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
injm_pnElem p n (E : {group aT}) : E \subset D -> ((f @* E)%G \in 'E_p^n(f @* G)) = (E \in 'E_p^n(G)).	Proof. by move=> sED; rewrite inE injm_pElem // card_injm ?inE. Qed.	Lemma	injm_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...	[ "aT", "card_injm", "group", "inE", "injm_pElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_nElem n (E : {group aT}) : E \subset D -> ((f @* E)%G \in 'E^n(f @* G)) = (E \in 'E^n(G)).	Proof. move=> sED; apply/nElemP/nElemP=> [] [p EpE]; by exists p; rewrite injm_pnElem in EpE *. Qed.	Lemma	injm_nElem	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", "group", "injm_pnElem", "nElemP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_pmaxElem p (E : {group aT}) : E \subset D -> ((f @* E)%G \in 'E_p(f @ G)) = (E \in 'E*_p(G)).	Proof. move=> sED; have defE := morphim_invm injf sED. apply/pmaxElemP/pmaxElemP=> [] [EpE maxE]. split=> [\|H EpH sEH]; first by rewrite injm_pElem in EpE. have sHD: H \subset D by apply: subset_trans (sGD); case/pElemP: EpH. by rewrite -(morphim_invm injf sHD) [f @* H]maxE ?morphimS ?injm_pElem. rewrite injm_pEl...	Lemma	injm_pmaxElem	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", "fH", "group", "group_inj", "injf", "injm_pElem", "morphimS", "morphim_invm", "morphpreK", "pElemP", "pmaxElemP", "sGD", "sHD", "split", "sub_morphim_pre", "subsetIl", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_grank : 'm(f @* G) = 'm(G).	Proof. by apply/eqP; rewrite eqn_leq -{3}defG !morphim_grank ?morphimS. Qed.	Lemma	injm_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...	[ "apply", "defG", "eqn_leq", "morphimS", "morphim_grank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_p_rank p : 'r_p(f @* G) = 'r_p(G).	Proof. apply/eqP; rewrite eqn_leq; apply/andP; split. have [fE] := p_rank_witness p (f @* G); move: 'r_p(_) => n Ep_fE. apply/p_rank_geP; exists (f @*^-1 fE)%G. rewrite -injm_pnElem ?subsetIl ?(group_inj (morphpreK _)) //. by case/pnElemP: Ep_fE => sfEG _ _; rewrite (subset_trans sfEG) ?morphimS. have [E] := p_...	Lemma	injm_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...	[ "apply", "eqn_leq", "fE", "group_inj", "injm_pnElem", "morphimS", "morphpreK", "p_rank_geP", "p_rank_witness", "pnElemP", "split", "subsetIl", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_rank : 'r(f @* G) = 'r(G).	Proof. apply/eqP; rewrite eqn_leq; apply/andP; split. by have [p _ ->] := rank_witness (f @* G); rewrite injm_p_rank p_rank_le_rank. by have [p _ ->] := rank_witness G; rewrite -injm_p_rank p_rank_le_rank. Qed.	Lemma	injm_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...	[ "apply", "eqn_leq", "injm_p_rank", "p_rank_le_rank", "rank_witness", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isoGH : G \isog H.		Hypothesis	isoGH	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...	[ "isog" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_isog : exponent G = exponent H.	Proof. by case/isogP: isoGH => f injf <-; rewrite exponent_injm. Qed.	Lemma	exponent_isog	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", "exponent_injm", "injf", "isoGH", "isogP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_abelem p : p.-abelem G = p.-abelem H.	Proof. by case/isogP: isoGH => f injf <-; rewrite injm_abelem. Qed.	Lemma	isog_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", "injf", "injm_abelem", "isoGH", "isogP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_grank : 'm(G) = 'm(H).	Proof. by case/isogP: isoGH => f injf <-; rewrite [RHS]injm_grank. Qed.	Lemma	isog_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...	[ "injf", "injm_grank", "isoGH", "isogP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_p_rank p : 'r_p(G) = 'r_p(H).	Proof. by case/isogP: isoGH => f injf <-; rewrite injm_p_rank. Qed.	Lemma	isog_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...	[ "injf", "injm_p_rank", "isoGH", "isogP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_rank : 'r(G) = 'r(H).	Proof. by case/isogP: isoGH => f injf <-; rewrite injm_rank. Qed.	Lemma	isog_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...	[ "injf", "injm_rank", "isoGH", "isogP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_quotient G H : exponent (G / H) %\| exponent G.	Proof. exact: exponent_morphim. Qed.	Lemma	exponent_quotient	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", "exponent_morphim" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_LdivT n H : 'Ldiv_n() / H \subset 'Ldiv_n().	Proof. exact: morphim_LdivT. Qed.	Lemma	quotient_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...	[ "morphim_LdivT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_Ldiv n A H : 'Ldiv_n(A) / H \subset 'Ldiv_n(A / H).	Proof. exact: morphim_Ldiv. Qed.	Lemma	quotient_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...	[ "morphim_Ldiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_abelem G H : p.-abelem G -> p.-abelem (G / H).	Proof. exact: morphim_abelem. Qed.	Lemma	quotient_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", "morphim_abelem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_pElem G H E : E \in 'E_p(G) -> (E / H)%G \in 'E_p(G / H).	Proof. exact: morphim_pElem. Qed.	Lemma	quotient_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...	[ "morphim_pElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn_quotient G H : logn p #\|G / H\| <= logn p #\|G\|.	Proof. exact: logn_morphim. Qed.	Lemma	logn_quotient	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...	