sup (widen_ord leAB p)) ?p_rankS. Qed.	Lemma	rankS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "big_mkord", "bigmax_leqP", "bigmax_sup", "ltnS", "p_rankS", "rank", "subset_leq_card", "widen_ord" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_geP n G : reflect (exists E, E \in 'E^n(G)) (n <= 'r(G)).	Proof. apply: (iffP idP) => [\|[E]]. have [p _ ->] := rank_witness G; case/p_rank_geP=> E. by rewrite def_pnElem; case/setIP; exists E. case/nElemP=> p /[1!inE] /andP[EpG_E /eqP <-]. by rewrite (leq_trans (logn_le_p_rank EpG_E)) ?p_rank_le_rank. Qed.	Lemma	rank_geP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "def_pnElem", "inE", "leq_trans", "logn_le_p_rank", "nElemP", "p_rank_geP", "p_rank_le_rank", "rank_witness", "setIP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_morphim G : exponent (f @* G) %\| exponent G.	Proof. apply/exponentP=> _ /morphimP[x Dx Gx ->]. by rewrite -morphX // expg_exponent // morph1. Qed.	Lemma	exponent_morphim	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Dx", "apply", "expg_exponent", "exponent", "exponentP", "morph1", "morphX", "morphimP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_LdivT n : f @* 'Ldiv_n() \subset 'Ldiv_n().	Proof. apply/subsetP=> _ /morphimP[x Dx xn ->]; rewrite inE in xn. by rewrite inE -morphX // (eqP xn) morph1. Qed.	Lemma	morphim_LdivT	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Dx", "apply", "inE", "morph1", "morphX", "morphimP", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_Ldiv n A : f @* 'Ldiv_n(A) \subset 'Ldiv_n(f @* A).	Proof. by apply: subset_trans (morphimI f A _) (setIS _ _); apply: morphim_LdivT. Qed.	Lemma	morphim_Ldiv	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "morphimI", "morphim_LdivT", "setIS", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_abelem p G : p.-abelem G -> p.-abelem (f @* G).	Proof. case: (eqsVneq G 1) => [-> \| ntG] abelG; first by rewrite morphim1 abelem1. have [p_pr _ _] := pgroup_pdiv (abelem_pgroup abelG) ntG. case/abelemP: abelG => // abG elemG; apply/abelemP; rewrite ?morphim_abelian //. by split=> // _ /morphimP[x Dx Gx ->]; rewrite -morphX // elemG ?morph1. Qed.	Lemma	morphim_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Dx", "abelem", "abelem1", "abelemP", "abelem_pgroup", "apply", "eqsVneq", "morph1", "morphX", "morphim1", "morphimP", "morphim_abelian", "p_pr", "pgroup_pdiv", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pElem p G E : E \in 'E_p(G) -> (f @* E)%G \in 'E_p(f @* G).	Proof. by rewrite !inE => /andP[sEG abelE]; rewrite morphimS // morphim_abelem. Qed.	Lemma	morphim_pElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelE", "inE", "morphimS", "morphim_abelem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pnElem p n G E : E \in 'E_p^n(G) -> {m \| m <= n & (f @* E)%G \in 'E_p^m(f @* G)}.	Proof. rewrite inE => /andP[EpE /eqP <-]. by exists (logn p #\|f @* E\|); rewrite ?logn_morphim // inE morphim_pElem /=. Qed.	Lemma	morphim_pnElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "inE", "logn", "logn_morphim", "morphim_pElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_grank G : G \subset D -> 'm(f @* G) <= 'm(G).	Proof. have [B defG <-] := grank_witness G; rewrite -defG gen_subG => sBD. by rewrite morphim_gen ?morphimEsub ?(leq_trans (grank_min _)) ?leq_imset_card. Qed.	Lemma	morphim_grank	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "defG", "gen_subG", "grank_min", "grank_witness", "leq_imset_card", "leq_trans", "morphimEsub", "morphim_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
defG : invm injf @* (f @* G) = G	:= morphim_invm injf sGD.	Let	defG	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "injf", "invm", "morphim_invm", "sGD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_injm : exponent (f @* G) = exponent G.	Proof. by apply/eqP; rewrite eqn_dvd -{3}defG !exponent_morphim. Qed.	Lemma	exponent_injm	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "defG", "eqn_dvd", "exponent", "exponent_morphim" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Ldiv n A : f @* 'Ldiv_n(A) = 'Ldiv_n(f @* A).	Proof. apply/eqP; rewrite eqEsubset morphim_Ldiv. rewrite -[f @* 'Ldiv_n(A)](morphpre_invm injf). rewrite -sub_morphim_pre; first by rewrite subIset ?morphim_sub. rewrite injmI ?injm_invm // setISS ?morphim_LdivT //. by rewrite sub_morphim_pre ?morphim_sub // morphpre_invm. Qed.	Lemma	injm_Ldiv	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "eqEsubset", "injf", "injmI", "injm_invm", "morphim_Ldiv", "morphim_LdivT", "morphim_sub", "morphpre_invm", "setISS", "subIset", "sub_morphim_pre" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_abelem p : p.-abelem (f @* G) = p.-abelem G.	Proof. by apply/idP/idP; first rewrite -{2}defG; apply: morphim_abelem. Qed.	Lemma	injm_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "apply", "defG", "morphim_abelem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_pElem p (E : {group aT}) : E \subset D -> ((f @* E)%G \in 'E_p(f @* G)) = (E \in 'E_p(G)).	Proof. move=> sED; apply/idP/idP=> EpE; last exact: morphim_pElem. by rewrite -defG -(group_inj (morphim_invm injf sED)) morphim_pElem. Qed.	Lemma	injm_pElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "aT", "apply", "defG", "group", "group_inj", "injf", "last", "morphim_invm", "morphim_pElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_cyclic X : cyclic X -> exponent X = #|X|.

Proof. by case/cyclicP=> x ->; apply: exponent_cycle. Qed.

Lemma

exponent_cyclic

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "cyclic", "cyclicP", "exponent", "exponent_cycle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primes_exponent G : primes (exponent G) = primes (#|G|).

Proof. apply/eq_primes => p; rewrite !mem_primes exponent_gt0 cardG_gt0 /=. by apply: andb_id2l => p_pr; apply: negb_inj; rewrite -!p'natE // pnat_exponent. Qed.

Lemma

primes_exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "cardG_gt0", "eq_primes", "exponent", "exponent_gt0", "mem_primes", "p'natE", "p_pr", "pnat_exponent", "primes" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_of_exponent G : \pi(exponent G) = \pi(G).

Proof. by rewrite /pi_of primes_exponent. Qed.

Lemma

pi_of_exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "exponent", "pi", "pi_of", "primes_exponent" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

partn_exponentS pi H G : H \subset G -> #|G|`_pi %| #|H| -> ((exponent H)`_pi = (exponent G)`_pi)%N.