[ "logn", "logn_morphim" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_pnElem G H n E : E \in 'E_p^n(G) -> {m \| m <= n & (E / H)%G \in 'E_p^m(G / H)}.	Proof. exact: morphim_pnElem. Qed.	Lemma	quotient_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...	[ "morphim_pnElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_grank G H : G \subset 'N(H) -> 'm(G / H) <= 'm(G).	Proof. exact: morphim_grank. Qed.	Lemma	quotient_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...	[ "morphim_grank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_quotient G H : G \subset 'N(H) -> 'r_p(G) - 'r_p(H) <= 'r_p(G / H).	Proof. move=> nHG; rewrite leq_subLR. have [E EpE] := p_rank_witness p G; have{EpE} [sEG abelE <-] := pnElemP EpE. rewrite -(LagrangeI E H) lognM ?cardG_gt0 //. rewrite -card_quotient ?(subset_trans sEG) // leq_add ?logn_le_p_rank // !inE. by rewrite subsetIr (abelemS (subsetIl E H)). by rewrite quotientS ?quotient_a...	Lemma	p_rank_quotient	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...	[ "LagrangeI", "abelE", "abelemS", "cardG_gt0", "card_quotient", "inE", "leq_add", "leq_subLR", "lognM", "logn_le_p_rank", "nHG", "p_rank_witness", "pnElemP", "quotientS", "quotient_abelem", "subsetIl", "subsetIr", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_dprod K H G : K \x H = G -> 'r_p(K) + 'r_p(H) = 'r_p(G).	Proof. move=> defG; apply/eqP; rewrite eqn_leq -leq_subLR andbC. have [_ defKH cKH tiKH] := dprodP defG; have nKH := cents_norm cKH. rewrite {1}(isog_p_rank (quotient_isog nKH tiKH)) /= -quotientMidl defKH. rewrite p_rank_quotient; first by rewrite -defKH mul_subG ?normG. have [[E EpE] [F EpF]] := (p_rank_witness p K, ...	Lemma	p_rank_dprod	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", "cKH", "centSS", "cents_norm", "defG", "dprodEY", "dprodP", "dprod_abelem", "dprod_card", "eqn_leq", "eqxx", "genGid", "genM_join", "genS", "inE", "isog_p_rank", "leq_subLR", "lognM", "mul_subG", "nKH", "normG", "p_rank_geP", "p_rank_quotient", "p_ra...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_p'quotient G H : (p : nat)^'.-group H -> G \subset 'N(H) -> 'r_p(G / H) = 'r_p(G).	Proof. move=> p'H nHG; have [P sylP] := Sylow_exists p G. have [sPG pP _] := and3P sylP; have nHP := subset_trans sPG nHG. have tiHP: H :&: P = 1 := coprime_TIg (p'nat_coprime p'H pP). rewrite -(p_rank_Sylow sylP) -(p_rank_Sylow (quotient_pHall nHP sylP)). by rewrite (isog_p_rank (quotient_isog nHP tiHP)). Qed.	Lemma	p_rank_p'quotient	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", "coprime_TIg", "group", "isog_p_rank", "nHG", "nat", "p'nat_coprime", "pP", "p_rank_Sylow", "quotient_isog", "quotient_pHall", "subset_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_sub G : 'Ohm_n(G) \subset G.	Proof. by rewrite gen_subG; apply/subsetP=> x /setIdP[]. Qed.	Lemma	Ohm_sub	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", "gen_subG", "setIdP", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1 : 'Ohm_n([1 gT]) = 1.	Proof. exact: (trivgP (Ohm_sub _)). Qed.	Lemma	Ohm1	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...	[ "Ohm_sub", "gT", "trivgP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_id G : 'Ohm_n('Ohm_n(G)) = 'Ohm_n(G).	Proof. apply/eqP; rewrite eqEsubset Ohm_sub genS //. by apply/subsetP=> x /setIdP[Gx oxn]; rewrite inE mem_gen // inE Gx. Qed.	Lemma	Ohm_id	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...	[ "Ohm_sub", "apply", "eqEsubset", "genS", "inE", "mem_gen", "setIdP", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_cont rT G (f : {morphism G >-> rT}) : f @* 'Ohm_n(G) \subset 'Ohm_n(f @* G).	Proof. rewrite morphim_gen ?genS //; first by rewrite -gen_subG Ohm_sub. apply/subsetP=> fx /morphimP[x Gx]; rewrite inE Gx /=. case/OhmPredP=> p p_pr xpn_1 -> {fx}. rewrite inE morphimEdom imset_f //=; apply/OhmPredP; exists p => //. by rewrite -morphX // xpn_1 morph1. Qed.	Lemma	Ohm_cont	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...	[ "OhmPredP", "Ohm_sub", "apply", "genS", "gen_subG", "imset_f", "inE", "morph1", "morphX", "morphimEdom", "morphimP", "morphim_gen", "morphism", "p_pr", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
OhmS H G : H \subset G -> 'Ohm_n(H) \subset 'Ohm_n(G).	Proof. move=> sHG; apply: genS; apply/subsetP=> x /[!inE] /andP[Hx ->]. by rewrite (subsetP sHG). Qed.	Lemma	OhmS	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", "genS", "inE", "sHG", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
OhmE p G : p.-group G -> 'Ohm_n(G) = <<'Ldiv_(p ^ n)(G)>>.	Proof. move=> pG; congr <<_>>; apply/setP=> x /[!inE]; apply: andb_id2l => Gx. have [-> \| ntx] := eqVneq x 1; first by rewrite !expg1n. by rewrite (pdiv_p_elt (mem_p_elt pG Gx)). Qed.	Lemma	OhmE	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", "eqVneq", "expg1n", "group", "inE", "mem_p_elt", "pG", "pdiv_p_elt", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