Proof. move=> sHG Gpi_dvd_H; apply/eqP; rewrite eqn_dvd. rewrite partn_dvd ?exponentS ?exponent_gt0 //=; apply/dvdn_partP=> // p. rewrite pi_of_part ?exponent_gt0 // => /andP[_ /= pi_p]. have sppi: {subset (p : nat_pred) <= pi} by move=> q /eqnP->. have [P sylP] := Sylow_exists p H; have sPH := pHall_sub sylP. have{} s...

Lemma

partn_exponentS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Sylow", "Sylow_Jsub", "Sylow_exists", "apply", "bigD1", "big_ind", "cardG_gt0", "cardSg", "card_Hall", "cycleJ", "cycle_subG", "dvd1n", "dvdn", "dvdn_lcm", "dvdn_lcml", "dvdn_partP", "eqnP", "eqn_dvd", "exponent", "exponentS", "exponent_gt0", "groupX", "nat_pred", "ord...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_Hall pi G H : pi.-Hall(G) H -> exponent H = ((exponent G)`_pi)%N.

Proof. move=> hallH; have [sHG piH _] := and3P hallH. rewrite -(partn_exponentS sHG) -?(card_Hall hallH) ?part_pnat_id //. by apply: pnat_dvd piH; apply: exponent_dvdn. Qed.

Lemma

exponent_Hall

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Hall", "apply", "card_Hall", "exponent", "exponent_dvdn", "part_pnat_id", "partn_exponentS", "pi", "pnat_dvd", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_Zgroup G : Zgroup G -> exponent G = #|G|.

Proof. move/forall_inP=> ZgG; apply/eqP; rewrite eqn_dvd exponent_dvdn. apply/(dvdn_partP _ (cardG_gt0 _)) => p _. have [S sylS] := Sylow_exists p G; rewrite -(card_Hall sylS). have /cyclicP[x defS]: cyclic S by rewrite ZgG ?(p_Sylow sylS). by rewrite defS dvdn_exponent // -cycle_subG -defS (pHall_sub sylS). Qed.

Lemma

exponent_Zgroup

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Sylow_exists", "Zgroup", "apply", "cardG_gt0", "card_Hall", "cycle_subG", "cyclic", "cyclicP", "dvdn_exponent", "dvdn_partP", "eqn_dvd", "exponent", "exponent_dvdn", "forall_inP", "pHall_sub", "p_Sylow" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprod_exponent A B G : A \* B = G -> lcmn (exponent A) (exponent B) = (exponent G).

Proof. case/cprodP=> [[K H -> ->{A B}] <- cKH]. apply/eqP; rewrite eqn_dvd dvdn_lcm !exponentS ?mulG_subl ?mulG_subr //=. apply/exponentP=> _ /imset2P[x y Kx Hy ->]. rewrite -[1]mulg1 expgMn; first by red; rewrite -(centsP cKH). congr (_ * _); apply/eqP; rewrite -order_dvdn. by rewrite (dvdn_trans (dvdn_exponent Kx))...

Lemma

cprod_exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "cKH", "centsP", "cprodP", "dvdn_exponent", "dvdn_lcm", "dvdn_lcml", "dvdn_lcmr", "dvdn_trans", "eqn_dvd", "expgMn", "exponent", "exponentP", "exponentS", "imset2P", "lcmn", "mulG_subl", "mulG_subr", "mulg1", "order_dvdn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dprod_exponent A B G : A \x B = G -> lcmn (exponent A) (exponent B) = (exponent G).

Proof. case/dprodP=> [[K H -> ->{A B}] defG cKH _]. by apply: cprod_exponent; rewrite cprodE. Qed.

Lemma

dprod_exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "cKH", "cprodE", "cprod_exponent", "defG", "dprodP", "exponent", "lcmn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_LdivT A n : (A \subset 'Ldiv_n()) = (exponent A %| n).

Proof. by apply/subsetP/exponentP=> eAn x /eAn /[1!inE] /eqP. Qed.

Lemma

sub_LdivT

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "exponent", "exponentP", "inE", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

LdivT_J n x : 'Ldiv_n() :^ x = 'Ldiv_n().

Proof. apply/setP=> y; rewrite !inE mem_conjg inE -conjXg. by rewrite (canF_eq (conjgKV x)) conj1g. Qed.

Lemma

LdivT_J

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "canF_eq", "conj1g", "conjXg", "conjgKV", "inE", "mem_conjg", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

LdivJ n A x : 'Ldiv_n(A :^ x) = 'Ldiv_n(A) :^ x.

Proof. by rewrite conjIg LdivT_J. Qed.

Lemma

LdivJ

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "LdivT_J", "conjIg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_Ldiv A n : (A \subset 'Ldiv_n(A)) = (exponent A %| n).

Proof. by rewrite subsetI subxx sub_LdivT. Qed.

Lemma

sub_Ldiv

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "exponent", "sub_LdivT", "subsetI", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_Ldiv G n : abelian G -> group_set 'Ldiv_n(G).

Proof. move=> cGG; apply/group_setP. split=> [|x y]; rewrite !inE ?group1 ?expg1n //=. case/andP=> Gx /eqP xn /andP[Gy /eqP yn]. by rewrite groupM //= expgMn ?xn ?yn ?mulg1 //; apply: (centsP cGG). Qed.

Lemma

group_Ldiv

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelian", "apply", "cGG", "centsP", "expg1n", "expgMn", "group1", "groupM", "group_set", "group_setP", "inE", "mulg1", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_exponent_gen A : abelian A -> exponent <<A>> = exponent A.

Proof. rewrite -abelian_gen; set n := exponent A; set G := <<A>> => cGG. apply/eqP; rewrite eqn_dvd andbC exponentS ?subset_gen //= -sub_Ldiv. rewrite -(gen_set_id (group_Ldiv n cGG)) genS // subsetI subset_gen /=. by rewrite sub_LdivT. Qed.

Lemma

abelian_exponent_gen

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelian", "abelian_gen", "apply", "cGG", "eqn_dvd", "exponent", "exponentS", "genS", "gen_set_id", "group_Ldiv", "sub_Ldiv", "sub_LdivT", "subsetI", "subset_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_pgroup p A : p.-abelem A -> p.-group A.

Proof. by case/andP. Qed.

Lemma

abelem_pgroup

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_abelian p A : p.-abelem A -> abelian A.

Proof. by case/and3P. Qed.

Lemma

abelem_abelian

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelian" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem1 p : p.-abelem [1 gT].

Proof. by rewrite /abelem pgroup1 abelian1 exponent1 dvd1n. Qed.

Lemma

abelem1

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelian1", "dvd1n", "exponent1", "gT", "pgroup1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelemE p G : prime p -> p.-abelem G = abelian G && (exponent G %| p).

Proof. move=> p_pr; rewrite /abelem -pnat_exponent andbA -!(andbC (_ %| _)). by case: (dvdn_pfactor _ 1 p_pr) => // [[k _ ->]]; rewrite pnatX pnat_id. Qed.

Lemma

abelemE

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelian", "dvdn_pfactor", "exponent", "p_pr", "pnatX", "pnat_exponent", "pnat_id", "prime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelemP p G : prime p -> reflect (abelian G /\ forall x, x \in G -> x ^+ p = 1) (p.-abelem G).

Proof. by move=> p_pr; rewrite abelemE //; apply: (iffP andP) => [] [-> /exponentP]. Qed.

Lemma

abelemP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelemE", "abelian", "apply", "exponentP", "p_pr", "prime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_order_p p G x : p.-abelem G -> x \in G -> x != 1 -> #[x] = p.

Proof. case/and3P=> pG _ eG Gx; rewrite -cycle_eq1 => ntX. have{ntX} [p_pr p_x _] := pgroup_pdiv (mem_p_elt pG Gx) ntX. by apply/eqP; rewrite eqn_dvd p_x andbT order_dvdn (exponentP eG). Qed.

Lemma

abelem_order_p

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "apply", "cycle_eq1", "eqn_dvd", "exponentP", "mem_p_elt", "order_dvdn", "pG", "p_pr", "pgroup_pdiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cyclic_abelem_prime p X : p.-abelem X -> cyclic X -> X :!=: 1 -> #|X| = p.

Proof. move=> abelX cycX; case/cyclicP: cycX => x -> in abelX *. by rewrite cycle_eq1; apply: abelem_order_p abelX (cycle_id x). Qed.

Lemma

cyclic_abelem_prime

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelem_order_p", "apply", "cycle_eq1", "cycle_id", "cyclic", "cyclicP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle_abelem p x : p.-elt x || prime p -> p.-abelem <[x]> = (#[x] %| p).

Proof. move=> p_xVpr; rewrite /abelem cycle_abelian /=. apply/andP/idP=> [[_ xp1] | x_dvd_p]. by rewrite order_dvdn (exponentP xp1) ?cycle_id. split; last exact: dvdn_trans (exponent_dvdn _) x_dvd_p. by case/orP: p_xVpr => // /pnat_id; apply: pnat_dvd. Qed.

Lemma

cycle_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "apply", "cycle_abelian", "cycle_id", "dvdn_trans", "exponentP", "exponent_dvdn", "last", "order_dvdn", "pnat_dvd", "pnat_id", "prime", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent2_abelem G : exponent G %| 2 -> 2.-abelem G.

Proof. move/exponentP=> expG; apply/abelemP=> //; split=> //. apply/centsP=> x Gx y Gy; apply: (mulIg x); apply: (mulgI y). by rewrite -!mulgA !(mulgA y) -!(expgS _ 1) !expG ?mulg1 ?groupM. Qed.

Lemma

exponent2_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelemP", "apply", "centsP", "expgS", "exponent", "exponentP", "groupM", "mulIg", "mulg1", "mulgA", "mulgI", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime_abelem p G : prime p -> #|G| = p -> p.-abelem G.

Proof. move=> p_pr oG; rewrite /abelem -oG exponent_dvdn. by rewrite /pgroup cyclic_abelian ?prime_cyclic ?oG ?pnat_id. Qed.

Lemma

prime_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "cyclic_abelian", "exponent_dvdn", "p_pr", "pgroup", "pnat_id", "prime", "prime_cyclic" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_cyclic p G : p.-abelem G -> cyclic G = (logn p #|G| <= 1).

Proof. move=> abelG; have [pG _ expGp] := and3P abelG. case: (eqsVneq G 1) => [-> | ntG]; first by rewrite cyclic1 cards1 logn1. have [p_pr _ [e oG]] := pgroup_pdiv pG ntG; apply/idP/idP. case/cyclicP=> x defG; rewrite -(pfactorK 1 p_pr) dvdn_leq_log ?prime_gt0 //. by rewrite defG order_dvdn (exponentP expGp) // de...

Lemma

abelem_cyclic

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "apply", "cards1", "cycle_id", "cyclic", "cyclic1", "cyclicP", "defG", "dvdn_leq_log", "e0", "eqsVneq", "exponentP", "leqn0", "logn", "logn1", "ltnS", "order_dvdn", "pG", "p_pr", "pfactorK", "pgroup_pdiv", "prime_cyclic", "prime_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelemS p H G : H \subset G -> p.-abelem G -> p.-abelem H.

Proof. move=> sHG /and3P[cGG pG Gp1]; rewrite /abelem. by rewrite (pgroupS sHG) // (abelianS sHG) // (dvdn_trans (exponentS sHG)). Qed.

Lemma

abelemS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelianS", "cGG", "dvdn_trans", "exponentS", "pG", "pgroupS", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelemJ p G x : p.-abelem (G :^ x) = p.-abelem G.

Proof. by rewrite /abelem pgroupJ abelianJ exponentJ. Qed.

Lemma

abelemJ

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelianJ", "exponentJ", "pgroupJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprod_abelem p A B G : A \* B = G -> p.-abelem G = p.-abelem A && p.-abelem B.

Proof. case/cprodP=> [[H K -> ->{A B}] defG cHK]. apply/idP/andP=> [abelG | []]. by rewrite !(abelemS _ abelG) // -defG (mulG_subl, mulG_subr). case/and3P=> pH cHH expHp; case/and3P=> pK cKK expKp. rewrite -defG /abelem pgroupM pH pK abelianM cHH cKK cHK /=. apply/exponentP=> _ /imset2P[x y Hx Ky ->]. rewrite expgMn;...

Lemma

cprod_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelemS", "abelianM", "apply", "centsP", "cprodP", "defG", "expgMn", "exponentP", "imset2P", "mul1g", "mulG_subl", "mulG_subr", "pgroupM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dprod_abelem p A B G : A \x B = G -> p.-abelem G = p.-abelem A && p.-abelem B.

Proof. move=> defG; case/dprodP: (defG) => _ _ _ tiHK. by apply: cprod_abelem; rewrite -dprodEcp. Qed.

Lemma

dprod_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "apply", "cprod_abelem", "defG", "dprodEcp", "dprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

is_abelem_pgroup p G : p.-group G -> is_abelem G = p.-abelem G.

Proof. rewrite /is_abelem => pG. case: (eqsVneq G 1) => [-> | ntG]; first by rewrite !abelem1. by have [p_pr _ [k ->]] := pgroup_pdiv pG ntG; rewrite pdiv_pfactor. Qed.