OhmEabelian p G : p.-group G -> abelian 'Ohm_n(G) -> 'Ohm_n(G) = 'Ldiv_(p ^ n)(G).	Proof. move=> pG; rewrite (OhmE pG) abelian_gen => cGGn; rewrite gen_set_id //. rewrite -(setIidPr (subset_gen 'Ldiv_(p ^ n)(G))) setIA. by rewrite [_ :&: G](setIidPl _) ?gen_subG ?subsetIl // group_Ldiv ?abelian_gen. Qed.	Lemma	OhmEabelian	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...	[ "OhmE", "abelian", "abelian_gen", "gen_set_id", "gen_subG", "group", "group_Ldiv", "pG", "setIA", "setIidPl", "setIidPr", "subsetIl", "subset_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_p_cycle p x : p.-elt x -> 'Ohm_n(<[x]>) = <[x ^+ (p ^ (logn p #[x] - n))]>.	Proof. move=> p_x; apply/eqP; rewrite (OhmE p_x) eqEsubset cycle_subG mem_gen. rewrite !inE mem_cycle -expgM -expnD addnC -maxnE -order_dvdn. by rewrite -{1}(part_pnat_id p_x) p_part dvdn_exp2l ?leq_maxr. rewrite gen_subG andbT; apply/subsetP=> y /LdivP[x_y ypn]. case: (leqP (logn p #[x]) n) => [\|lt_n_x]. by rewr...	Lemma	Ohm_p_cycle	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...	[ "LdivP", "OhmE", "addnC", "apply", "cardSg", "congr_group", "cycle_subG", "cycle_sub_group", "dvdn_exp2l", "eqEsubset", "expgM", "expnD", "gen_subG", "inE", "leqP", "leq_maxr", "logn", "lognE", "ltnW", "maxnE", "mem_cycle", "mem_gen", "muln_divA", "order_dvdn", "p_par...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_dprod A B G : A \x B = G -> 'Ohm_n(A) \x 'Ohm_n(B) = 'Ohm_n(G).	Proof. case/dprodP => [[H K -> ->{A B}]] <- cHK tiHK. rewrite dprodEY //. - by rewrite (subset_trans (subset_trans _ cHK)) ?centS ?Ohm_sub. - by apply/trivgP; rewrite -tiHK setISS ?Ohm_sub. apply/eqP; rewrite -(cent_joinEr cHK) eqEsubset join_subG /=. rewrite !OhmS ?joing_subl ?joing_subr //= cent_joinEr //= -genM_join...	Lemma	Ohm_dprod	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...	[ "OhmPredP", "OhmS", "Ohm_sub", "apply", "centS", "cent_joinEr", "centsP", "commute", "dprodEY", "dprodP", "eqEsubset", "eq_invg_mul", "expgMn", "genM_join", "genS", "groupV", "groupX", "imset2P", "inE", "invg1", "invgK", "join_subG", "joing_subl", "joing_subr", "mem_g...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_sub G : 'Mho^n(G) \subset G.	Proof. rewrite gen_subG; apply/subsetP=> _ /imsetP[x /setIdP[Gx _] ->]. exact: groupX. Qed.	Lemma	Mho_sub	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...	[ "Mho", "apply", "gen_subG", "groupX", "imsetP", "setIdP", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho1 : 'Mho^n([1 gT]) = 1.	Proof. exact: (trivgP (Mho_sub _)). Qed.	Lemma	Mho1	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...	[ "Mho", "Mho_sub", "gT", "trivgP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_Mho rT D G (f : {morphism D >-> rT}) : G \subset D -> f @* 'Mho^n(G) = 'Mho^n(f @* G).	Proof. move=> sGD; have sGnD := subset_trans (Mho_sub G) sGD. apply/eqP; rewrite eqEsubset {1}morphim_gen -1?gen_subG // !gen_subG. apply/andP; split; apply/subsetP=> y. case/morphimP=> xpn _ /imsetP[x /setIdP[Gx]]. set p := pdiv _ => p_x -> -> {xpn y}; have Dx := subsetP sGD x Gx. by rewrite morphX // Mho_p_elt ...	Lemma	morphim_Mho	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", "Mho", "Mho_p_elt", "Mho_sub", "apply", "constt_p_elt", "eqEsubset", "gen_subG", "groupX", "imsetP", "mem_morphim", "morphX", "morph_constt", "morph_p_elt", "morphimP", "morphim_gen", "morphism", "p_elt_constt", "pdiv", "sGD", "setIdP", "split", "subsetP", "subset...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_cont rT G (f : {morphism G >-> rT}) : f @* 'Mho^n(G) \subset 'Mho^n(f @* G).	Proof. by rewrite morphim_Mho. Qed.	Lemma	Mho_cont	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...	[ "Mho", "morphim_Mho", "morphism" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
MhoS H G : H \subset G -> 'Mho^n(H) \subset 'Mho^n(G).	Proof. move=> sHG; apply: genS; apply: imsetS; apply/subsetP=> x. by rewrite !inE => /andP[Hx]; rewrite (subsetP sHG). Qed.	Lemma	MhoS	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...	[ "Mho", "apply", "genS", "imsetS", "inE", "sHG", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
MhoE p G : p.-group G -> 'Mho^n(G) = <<[set x ^+ (p ^ n) \| x in G]>>.	Proof. move=> pG; apply/eqP; rewrite eqEsubset !gen_subG; apply/andP. do [split; apply/subsetP => xpn; case/imsetP => x] => [\|Gx ->]; last first. by rewrite Mho_p_elt ?(mem_p_elt pG). case/setIdP=> Gx _ ->; have [-> \| ntx] := eqVneq x 1; first by rewrite expg1n. by rewrite (pdiv_p_elt (mem_p_elt pG Gx) ntx) mem_gen /...	Lemma	MhoE	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...	[ "Mho", "Mho_p_elt", "apply", "eqEsubset", "eqVneq", "expg1n", "gen_subG", "group", "imsetP", "imset_f", "last", "mem_gen", "mem_p_elt", "pG", "pdiv_p_elt", "setIdP", "split", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
MhoEabelian p G : p.-group G -> abelian G -> 'Mho^n(G) = [set x ^+ (p ^ n) \| x in G].	Proof. move=> pG cGG; rewrite (MhoE pG); rewrite gen_set_id //; apply/group_setP. split=> [\|xn yn]; first by apply/imsetP; exists 1; rewrite ?expg1n. case/imsetP=> x Gx ->; case/imsetP=> y Gy ->. by rewrite -expgMn; [apply: (centsP cGG) \| apply: imset_f; rewrite groupM]. Qed.	Lemma	MhoEabelian	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...	[ "Mho", "MhoE", "abelian", "apply", "cGG", "centsP", "expg1n", "expgMn", "gen_set_id", "group", "groupM", "group_setP", "imsetP", "imset_f", "pG", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_Mho G : 'Mho^n(G) == 1 -> 'Ohm_n(G) == G.	Proof. rewrite -subG1 gen_subG eqEsubset Ohm_sub /= => Gp1. rewrite -{1}(Sylow_gen G) genS //; apply/bigcupsP=> P. case/SylowP=> p p_pr /and3P[sPG pP _]; apply/subsetP=> x Px. have Gx := subsetP sPG x Px; rewrite inE Gx //=. rewrite (sameP eqP set1P) (subsetP Gp1) ?mem_gen //; apply: imset_f. by rewrite inE Gx; apply: ...	Lemma	trivg_Mho	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...	[ "Mho", "Ohm_sub", "Px", "SylowP", "Sylow_gen", "apply", "bigcupsP", "eqEsubset", "genS", "gen_subG", "imset_f", "inE", "mem_gen", "mem_p_elt", "pP", "p_pr", "pgroup_p", "set1P", "subG1", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_p_cycle p x : p.-elt x -> 'Mho^n(<[x]>) = <[x ^+ (p ^ n)]>.	