Lemma

is_abelem_pgroup

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelem1", "eqsVneq", "group", "is_abelem", "pG", "p_pr", "pdiv_pfactor", "pgroup_pdiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

is_abelemP G : reflect (exists2 p, prime p & p.-abelem G) (is_abelem G).

Proof. apply: (iffP idP) => [abelG | [p p_pr abelG]]. case: (eqsVneq G 1) => [-> | ntG]; first by exists 2; rewrite ?abelem1. by exists (pdiv #|G|); rewrite ?pdiv_prime // ltnNge -trivg_card_le1. by rewrite (is_abelem_pgroup (abelem_pgroup abelG)). Qed.

Lemma

is_abelemP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelem1", "abelem_pgroup", "apply", "eqsVneq", "is_abelem", "is_abelem_pgroup", "ltnNge", "p_pr", "pdiv", "pdiv_prime", "prime", "trivg_card_le1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pElemP p A E : reflect (E \subset A /\ p.-abelem E) (E \in 'E_p(A)).

Proof. by rewrite inE; apply: andP. Qed.

Lemma

pElemP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "apply", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pElemS p A B : A \subset B -> 'E_p(A) \subset 'E_p(B).

Proof. by move=> sAB; apply/subsetP=> E /[!inE] /andP[/subset_trans->]. Qed.

Lemma

pElemS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "inE", "subsetP", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pElemI p A B : 'E_p(A :&: B) = 'E_p(A) :&: subgroups B.

Proof. by apply/setP=> E; rewrite !inE subsetI andbAC. Qed.

Lemma

pElemI

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "inE", "setP", "subgroups", "subsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pElemJ x p A E : ((E :^ x)%G \in 'E_p(A :^ x)) = (E \in 'E_p(A)).

Proof. by rewrite !inE conjSg abelemJ. Qed.

Lemma

pElemJ

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelemJ", "conjSg", "inE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElemP p n A E : reflect [/\ E \subset A, p.-abelem E & logn p #|E| = n] (E \in 'E_p^n(A)).

Proof. by rewrite !inE -andbA; apply: (iffP and3P) => [] [-> -> /eqP]. Qed.

Lemma

pnElemP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "apply", "inE", "logn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElemPcard p n A E : E \in 'E_p^n(A) -> [/\ E \subset A, p.-abelem E & #|E| = p ^ n]%N.

Proof. by case/pnElemP=> -> abelE <-; rewrite -card_pgroup // abelem_pgroup. Qed.

Lemma

pnElemPcard

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelE", "abelem", "abelem_pgroup", "card_pgroup", "pnElemP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_pnElem p n A E : E \in 'E_p^n(A) -> #|E| = (p ^ n)%N.

Proof. by case/pnElemPcard. Qed.

Lemma

card_pnElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "pnElemPcard" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElem0 p G : 'E_p^0(G) = [set 1%G].

Proof. apply/setP=> E; rewrite !inE -andbA; apply/and3P/idP=> [[_ pE] | /eqP->]. apply: contraLR; case/(pgroup_pdiv (abelem_pgroup pE)) => p_pr _ [k ->]. by rewrite pfactorK. by rewrite sub1G abelem1 cards1 logn1. Qed.

Lemma

pnElem0

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem1", "abelem_pgroup", "apply", "cards1", "inE", "logn1", "pE", "p_pr", "pfactorK", "pgroup_pdiv", "setP", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElem_prime p n A E : E \in 'E_p^n.+1(A) -> prime p.

Proof. by case/pnElemP=> _ _; rewrite lognE; case: prime. Qed.

Lemma

pnElem_prime

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "lognE", "pnElemP", "prime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElemE p n A : prime p -> 'E_p^n(A) = [set E in 'E_p(A) | #|E| == (p ^ n)%N].

Proof. move/pfactorK=> pnK; apply/setP=> E; rewrite 3!inE. case: (@andP (E \subset A)) => //= [[_]] /andP[/p_natP[k ->] _]. by rewrite pnK (can_eq pnK). Qed.

Lemma

pnElemE

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "can_eq", "inE", "p_natP", "pfactorK", "prime", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElemS p n A B : A \subset B -> 'E_p^n(A) \subset 'E_p^n(B).

Proof. move=> sAB; apply/subsetP=> E. by rewrite !inE -!andbA => /andP[/subset_trans->]. Qed.

Lemma

pnElemS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "inE", "subsetP", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElemI p n A B : 'E_p^n(A :&: B) = 'E_p^n(A) :&: subgroups B.

Proof. by apply/setP=> E; rewrite !inE subsetI -!andbA; do !bool_congr. Qed.

Lemma

pnElemI

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "inE", "setP", "subgroups", "subsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElemJ x p n A E : ((E :^ x)%G \in 'E_p^n(A :^ x)) = (E \in 'E_p^n(A)).

Proof. by rewrite inE pElemJ cardJg !inE. Qed.

Lemma

pnElemJ

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "cardJg", "inE", "pElemJ" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_pnElem p n G : p.-abelem G -> n <= logn p #|G| -> exists E, E \in 'E_p^n(G).

Proof. case: n => [|n] abelG lt_nG; first by exists 1%G; rewrite pnElem0 set11. have p_pr: prime p by move: lt_nG; rewrite lognE; case: prime. case/(normal_pgroup (abelem_pgroup abelG)): lt_nG => // E [sEG _ oE]. by exists E; rewrite pnElemE // !inE oE sEG (abelemS sEG) /=. Qed.

Lemma

abelem_pnElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelemS", "abelem_pgroup", "inE", "logn", "lognE", "normal_pgroup", "p_pr", "pnElem0", "pnElemE", "prime", "set11" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_p1Elem p A X : X \in 'E_p^1(A) -> #|X| = p.

Proof. exact: card_pnElem. Qed.

Lemma

card_p1Elem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "card_pnElem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p1ElemE p A : prime p -> 'E_p^1(A) = [set X in subgroups A | #|X| == p].

Proof. move=> p_pr; apply/setP=> X; rewrite pnElemE // !inE -andbA; congr (_ && _). by apply: andb_idl => /eqP oX; rewrite prime_abelem ?oX. Qed.

Lemma

p1ElemE

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "inE", "p_pr", "pnElemE", "prime", "prime_abelem", "setP", "subgroups" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

TIp1ElemP p A X Y : X \in 'E_p^1(A) -> Y \in 'E_p^1(A) -> reflect (X :&: Y = 1) (X :!=: Y).

Proof. move=> EpX EpY; have p_pr := pnElem_prime EpX. have [oX oY] := (card_p1Elem EpX, card_p1Elem EpY). have [<-|] := eqVneq. by right=> X1; rewrite -oX -(setIid X) X1 cards1 in p_pr. by rewrite eqEcard oX oY leqnn andbT; left; rewrite prime_TIg ?oX. Qed.