Proof. move=> p_x. apply/eqP; rewrite (MhoE p_x) eqEsubset cycle_subG mem_gen. by apply: imset_f; apply: cycle_id. rewrite gen_subG andbT; apply/subsetP=> _ /imsetP[_ /cycleP[k ->] ->]. by rewrite -expgM mulnC expgM mem_cycle. Qed.	Lemma	Mho_p_cycle	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...	[ "Mho", "MhoE", "apply", "cycleP", "cycle_id", "cycle_subG", "eqEsubset", "expgM", "gen_subG", "imsetP", "imset_f", "mem_cycle", "mem_gen", "mulnC", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_cprod A B G : A \* B = G -> 'Mho^n(A) \* 'Mho^n(B) = 'Mho^n(G).	Proof. case/cprodP => [[H K -> ->{A B}]] <- cHK; rewrite cprodEY //. by rewrite (subset_trans (subset_trans _ cHK)) ?centS ?Mho_sub. apply/eqP; rewrite -(cent_joinEr cHK) eqEsubset join_subG /=. rewrite !MhoS ?joing_subl ?joing_subr //= cent_joinEr // -genM_join. apply: genS; apply/subsetP=> xypn /imsetP[_ /setIdP[/i...	Lemma	Mho_cprod	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...	[ "Mho", "MhoS", "Mho_p_elt", "Mho_sub", "apply", "centS", "cent_joinEr", "centsP", "commute", "commuteX2", "consttM", "constt_p_elt", "cprodEY", "cprodP", "eqEsubset", "expgMn", "genM_join", "genS", "groupX", "imset2P", "imsetP", "join_subG", "joing_subl", "joing_subr", ...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_dprod A B G : A \x B = G -> 'Mho^n(A) \x 'Mho^n(B) = 'Mho^n(G).	Proof. case/dprodP => [[H K -> ->{A B}]] defG cHK tiHK. rewrite dprodEcp; last by apply: Mho_cprod; rewrite cprodE. by apply/trivgP; rewrite -tiHK setISS ?Mho_sub. Qed.	Lemma	Mho_dprod	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...	[ "Mho", "Mho_cprod", "Mho_sub", "apply", "cprodE", "defG", "dprodEcp", "dprodP", "last", "setISS", "trivgP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_igFun i	:= [igFun by Ohm_sub i & Ohm_cont i].	Canonical	Ohm_igFun	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...	[ "Ohm_cont", "Ohm_sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_gFun i	:= [gFun by Ohm_cont i].	Canonical	Ohm_gFun	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...	[ "Ohm_cont" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_mgFun i	:= [mgFun by OhmS i].	Canonical	Ohm_mgFun	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...	[ "OhmS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_igFun i	:= [igFun by Mho_sub i & Mho_cont i].	Canonical	Mho_igFun	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...	[ "Mho_cont", "Mho_sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_gFun i	:= [gFun by Mho_cont i].	Canonical	Mho_gFun	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...	[ "Mho_cont" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_mgFun i	:= [mgFun by MhoS i].	Canonical	Mho_mgFun	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...	[ "MhoS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_char : 'Ohm_n(G) \char G.	Proof. exact: gFchar. Qed.	Lemma	Ohm_char	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...	[ "char", "gFchar" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_normal : 'Ohm_n(G) <\| G.	Proof. exact: gFnormal. Qed.	Lemma	Ohm_normal	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...	[ "gFnormal" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_char : 'Mho^n(G) \char G.	Proof. exact: gFchar. Qed.	Lemma	Mho_char	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...	[ "Mho", "char", "gFchar" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_normal : 'Mho^n(G) <\| G.	Proof. exact: gFnormal. Qed.	Lemma	Mho_normal	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...	[ "Mho", "gFnormal" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_Ohm (f : {morphism D >-> rT}) : G \subset D -> f @* 'Ohm_n(G) \subset 'Ohm_n(f @* G).	Proof. exact: morphimF. Qed.	Lemma	morphim_Ohm	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...	[ "morphimF", "morphism" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Ohm (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> f @* 'Ohm_n(G) = 'Ohm_n(f @* G).	Proof. by move=> injf; apply: injmF. Qed.	Lemma	injm_Ohm	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", "injf", "injmF", "morphism" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_Ohm (H : {group rT}) : G \isog H -> 'Ohm_n(G) \isog 'Ohm_n(H).	Proof. exact: gFisog. Qed.	Lemma	isog_Ohm	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...	[ "gFisog", "group", "isog" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_Mho (H : {group rT}) : G \isog H -> 'Mho^n(G) \isog 'Mho^n(H).	Proof. exact: gFisog. Qed.	Lemma	isog_Mho	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...	[ "Mho", "gFisog", "group", "isog" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm0 G : 'Ohm_0(G) = 1.	Proof. by apply/trivgP; rewrite /= gen_subG; apply/subsetP=> x /setIdP[_] /[1!inE]. Qed.	Lemma	Ohm0	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", "gen_subG", "inE", "setIdP", "subsetP", "trivgP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_leq m n G : m <= n -> 'Ohm_m(G) \subset 'Ohm_n(G).	Proof. move/subnKC <-; rewrite genS //; apply/subsetP=> y. by rewrite !inE expnD expgM => /andP[-> /eqP->]; rewrite expg1n /=. Qed.	Lemma	Ohm_leq	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", "expg1n", "expgM", "expnD", "genS", "inE", "subnKC", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
OhmJ n G x : 'Ohm_n(G :^ x) = 'Ohm_n(G) :^ x.	Proof. rewrite -{1}(setIid G) -(setIidPr (Ohm_sub n G)). by rewrite -!morphim_conj injm_Ohm ?injm_conj. Qed.	Lemma	OhmJ	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...	[ "Ohm_sub", "injm_Ohm", "injm_conj", "morphim_conj", "setIid", "setIidPr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho0 G : 'Mho^0(G) = G.	Proof. apply/eqP; rewrite eqEsubset Mho_sub /=. apply/subsetP=> x Gx; rewrite -[x]prod_constt group_prod // => p _. exact: Mho_p_elt (groupX _ Gx) (p_elt_constt _ _). Qed.	Lemma	Mho0	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...	[ "Mho", "Mho_p_elt", "Mho_sub", "apply", "eqEsubset", "groupX", "group_prod", "p_elt_constt", "prod_constt", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_leq m n G : m <= n -> 'Mho^n(G) \subset 'Mho^m(G).	Proof. move/subnKC <-; rewrite gen_subG //. apply/subsetP=> _ /imsetP[x /setIdP[Gx p_x] ->]. by rewrite expnD expgM groupX ?(Mho_p_elt _ _ p_x). Qed.	Lemma	Mho_leq	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...	[ "Mho", "Mho_p_elt", "apply", "expgM", "expnD", "gen_subG", "groupX", "imsetP", "setIdP", "subnKC", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
MhoJ n G x : 'Mho^n(G :^ x) = 'Mho^n(G) :^ x.	Proof. by rewrite -{1}(setIid G) -(setIidPr (Mho_sub n G)) -!morphim_conj morphim_Mho. Qed.	Lemma	MhoJ	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...	[ "Mho", "Mho_sub", "morphim_Mho", "morphim_conj", "setIid", "setIidPr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extend_cyclic_Mho G p x : p.-group G -> x \in G -> 'Mho^1(G) = <[x ^+ p]> -> forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (p ^ k)]>.	Proof. move=> pG Gx defG1 [//\|k _]; have pX := mem_p_elt pG Gx. apply/eqP; rewrite eqEsubset cycle_subG (Mho_p_elt _ Gx pX) andbT. rewrite (MhoE _ pG) gen_subG; apply/subsetP=> ypk; case/imsetP=> y Gy ->{ypk}. have: y ^+ p \in <[x ^+ p]> by rewrite -defG1 (Mho_p_elt 1 _ (mem_p_elt pG Gy)). rewrite !expnS /= !expgM => /...	Lemma	extend_cyclic_Mho	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...	[ "Mho", "MhoE", "Mho_p_elt", "apply", "cycleP", "cycle_subG", "eqEsubset", "expgM", "expnS", "gen_subG", "group", "imsetP", "mem_cycle", "mem_p_elt", "mulnC", "mulnCA", "pG", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1Eprime G : 'Ohm_1(G) = <<[set x in G \| prime #[x]]>>.	Proof. rewrite -['Ohm_1(G)](genD1 (group1 _)); congr <<_>>. apply/setP=> x; rewrite !inE andbCA -order_dvdn -order_gt1; congr (_ && _). apply/andP/idP=> [[p_gt1] \| p_pr]; last by rewrite prime_gt1 ?pdiv_id. set p := pdiv _ => ox_p; have p_pr: prime p by rewrite pdiv_prime. by have [_ dv_p] := primeP p_pr; case/pred2P: ...	Lemma	Ohm1Eprime	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", "genD1", "group1", "inE", "last", "order_dvdn", "order_gt1", "p_gt1", "p_pr", "pdiv", "pdiv_id", "pdiv_prime", "pred2P", "prime", "primeP", "prime_gt1", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_Ohm1 p G : p.-group G -> p.-abelem 'Ohm_1(G) = abelian 'Ohm_1(G).	Proof. move=> pG; rewrite /abelem (pgroupS (Ohm_sub 1 G)) //. case abG1: (abelian _) => //=; apply/exponentP=> x. by rewrite (OhmEabelian pG abG1); case/LdivP. Qed.	Lemma	abelem_Ohm1	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...	[ "LdivP", "OhmEabelian", "Ohm_sub", "abelem", "abelian", "apply", "exponentP", "group", "pG", "pgroupS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1_abelem p G : p.-group G -> abelian G -> p.-abelem ('Ohm_1(G)).	Proof. by move=> pG cGG; rewrite abelem_Ohm1 ?(abelianS (Ohm_sub 1 G)). Qed.	Lemma	Ohm1_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...	[ "Ohm_sub", "abelem", "abelem_Ohm1", "abelian", "abelianS", "cGG", "group", "pG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1_id p G : p.-abelem G -> 'Ohm_1(G) = G.	Proof. case/and3P=> pG cGG /exponentP Gp. apply/eqP; rewrite eqEsubset Ohm_sub (OhmE 1 pG) sub_gen //. by apply/subsetP=> x Gx; rewrite !inE Gx Gp /=. Qed.	Lemma	Ohm1_id	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...	[ "OhmE", "Ohm_sub", "abelem", "apply", "cGG", "eqEsubset", "exponentP", "inE", "pG", "sub_gen", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_Ohm1P p G : abelian G -> p.-group G -> reflect ('Ohm_1(G) = G) (p.-abelem G).	Proof. move=> cGG pG. by apply: (iffP idP) => [\| <-]; [apply: Ohm1_id \| apply: Ohm1_abelem]. Qed.	Lemma	abelem_Ohm1P	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...	[ "Ohm1_abelem", "Ohm1_id", "abelem", "abelian", "apply", "cGG", "group", "pG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
TI_Ohm1 G H : H :&: 'Ohm_1(G) = 1 -> H :&: G = 1.	Proof. move=> tiHG1; case: (trivgVpdiv (H :&: G)) => // [[p pr_p]]. case/Cauchy=> // x /setIP[Hx Gx] ox. suffices x1: x \in [1] by rewrite -ox (set1P x1) order1 in pr_p. by rewrite -{}tiHG1 inE Hx Ohm1Eprime mem_gen // inE Gx ox. Qed.	Lemma	TI_Ohm1	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", "Ohm1Eprime", "inE", "mem_gen", "order1", "pr_p", "set1P", "setIP", "trivgVpdiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1_eq1 G : ('Ohm_1(G) == 1) = (G :==: 1).	Proof. apply/idP/idP => [/eqP G1_1 \| /eqP->]; last by rewrite -subG1 Ohm_sub. by rewrite -(setIid G) TI_Ohm1 // G1_1 setIg1. Qed.	Lemma	Ohm1_eq1	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...	[ "Ohm_sub", "TI_Ohm1", "apply", "last", "setIg1", "setIid", "subG1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_Ohm1 G H : G :&: H != 1 -> G :&: 'Ohm_1(H) != 1.	Proof. by apply: contraNneq => /TI_Ohm1->. Qed.	Lemma	meet_Ohm1	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_Ohm1", "apply", "contraNneq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1_cent_max G E p : E \in 'E*_p(G) -> p.-group G -> 'Ohm_1('C_G(E)) = E.	