Lemma

TIp1ElemP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "card_p1Elem", "cards1", "eqEcard", "eqVneq", "leqnn", "p_pr", "pnElem_prime", "prime_TIg", "setIid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_p1Elem_pnElem p n A E : E \in 'E_p^n(A) -> #|'E_p^1(E)| = (\sum_(i < n) p ^ i)%N.

Proof. case/pnElemP=> _ {A} abelE dimE; have [pE cEE _] := and3P abelE. have [E1 | ntE] := eqsVneq E 1. rewrite -dimE E1 cards1 logn1 big_ord0 eq_card0 // => X. by rewrite !inE subG1 trivg_card1; case: eqP => // ->; rewrite logn1 andbF. have [p_pr _ _] := pgroup_pdiv pE ntE; have p_gt1 := prime_gt1 p_pr. apply/eqP;...

Lemma

card_p1Elem_pnElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "TIp1ElemP", "X'", "abelE", "abelem_order_p", "apply", "big_ord0", "bigcupP", "bigcupsP", "card_imset", "card_p1Elem", "card_pgroup", "card_uniform_partition", "cards0", "cards1", "cardsD1", "cycle_id", "cycle_subG", "eqEsubset", "eq_card0", "eq_sym", "eqn_pmul2l", "eqsVneq...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_p1Elem_p2Elem p A E : E \in 'E_p^2(A) -> #|'E_p^1(E)| = p.+1.

Proof. by move/card_p1Elem_pnElem->; rewrite big_ord_recl big_ord1. Qed.

Lemma

card_p1Elem_p2Elem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "big_ord1", "big_ord_recl", "card_p1Elem_pnElem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p2Elem_dprodP p A E X Y : E \in 'E_p^2(A) -> X \in 'E_p^1(E) -> Y \in 'E_p^1(E) -> reflect (X \x Y = E) (X :!=: Y).

Proof. move=> Ep2E EpX EpY; have [_ abelE oE] := pnElemPcard Ep2E. apply: (iffP (TIp1ElemP EpX EpY)) => [tiXY|]; last by case/dprodP. have [[sXE _ oX] [sYE _ oY]] := (pnElemPcard EpX, pnElemPcard EpY). rewrite dprodE ?(sub_abelian_cent2 (abelem_abelian abelE)) //. by apply/eqP; rewrite eqEcard mul_subG //= TI_cardMg //...

Lemma

p2Elem_dprodP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "TI_cardMg", "TIp1ElemP", "abelE", "abelem_abelian", "apply", "dprodE", "dprodP", "eqEcard", "last", "mul_subG", "pnElemPcard", "sub_abelian_cent2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nElemP n G E : reflect (exists p, E \in 'E_p^n(G)) (E \in 'E^n(G)).

Proof. rewrite ['E^n(G)]big_mkord. apply: (iffP bigcupP) => [[[p /= _] _] | [p]]; first by exists p. case: n => [|n EpnE]; first by rewrite pnElem0; exists ord0; rewrite ?pnElem0. suffices lepG: p < #|G|.+1 by exists (Ordinal lepG). have:= EpnE; rewrite pnElemE ?(pnElem_prime EpnE) // !inE -andbA ltnS. case/and3P=> sE...

Lemma

nElemP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "big_mkord", "bigcupP", "cardSg", "dvdn_exp", "dvdn_leq", "dvdn_trans", "inE", "ltnS", "ord0", "pnElem0", "pnElemE", "pnElem_prime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nElem0 G : 'E^0(G) = [set 1%G].

Proof. apply/setP=> E; apply/nElemP/idP=> [[p] |]; first by rewrite pnElem0. by exists 2; rewrite pnElem0. Qed.

Lemma

nElem0

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "nElemP", "pnElem0", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nElem1P G E : reflect (E \subset G /\ exists2 p, prime p & #|E| = p) (E \in 'E^1(G)).

Proof. apply: (iffP nElemP) => [[p pE] | [sEG [p p_pr oE]]]. have p_pr := pnElem_prime pE; rewrite pnElemE // !inE -andbA in pE. by case/and3P: pE => -> _ /eqP; split; last exists p. exists p; rewrite pnElemE // !inE sEG oE eqxx abelemE // -oE exponent_dvdn. by rewrite cyclic_abelian // prime_cyclic // oE. Qed.

Lemma

nElem1P

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelemE", "apply", "cyclic_abelian", "eqxx", "exponent_dvdn", "inE", "last", "nElemP", "pE", "p_pr", "pnElemE", "pnElem_prime", "prime", "prime_cyclic", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nElemS n G H : G \subset H -> 'E^n(G) \subset 'E^n(H).

Proof. move=> sGH; apply/subsetP=> E /nElemP[p EpnG_E]. by apply/nElemP; exists p; rewrite // (subsetP (pnElemS _ _ sGH)). Qed.

Lemma

nElemS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "nElemP", "pnElemS", "sGH", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nElemI n G H : 'E^n(G :&: H) = 'E^n(G) :&: subgroups H.

Proof. apply/setP=> E; apply/nElemP/setIP=> [[p] | []]. by rewrite pnElemI; case/setIP; split=> //; apply/nElemP; exists p. by case/nElemP=> p EpnG_E sHE; exists p; rewrite pnElemI inE EpnG_E. Qed.

Lemma

nElemI

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "inE", "nElemP", "pnElemI", "setIP", "setP", "split", "subgroups" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

def_pnElem p n G : 'E_p^n(G) = 'E_p(G) :&: 'E^n(G).

Proof. apply/setP=> E; rewrite inE in_setI; apply: andb_id2l => /pElemP[sEG abelE]. apply/idP/nElemP=> [|[q]]; first by exists p; rewrite !inE sEG abelE. rewrite !inE -2!andbA => /and4P[_ /pgroupP qE _]. have [->|] := eqVneq E 1%G; first by rewrite cards1 !logn1. case/(pgroup_pdiv (abelem_pgroup abelE)) => p_pr pE _. b...

Lemma

def_pnElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelE", "abelem_pgroup", "apply", "cards1", "eqVneq", "eqnP", "inE", "in_setI", "logn1", "nElemP", "pE", "pElemP", "p_pr", "pgroupP", "pgroup_pdiv", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmaxElemP p A E : reflect (E \in 'E_p(A) /\ forall H, H \in 'E_p(A) -> E \subset H -> H :=: E) (E \in 'E*_p(A)).

Proof. by rewrite [E \in 'E*_p(A)]inE; apply: (iffP maxgroupP). Qed.

Lemma

pmaxElemP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "inE", "maxgroupP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmaxElem_exists p A D : D \in 'E_p(A) -> {E | E \in 'E*_p(A) & D \subset E}.