Proof. move=> EpmE pG; have [G1 \| ntG]:= eqsVneq G 1. case/pmaxElemP: EpmE; case/pElemP; rewrite G1 => /trivgP-> _ _. by apply/trivgP; rewrite cent1T setIT Ohm_sub. have [p_pr _ _] := pgroup_pdiv pG ntG. by rewrite (OhmE 1 (pgroupS (subsetIl G _) pG)) (pmaxElem_LdivP _ _) ?genGid. Qed.	Lemma	Ohm1_cent_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...	[ "G1", "OhmE", "Ohm_sub", "apply", "cent1T", "eqsVneq", "genGid", "group", "pElemP", "pG", "p_pr", "pgroupS", "pgroup_pdiv", "pmaxElemP", "pmaxElem_LdivP", "setIT", "subsetIl", "trivgP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1_cyclic_pgroup_prime p G : cyclic G -> p.-group G -> G :!=: 1 -> #\|'Ohm_1(G)\| = p.	Proof. move=> cycG pG ntG; set K := 'Ohm_1(G). have abelK: p.-abelem K by rewrite Ohm1_abelem ?cyclic_abelian. have sKG: K \subset G := Ohm_sub 1 G. case/cyclicP: (cyclicS sKG cycG) => x /=; rewrite -/K => defK. rewrite defK -orderE (abelem_order_p abelK) //= -/K ?defK ?cycle_id //. rewrite -cycle_eq1 -defK -(setIidPr ...	Lemma	Ohm1_cyclic_pgroup_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...	[ "Ohm1_abelem", "Ohm_sub", "TI_Ohm1", "abelem", "abelem_order_p", "apply", "contraNneq", "cycle_eq1", "cycle_id", "cyclic", "cyclicP", "cyclicS", "cyclic_abelian", "group", "orderE", "pG", "sKG", "setIid", "setIidPr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cyclic_pgroup_dprod_trivg p A B C : p.-group C -> cyclic C -> A \x B = C -> A = 1 /\ B = C \/ B = 1 /\ A = C.	Proof. move=> pC cycC; case/cyclicP: cycC pC => x ->{C} pC defC. case/dprodP: defC => [] [G H -> ->{A B}] defC _ tiGH; rewrite -defC. have [/trivgP \| ntC] := eqVneq <[x]> 1. by rewrite -defC mulG_subG => /andP[/trivgP-> _]; rewrite mul1g; left. have [pr_p _ _] := pgroup_pdiv pC ntC; pose K := 'Ohm_1(<[x]>). have prK ...	Lemma	cyclic_pgroup_dprod_trivg	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...	[ "Ohm1_cyclic_pgroup_prime", "TI_Ohm1", "apply", "cards1", "cycle_cyclic", "cyclic", "cyclicP", "dprodP", "eqVneq", "group", "last", "mul1g", "mulG_subG", "mulG_subl", "mulG_subr", "mulg1", "pgroup_pdiv", "pr_p", "prime", "prime_subgroupVti", "sKG", "setIidPl", "subsetI", ...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
piOhm1 G : \pi('Ohm_1(G)) = \pi(G).	Proof. apply/eq_piP => p; apply/idP/idP; first exact: (piSg (Ohm_sub 1 G)). rewrite !mem_primes !cardG_gt0 => /andP[p_pr /Cauchy[] // x Gx oxp]. by rewrite p_pr -oxp order_dvdG //= Ohm1Eprime mem_gen // inE Gx oxp. Qed.	Lemma	piOhm1	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", "Ohm1Eprime", "Ohm_sub", "apply", "cardG_gt0", "eq_piP", "inE", "mem_gen", "mem_primes", "order_dvdG", "p_pr", "pi", "piSg" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1Eexponent p G : prime p -> exponent 'Ohm_1(G) %\| p -> 'Ohm_1(G) = 'Ldiv_p(G).	Proof. move=> p_pr expG1p; have pG: p.-group G. apply: sub_in_pnat (pnat_pi (cardG_gt0 G)) => q _. rewrite -piOhm1 mem_primes; case/and3P=> q_pr _; apply: pgroupP q_pr. by rewrite -pnat_exponent (pnat_dvd expG1p) ?pnat_id. apply/eqP; rewrite eqEsubset {2}(OhmE 1 pG) subset_gen subsetI Ohm_sub. by rewrite sub_Ldiv...	Lemma	Ohm1Eexponent	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...	[ "OhmE", "Ohm_sub", "apply", "cardG_gt0", "eqEsubset", "exponent", "group", "mem_primes", "pG", "p_pr", "pgroupP", "piOhm1", "pnat_dvd", "pnat_exponent", "pnat_id", "pnat_pi", "prime", "sub_LdivT", "sub_in_pnat", "subsetI", "subset_gen" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_Ohm1 p G : 'r_p('Ohm_1(G)) = 'r_p(G).	Proof. apply/eqP; rewrite eqn_leq p_rankS ?Ohm_sub //. apply/bigmax_leqP=> E /setIdP[sEG abelE]. by rewrite (bigmax_sup E) // inE -{1}(Ohm1_id abelE) OhmS. Qed.	Lemma	p_rank_Ohm1	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...	[ "Ohm1_id", "OhmS", "Ohm_sub", "abelE", "apply", "bigmax_leqP", "bigmax_sup", "eqn_leq", "inE", "p_rankS", "setIdP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_Ohm1 G : 'r('Ohm_1(G)) = 'r(G).	Proof. apply/eqP; rewrite eqn_leq rankS ?Ohm_sub //. by have [p _ ->] := rank_witness G; rewrite -p_rank_Ohm1 p_rank_le_rank. Qed.	Lemma	rank_Ohm1	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...	[ "Ohm_sub", "apply", "eqn_leq", "p_rank_Ohm1", "p_rank_le_rank", "rankS", "rank_witness" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank_abelian p G : abelian G -> 'r_p(G) = logn p #\|'Ohm_1(G)\|.	Proof. move=> cGG; have nilG := abelian_nil cGG; case p_pr: (prime p); last first. by apply/eqP; rewrite lognE p_pr eqn0Ngt p_rank_gt0 mem_primes p_pr. case/dprodP: (Ohm_dprod 1 (nilpotent_pcoreC p nilG)) => _ <- _ /TI_cardMg->. rewrite mulnC logn_Gauss. rewrite prime_coprime // -p'natE // -/(pgroup _ _). exact: ...	Lemma	p_rank_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...	[ "Ohm1_abelem", "Ohm_dprod", "Ohm_sub", "TI_cardMg", "abelian", "abelianS", "abelian_nil", "apply", "cGG", "dprodP", "eqn0Ngt", "last", "logn", "lognE", "logn_Gauss", "mem_primes", "mulnC", "nilpotent_pcoreC", "nilpotent_pcore_Hall", "p'natE", "p_pr", "p_rank_Ohm1", "p_ran...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_abelian_pgroup p G : p.-group G -> abelian G -> 'r(G) = logn p #\|'Ohm_1(G)\|.	Proof. by move=> pG cGG; rewrite (rank_pgroup pG) p_rank_abelian. Qed.	Lemma	rank_abelian_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...	[ "abelian", "cGG", "group", "logn", "pG", "p_rank_abelian", "rank_pgroup" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_splits x G : x \in G -> #[x] = exponent G -> abelian G -> [splits G, over <[x]>].	