Proof. move=> EpD; have [E maxE sDE] := maxgroup_exists (EpD : mem 'E_p(A) D). by exists E; rewrite // inE. Qed.

Lemma

pmaxElem_exists

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "inE", "maxgroup_exists" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmaxElem_LdivP p G E : prime p -> reflect ('Ldiv_p('C_G(E)) = E) (E \in 'E*_p(G)).

Proof. move=> p_pr; apply: (iffP (pmaxElemP p G E)) => [[] | defE]. case/pElemP=> sEG abelE maxE; have [_ cEE eE] := and3P abelE. apply/setP=> x; rewrite !inE -andbA; apply/and3P/idP=> [[Gx cEx xp] | Ex]. rewrite -(maxE (<[x]> <*> E)%G) ?joing_subr //; last first. by rewrite -cycle_subG joing_subl. re...

Lemma

pmaxElem_LdivP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelE", "abelemP", "abelian", "apply", "centsC", "centsS", "cprodEY", "cprod_abelem", "cycle_abelem", "cycle_subG", "eqEsubset", "exponentP", "inE", "join_subG", "joing_subl", "joing_subr", "last", "order_dvdn", "pElemP", "p_pr", "pmaxElemP", "prime", "sHG", "setIA", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmaxElemS p A B : A \subset B -> 'E*_p(B) :&: subgroups A \subset 'E*_p(A).

Proof. move=> sAB; apply/subsetP=> E /[!inE]. case/andP=> /maxgroupP[/pElemP[_ abelE] maxE] sEA. apply/maxgroupP; rewrite inE sEA; split=> // D EpD. by apply: maxE; apply: subsetP EpD; apply: pElemS. Qed.

Lemma

pmaxElemS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelE", "apply", "inE", "maxgroupP", "pElemP", "pElemS", "split", "subgroups", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmaxElemJ p A E x : ((E :^ x)%G \in 'E*_p(A :^ x)) = (E \in 'E*_p(A)).

Proof. apply/pmaxElemP/pmaxElemP=> [] [EpE maxE]. rewrite pElemJ in EpE; split=> //= H EpH sEH; apply: (act_inj 'Js x). by apply: maxE; rewrite ?conjSg ?pElemJ. rewrite pElemJ; split=> // H; rewrite -(actKV 'JG x H) pElemJ conjSg => EpHx'. by move/maxE=> /= ->. Qed.

Lemma

pmaxElemJ

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "actKV", "act_inj", "apply", "conjSg", "pElemJ", "pmaxElemP", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

grank_min B : 'm(<<B>>) <= #|B|.

Proof. by rewrite /gen_rank; case: arg_minnP => [|_ _ -> //]; rewrite genGid. Qed.

Lemma

grank_min

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "arg_minnP", "genGid", "gen_rank" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

grank_witness G : {B | <<B>> = G & #|B| = 'm(G)}.

Proof. rewrite /gen_rank; case: arg_minnP => [|B defG _]; first by rewrite genGid. by exists B; first apply/eqP. Qed.

Lemma

grank_witness

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "arg_minnP", "defG", "genGid", "gen_rank" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_witness p G : {E | E \in 'E_p^('r_p(G))(G)}.

Proof. have [E EG_E mE]: {E | E \in 'E_p(G) & 'r_p(G) = logn p #|E| }. by apply: eq_bigmax_cond; rewrite (cardD1 1%G) inE sub1G abelem1. by exists E; rewrite inE EG_E -mE /=. Qed.

Lemma

p_rank_witness

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem1", "apply", "cardD1", "eq_bigmax_cond", "inE", "logn", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_geP p n G : reflect (exists E, E \in 'E_p^n(G)) (n <= 'r_p(G)).

Proof. apply: (iffP idP) => [|[E]]; last first. by rewrite inE => /andP[Ep_E /eqP <-]; rewrite (bigmax_sup E). have [D /pnElemP[sDG abelD <-]] := p_rank_witness p G. by case/abelem_pnElem=> // E; exists E; apply: (subsetP (pnElemS _ _ sDG)). Qed.

Lemma

p_rank_geP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem_pnElem", "apply", "bigmax_sup", "inE", "last", "p_rank_witness", "pnElemP", "pnElemS", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_gt0 p H : ('r_p(H) > 0) = (p \in \pi(H)).

Proof. rewrite mem_primes cardG_gt0 /=; apply/p_rank_geP/andP=> [[E] | [p_pr]]. case/pnElemP=> sEG _; rewrite lognE; case: and3P => // [[-> _ pE] _]. by rewrite (dvdn_trans _ (cardSg sEG)). case/Cauchy=> // x Hx ox; exists <[x]>%G; rewrite 2!inE [#|_|]ox cycle_subG. by rewrite Hx (pfactorK 1) ?abelemE // cycle_abel...

Lemma

p_rank_gt0

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Cauchy", "abelemE", "apply", "cardG_gt0", "cardSg", "cycle_abelian", "cycle_subG", "dvdn_trans", "exponent_dvdn", "inE", "lognE", "mem_primes", "pE", "p_pr", "p_rank_geP", "pfactorK", "pi", "pnElemP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank1 p : 'r_p([1 gT]) = 0.

Proof. by apply/eqP; rewrite eqn0Ngt p_rank_gt0 /= cards1. Qed.

Lemma

p_rank1

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "cards1", "eqn0Ngt", "gT", "p_rank_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn_le_p_rank p A E : E \in 'E_p(A) -> logn p #|E| <= 'r_p(A).

Proof. by move=> EpA_E; rewrite (bigmax_sup E). Qed.

Lemma

logn_le_p_rank

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "bigmax_sup", "logn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_le_logn p G : 'r_p(G) <= logn p #|G|.

Proof. have [E EpE] := p_rank_witness p G. by have [sEG _ <-] := pnElemP EpE; apply: lognSg. Qed.

Lemma

p_rank_le_logn

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "logn", "lognSg", "p_rank_witness", "pnElemP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_abelem p G : p.-abelem G -> 'r_p(G) = logn p #|G|.

Proof. move=> abelG; apply/eqP; rewrite eqn_leq andbC (bigmax_sup G)//. by rewrite inE subxx. by apply/bigmax_leqP=> E /[1!inE] /andP[/lognSg->]. Qed.

Lemma

p_rank_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "apply", "bigmax_leqP", "bigmax_sup", "eqn_leq", "inE", "logn", "lognSg", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rankS p A B : A \subset B -> 'r_p(A) <= 'r_p(B).

Proof. move=> sAB; apply/bigmax_leqP=> E /(subsetP (pElemS p sAB)) EpB_E. by rewrite (bigmax_sup E). Qed.

Lemma

p_rankS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "bigmax_leqP", "bigmax_sup", "pElemS", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rankElem_max p A : 'E_p^('r_p(A))(A) \subset 'E*_p(A).