Proof. move=> Gx ox cGG; apply/splitsP; have [n] := ubnP #\|G\|. elim: n gT => // n IHn aT in x G Gx ox cGG * => /ltnSE-leGn. have: <[x]> \subset G by [rewrite cycle_subG]; rewrite subEproper. case/predU1P=> [<- \| /properP[sxG [y]]]. by exists 1%G; rewrite inE -subG1 subsetIr mulg1 /=. have [m] := ubnP #[y]; elim: m y ...	Lemma	abelian_splits	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", "abelian", "apply", "cGG", "card_quotient", "centsP", "complP", "coset", "cycleP", "cycle_eq1", "cycle_id", "cycle_subG", "cycle_sub_group", "divnA", "dvdn_exponent", "dvdn_leq", "eqEsubset", "eqn_dvd", "eqxx", "expgM", "expgMn", "expgVn", "exponent", "exponentP",...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem_splits p G H : p.-abelem G -> H \subset G -> [splits G, over H].	Proof. have [m] := ubnP #\|G\|; elim: m G H => // m IHm G H /ltnSE-leGm abelG sHG. have [-> \| ] := eqsVneq H 1. by apply/splitsP; exists G; rewrite inE mul1g -subG1 subsetIl /=. case/trivgPn=> x Hx ntx; have Gx := subsetP sHG x Hx. have [_ cGG eGp] := and3P abelG. have ox: #[x] = exponent G. by apply/eqP; rewrite eq...	Lemma	abelem_splits	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_order_p", "abelian_splits", "apply", "cGG", "complP", "cycle_eq1", "cycle_subG", "defG", "dvdn_exponent", "eqn_dvd", "eqsVneq", "exponent", "group_modl", "inE", "leq_trans", "ltnSE", "mul1g", "mulG_subr", "mulgA", "properP", "proper_card", "...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_type_subproof G : {H : {group gT} & abelian G -> {x \| #[x] = exponent G & <[x]> \x H = G}}.	Proof. case cGG: (abelian G); last by exists G. have [x Gx ox] := exponent_witness (abelian_nil cGG). case/splitsP/ex_mingroup: (abelian_splits Gx (esym ox) cGG) => H. case/mingroupp/complP=> tixH defG; exists H => _. exists x; rewrite ?dprodE // (sub_abelian_cent2 cGG) ?cycle_subG //. by rewrite -defG mulG_subr. Qed.	Fact	abelian_type_subproof	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_nil", "abelian_splits", "cGG", "complP", "cycle_subG", "defG", "dprodE", "ex_mingroup", "exponent", "exponent_witness", "gT", "group", "last", "mingroupp", "mulG_subr", "splitsP", "sub_abelian_cent2" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_type_rec n G	:= if n is n'.+1 then if abelian G && (G :!=: 1) then exponent G :: abelian_type_rec n' (tag (abelian_type_subproof G)) else [::] else [::].	Fixpoint	abelian_type_rec	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_type_subproof", "exponent", "n'" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_type (A : {set gT})	:= abelian_type_rec #\|A\| <<A>>.	Definition	abelian_type	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_type_rec", "gT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_type_dvdn_sorted A : sorted [rel m n \| n %\| m] (abelian_type A).	Proof. set R := SimplRel _; pose G := <<A>>%G; pose M := G. suffices: path R (exponent M) (abelian_type A) by case: (_ A) => // m t /andP[]. rewrite /abelian_type -/G; have: G \subset M by []. elim: {A}#\|A\| G M => //= n IHn G M sGM. case: andP => //= -[cGG ntG]; rewrite exponentS ?IHn //=. case: (abelian_type_subproof ...	Lemma	abelian_type_dvdn_sorted	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_type", "abelian_type_subproof", "cGG", "dprodP", "exponent", "exponentS", "mulG_subr", "path", "rel", "sorted" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_type_gt1 A : all [pred m \| m > 1] (abelian_type A).	Proof. rewrite /abelian_type; elim: {A}#\|A\| <<A>>%G => //= n IHn G. case: ifP => //= /andP[_ ntG]; rewrite {n}IHn. by rewrite ltn_neqAle exponent_gt0 eq_sym -dvdn1 -trivg_exponent ntG. Qed.	Lemma	abelian_type_gt1	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_type", "all", "dvdn1", "eq_sym", "exponent_gt0", "ltn_neqAle", "trivg_exponent" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_type_sorted A : sorted geq (abelian_type A).	Proof. have:= abelian_type_dvdn_sorted A; have:= abelian_type_gt1 A. case: (abelian_type A) => //= m t; elim: t m => //= n t IHt m /andP[]. by move/ltnW=> m_gt0 t_gt1 /andP[n_dv_m /IHt->]; rewrite // dvdn_leq. Qed.	Lemma	abelian_type_sorted	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_type", "abelian_type_dvdn_sorted", "abelian_type_gt1", "dvdn_leq", "geq", "ltnW", "sorted" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_structure G : abelian G -> {b \| \big[dprod/1]_(x <- b) <[x]> = G & map order b = abelian_type G}.	Proof. rewrite /abelian_type genGidG; have [n] := ubnPleq #\|G\|. elim: n G => /= [\|n IHn] G leGn cGG; first by rewrite leqNgt cardG_gt0 in leGn. rewrite [in _ && _]cGG /=; case: ifP => [ntG\|/eqP->]; last first. by exists [::]; rewrite ?big_nil. case: (abelian_type_subproof G) => H /= [//\|x ox xdefG]; rewrite -ox. have...	Theorem	abelian_structure	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", "abelian", "abelianS", "abelian_type", "abelian_type_subproof", "big_cons", "big_nil", "cGG", "cardG_gt0", "defG", "dprod", "dprodP", "dvdn1", "eq_sym", "genGidG", "last", "leqNgt", "leq_trans", "ltnS", "ltn_Pmull", "ltn_neqAle", "map", "mulG_subr", "order"...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