Proof. apply/subsetP=> E /setIdP[EpE dimE]. apply/pmaxElemP; split=> // F EpF sEF; apply/eqP. have pF: p.-group F by case/pElemP: EpF => _ /and3P[]. have pE: p.-group E by case/pElemP: EpE => _ /and3P[]. rewrite eq_sym eqEcard sEF dvdn_leq // (card_pgroup pE) (card_pgroup pF). by rewrite (eqP dimE) dvdn_exp2l // logn_l...

Lemma

p_rankElem_max

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "card_pgroup", "dvdn_exp2l", "dvdn_leq", "eqEcard", "eq_sym", "group", "logn_le_p_rank", "pE", "pElemP", "pmaxElemP", "setIdP", "split", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rankJ p A x : 'r_p(A :^ x) = 'r_p(A).

Proof. rewrite /p_rank (reindex_inj (act_inj 'JG x)). by apply: eq_big => [E | E _]; rewrite ?cardJg ?pElemJ. Qed.

Lemma

p_rankJ

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "act_inj", "apply", "cardJg", "eq_big", "pElemJ", "p_rank", "reindex_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_Sylow p G H : p.-Sylow(G) H -> 'r_p(H) = 'r_p(G).

Proof. move=> sylH; apply/eqP; rewrite eqn_leq (p_rankS _ (pHall_sub sylH)) /=. apply/bigmax_leqP=> E /[1!inE] /andP[sEG abelE]. have [P sylP sEP] := Sylow_superset sEG (abelem_pgroup abelE). have [x _ ->] := Sylow_trans sylP sylH. by rewrite p_rankJ -(p_rank_abelem abelE) (p_rankS _ sEP). Qed.

Lemma

p_rank_Sylow

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Sylow", "Sylow_superset", "Sylow_trans", "abelE", "abelem_pgroup", "apply", "bigmax_leqP", "eqn_leq", "inE", "pHall_sub", "p_rankJ", "p_rankS", "p_rank_abelem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_Hall pi p G H : pi.-Hall(G) H -> p \in pi -> 'r_p(H) = 'r_p(G).

Proof. move=> hallH pi_p; have [P sylP] := Sylow_exists p H. by rewrite -(p_rank_Sylow sylP) (p_rank_Sylow (subHall_Sylow hallH pi_p sylP)). Qed.

Lemma

p_rank_Hall

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Hall", "Sylow_exists", "p_rank_Sylow", "pi", "pi_p", "subHall_Sylow" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_pmaxElem_exists p r G : 'r_p(G) >= r -> exists2 E, E \in 'E*_p(G) & 'r_p(E) >= r.

Proof. case/p_rank_geP=> D /setIdP[EpD /eqP <- {r}]. have [E EpE sDE] := pmaxElem_exists EpD; exists E => //. case/pmaxElemP: EpE => /setIdP[_ abelE] _. by rewrite (p_rank_abelem abelE) lognSg. Qed.

Lemma

p_rank_pmaxElem_exists

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelE", "lognSg", "p_rank_abelem", "p_rank_geP", "pmaxElemP", "pmaxElem_exists", "setIdP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank1 : 'r([1 gT]) = 0.

Proof. by rewrite ['r(1)]big1_seq // => p _; rewrite p_rank1. Qed.

Lemma

rank1

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "big1_seq", "gT", "p_rank1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_le_rank p G : 'r_p(G) <= 'r(G).

Proof. case: (posnP 'r_p(G)) => [-> //|]; rewrite p_rank_gt0 mem_primes. case/and3P=> p_pr _ pG; have lepg: p < #|G|.+1 by rewrite ltnS dvdn_leq. by rewrite ['r(G)]big_mkord (bigmax_sup (Ordinal lepg)). Qed.

Lemma

p_rank_le_rank

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "big_mkord", "bigmax_sup", "dvdn_leq", "ltnS", "mem_primes", "pG", "p_pr", "p_rank_gt0", "posnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_gt0 G : ('r(G) > 0) = (G :!=: 1).

Proof. case: (eqsVneq G 1) => [-> |]; first by rewrite rank1. case: (trivgVpdiv G) => [/eqP->// | [p p_pr]]. case/Cauchy=> // x Gx oxp _; apply: leq_trans (p_rank_le_rank p G). have EpGx: <[x]>%G \in 'E_p(G). by rewrite inE cycle_subG Gx abelemE // cycle_abelian -oxp exponent_dvdn. by apply: leq_trans (logn_le_p_rank...

Lemma

rank_gt0

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Cauchy", "abelemE", "apply", "cycle_abelian", "cycle_subG", "eqsVneq", "eqxx", "exponent_dvdn", "inE", "leq_trans", "logn_le_p_rank", "logn_prime", "orderE", "p_pr", "p_rank_le_rank", "rank1", "trivgVpdiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_witness G : {p | prime p & 'r(G) = 'r_p(G)}.

Proof. have [p _ defmG]: {p : 'I_(#|G|.+1) | true & 'r(G) = 'r_p(G)}. by rewrite ['r(G)]big_mkord; apply: eq_bigmax_cond; rewrite card_ord. case: (eqsVneq G 1) => [-> | ]; first by exists 2; rewrite // rank1 p_rank1. by rewrite -rank_gt0 defmG p_rank_gt0 mem_primes; case/andP; exists p. Qed.

Lemma

rank_witness

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "big_mkord", "card_ord", "eq_bigmax_cond", "eqsVneq", "mem_primes", "p_rank1", "p_rank_gt0", "prime", "rank1", "rank_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_pgroup p G : p.-group G -> 'r(G) = 'r_p(G).

Proof. move=> pG; apply/eqP; rewrite eqn_leq p_rank_le_rank andbT. rewrite ['r(G)]big_mkord; apply/bigmax_leqP=> [[q /= _] _]. case: (posnP 'r_q(G)) => [-> // |]; rewrite p_rank_gt0 mem_primes. by case/and3P=> q_pr _ qG; rewrite (eqnP (pgroupP pG q q_pr qG)). Qed.

Lemma

rank_pgroup

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "big_mkord", "bigmax_leqP", "eqnP", "eqn_leq", "group", "mem_primes", "pG", "p_rank_gt0", "p_rank_le_rank", "pgroupP", "posnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_Sylow p G P : p.-Sylow(G) P -> 'r(P) = 'r_p(G).

Proof. move=> sylP; have pP := pHall_pgroup sylP. by rewrite -(p_rank_Sylow sylP) -(rank_pgroup pP). Qed.

Lemma

rank_Sylow

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Sylow", "pHall_pgroup", "pP", "p_rank_Sylow", "rank_pgroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_abelem p G : p.-abelem G -> 'r(G) = logn p #|G|.

Proof. by move=> abelG; rewrite (rank_pgroup (abelem_pgroup abelG)) p_rank_abelem. Qed.

Lemma

rank_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "abelem_pgroup", "logn", "p_rank_abelem", "rank_pgroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nt_pnElem p n E A : E \in 'E_p^n(A) -> n > 0 -> E :!=: 1.

Proof. by case/pnElemP=> _ /rank_abelem <- <-; rewrite rank_gt0. Qed.

Lemma

nt_pnElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "pnElemP", "rank_abelem", "rank_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rankJ A x : 'r(A :^ x) = 'r(A).

Proof. by rewrite /rank cardJg; apply: eq_bigr => p _; rewrite p_rankJ. Qed.

Lemma

rankJ

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "cardJg", "eq_bigr", "p_rankJ", "rank" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rankS A B : A \subset B -> 'r(A) <= 'r(B).

Proof. move=> sAB; rewrite /rank !big_mkord; apply/bigmax_leqP=> p _. have leAB: #|A| < #|B|.+1 by rewrite ltnS subset_leq_card. by rewrite (bigmax_sup (widen_ord leAB p)) ?p_rankS. Qed.

Lemma

rankS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "big_mkord", "bigmax_leqP", "bigmax_sup", "ltnS", "p_rankS", "rank", "subset_leq_card", "widen_ord" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_geP n G : reflect (exists E, E \in 'E^n(G)) (n <= 'r(G)).

Proof. apply: (iffP idP) => [|[E]]. have [p _ ->] := rank_witness G; case/p_rank_geP=> E. by rewrite def_pnElem; case/setIP; exists E. case/nElemP=> p /[1!inE] /andP[EpG_E /eqP <-]. by rewrite (leq_trans (logn_le_p_rank EpG_E)) ?p_rank_le_rank. Qed.

Lemma

rank_geP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "def_pnElem", "inE", "leq_trans", "logn_le_p_rank", "nElemP", "p_rank_geP", "p_rank_le_rank", "rank_witness", "setIP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_morphim G : exponent (f @* G) %| exponent G.

Proof. apply/exponentP=> _ /morphimP[x Dx Gx ->]. by rewrite -morphX // expg_exponent // morph1. Qed.

Lemma

exponent_morphim

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Dx", "apply", "expg_exponent", "exponent", "exponentP", "morph1", "morphX", "morphimP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_LdivT n : f @* 'Ldiv_n() \subset 'Ldiv_n().

Proof. apply/subsetP=> _ /morphimP[x Dx xn ->]; rewrite inE in xn. by rewrite inE -morphX // (eqP xn) morph1. Qed.

Lemma

morphim_LdivT

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Dx", "apply", "inE", "morph1", "morphX", "morphimP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_Ldiv n A : f @* 'Ldiv_n(A) \subset 'Ldiv_n(f @* A).

Proof. by apply: subset_trans (morphimI f A _) (setIS _ _); apply: morphim_LdivT. Qed.

Lemma

morphim_Ldiv

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "morphimI", "morphim_LdivT", "setIS", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_abelem p G : p.-abelem G -> p.-abelem (f @* G).

Proof. case: (eqsVneq G 1) => [-> | ntG] abelG; first by rewrite morphim1 abelem1. have [p_pr _ _] := pgroup_pdiv (abelem_pgroup abelG) ntG. case/abelemP: abelG => // abG elemG; apply/abelemP; rewrite ?morphim_abelian //. by split=> // _ /morphimP[x Dx Gx ->]; rewrite -morphX // elemG ?morph1. Qed.

Lemma

morphim_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Dx", "abelem", "abelem1", "abelemP", "abelem_pgroup", "apply", "eqsVneq", "morph1", "morphX", "morphim1", "morphimP", "morphim_abelian", "p_pr", "pgroup_pdiv", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pElem p G E : E \in 'E_p(G) -> (f @* E)%G \in 'E_p(f @* G).

Proof. by rewrite !inE => /andP[sEG abelE]; rewrite morphimS // morphim_abelem. Qed.

Lemma

morphim_pElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelE", "inE", "morphimS", "morphim_abelem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pnElem p n G E : E \in 'E_p^n(G) -> {m | m <= n & (f @* E)%G \in 'E_p^m(f @* G)}.

Proof. rewrite inE => /andP[EpE /eqP <-]. by exists (logn p #|f @* E|); rewrite ?logn_morphim // inE morphim_pElem /=. Qed.

Lemma

morphim_pnElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "inE", "logn", "logn_morphim", "morphim_pElem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_grank G : G \subset D -> 'm(f @* G) <= 'm(G).

Proof. have [B defG <-] := grank_witness G; rewrite -defG gen_subG => sBD. by rewrite morphim_gen ?morphimEsub ?(leq_trans (grank_min _)) ?leq_imset_card. Qed.

Lemma

morphim_grank

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "defG", "gen_subG", "grank_min", "grank_witness", "leq_imset_card", "leq_trans", "morphimEsub", "morphim_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

defG : invm injf @* (f @* G) = G

:= morphim_invm injf sGD.

Let

defG

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "injf", "invm", "morphim_invm", "sGD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_injm : exponent (f @* G) = exponent G.

Proof. by apply/eqP; rewrite eqn_dvd -{3}defG !exponent_morphim. Qed.

Lemma

exponent_injm

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "defG", "eqn_dvd", "exponent", "exponent_morphim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Ldiv n A : f @* 'Ldiv_n(A) = 'Ldiv_n(f @* A).

Proof. apply/eqP; rewrite eqEsubset morphim_Ldiv. rewrite -[f @* 'Ldiv_n(A)](morphpre_invm injf). rewrite -sub_morphim_pre; first by rewrite subIset ?morphim_sub. rewrite injmI ?injm_invm // setISS ?morphim_LdivT //. by rewrite sub_morphim_pre ?morphim_sub // morphpre_invm. Qed.

Lemma

injm_Ldiv

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "eqEsubset", "injf", "injmI", "injm_invm", "morphim_Ldiv", "morphim_LdivT", "morphim_sub", "morphpre_invm", "setISS", "subIset", "sub_morphim_pre" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_abelem p : p.-abelem (f @* G) = p.-abelem G.

Proof. by apply/idP/idP; first rewrite -{2}defG; apply: morphim_abelem. Qed.

Lemma

injm_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "apply", "defG", "morphim_abelem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_pElem p (E : {group aT}) : E \subset D -> ((f @* E)%G \in 'E_p(f @* G)) = (E \in 'E_p(G)).

Proof. move=> sED; apply/idP/idP=> EpE; last exact: morphim_pElem. by rewrite -defG -(group_inj (morphim_invm injf sED)) morphim_pElem. Qed.

Lemma

injm_pElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "aT", "apply", "defG", "group", "group_inj", "injf", "last", "morphim_invm", "morphim_pElem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d