count_logn_dprod_cycle p n b G : \big[dprod/1]_(x <- b) <[x]> = G -> count [pred x \| logn p #[x] > n] b = logn p #\|'Ohm_n.+1(G) : 'Ohm_n(G)\|.	Proof. have sOn1 H: 'Ohm_n(H) \subset 'Ohm_n.+1(H) by apply: Ohm_leq. pose lnO i (A : {set gT}) := logn p #\|'Ohm_i(A)\|. have lnO_le H: lnO n H <= lnO n.+1 H. by rewrite dvdn_leq_log ?cardG_gt0 // cardSg ?sOn1. have lnOx i A B H: A \x B = H -> lnO i A + lnO i B = lnO i H. move=> defH; case/dprodP: defH (defH) => {A ...	Lemma	count_logn_dprod_cycle	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...	[ "Ohm_dprod", "Ohm_leq", "Ohm_p_cycle", "Ohm_sub", "TI_cardMg", "addn1", "addnBA", "addnC", "apply", "big_cons", "big_nil", "cardG_gt0", "cardSg", "centsP", "consttC", "coprime", "coprime_TIg", "coprime_partC", "count", "cycleM", "cycle_abelian", "cycle_subG", "defG", "d...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_type_pgroup p b G : p.-group G -> \big[dprod/1]_(x <- b) <[x]> = G -> 1 \notin b -> perm_eq (abelian_type G) (map order b).	Proof. rewrite perm_sym; move: b => b1 pG defG1 ntb1. have cGG: abelian G. elim: (b1) {pG}G defG1 => [_ <-\|x b IHb G]; first by rewrite big_nil abelian1. rewrite big_cons; case/dprodP=> [[_ H _ defH]] <-; rewrite defH => cxH _. by rewrite abelianM cycle_abelian IHb. have p_bG b: \big[dprod/1]_(x <- b) <[x]> = G -...	Lemma	abelian_type_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...	[ "abelian", "abelian1", "abelianM", "abelian_structure", "abelian_type", "abelian_type_gt1", "add0n", "all", "allP", "all_cat", "apply", "big_cons", "big_nil", "cGG", "count", "count_logn_dprod_cycle", "count_map", "cycle_abelian", "cycle_eq1", "defG", "dprod", "dprodP", "...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_abelian_type G : abelian G -> size (abelian_type G) = 'r(G).	Proof. move=> cGG; have [b defG def_t] := abelian_structure cGG. apply/eqP; rewrite -def_t size_map eqn_leq andbC; apply/andP; split. have [p p_pr ->] := rank_witness G; rewrite p_rank_abelian //. by rewrite -indexg1 -(Ohm0 G) -(count_logn_dprod_cycle _ _ defG) count_size. case/lastP def_b: b => // [b' x]; pose p :...	Lemma	size_abelian_type	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...	[ "Ohm0", "abelian", "abelian_structure", "abelian_type", "abelian_type_dvdn_sorted", "abelian_type_gt1", "all", "allP", "all_cat", "all_count", "apply", "cGG", "cat_path", "cat_rcons", "cats1", "count_logn_dprod_cycle", "count_size", "defG", "dvdn_trans", "eqn_leq", "headI", ...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_card_Ohm_Mho_abelian n G : abelian G -> (#\|'Ohm_n(G)\| * #\|'Mho^n(G)\|)%N = #\|G\|.	Proof. case/abelian_structure => b defG _. elim: b G defG => [_ <-\|x b IHb G]. by rewrite !big_nil (trivgP (Ohm_sub _ _)) (trivgP (Mho_sub _ _)) !cards1. rewrite big_cons => defG; rewrite -(dprod_card defG). rewrite -(dprod_card (Ohm_dprod n defG)) -(dprod_card (Mho_dprod n defG)) /=. rewrite mulnCA -!mulnA mulnCA mu...	Lemma	mul_card_Ohm_Mho_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...	[ "Mho", "Mho_dprod", "Mho_p_cycle", "Mho_sub", "Ohm_dprod", "Ohm_p_cycle", "Ohm_sub", "abelian", "abelian_structure", "apply", "big_cons", "big_nil", "card_pgroup", "cards1", "cent1P", "cent_cycle", "commute", "commuteX2", "consttC", "coprime", "coprime_TIg", "coprime_partC"...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
grank_abelian G : abelian G -> 'm(G) = 'r(G).	Proof. move=> cGG; apply/eqP; rewrite eqn_leq; apply/andP; split. rewrite -size_abelian_type //; case/abelian_structure: cGG => b defG <-. suffices <-: <<[set x in b]>> = G. by rewrite (leq_trans (grank_min _)) // size_map cardsE card_size. rewrite -{G defG}(bigdprodWY defG). elim: b => [\|x b IHb]; first by...	Lemma	grank_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...	[ "Mho", "Mho_p_elt", "Mho_sub", "abelian", "abelian_structure", "apply", "big_cons", "big_nil", "bigdprodWY", "cGG", "cardG_gt0", "card_quotient", "card_size", "cards1", "cardsD1", "cardsE", "cent_joinEl", "coset", "coset_id", "cycle_subG", "defG", "divgS", "dvdn_Pexp2l", ...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank_cycle (x : gT) : 'r(<[x]>) = (x != 1).	Proof. have [->\|ntx] := eqVneq x 1; first by rewrite cycle1 rank1. apply/eqP; rewrite eqn_leq rank_gt0 cycle_eq1 ntx andbT. by rewrite -grank_abelian ?cycle_abelian //= -(cards1 x) grank_min. Qed.	Lemma	rank_cycle	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", "cycle1", "cycle_abelian", "cycle_eq1", "eqVneq", "eqn_leq", "gT", "grank_abelian", "grank_min", "rank1", "rank_gt0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_rank1_cyclic G : abelian G -> cyclic G = ('r(G) <= 1).	Proof. move=> cGG; have [b defG atypG] := abelian_structure cGG. apply/idP/idP; first by case/cyclicP=> x ->; rewrite rank_cycle leq_b1. rewrite -size_abelian_type // -{}atypG -{}defG unlock. by case: b => [\|x []] //= _; rewrite ?cyclic1 // dprodg1 cycle_cyclic. Qed.	Lemma	abelian_rank1_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...	[ "abelian", "abelian_structure", "apply", "cGG", "cycle_cyclic", "cyclic", "cyclic1", "cyclicP", "defG", "dprodg1", "leq_b1", "rank_cycle", "size_abelian_type" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
homocyclic A	:= abelian A && constant (abelian_type A).	Definition	homocyclic	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_type", "constant" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
homocyclic_Ohm_Mho n p G : p.-group G -> homocyclic G -> 'Ohm_n(G) = 'Mho^(logn p (exponent G) - n)(G).	Proof. move=> pG /andP[cGG homoG]; set e := exponent G. have{pG} p_e: p.-nat e by apply: pnat_dvd pG; apply: exponent_dvdn. have{homoG}: all (pred1 e) (abelian_type G). move: homoG; rewrite /abelian_type -(prednK (cardG_gt0 G)) /=. by case: (_ && _) (tag _); rewrite //= genGid eqxx. have{cGG} [b defG <-] := abelian...	Lemma	homocyclic_Ohm_Mho	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...	[ "Mho", "Mho_dprod", "Mho_p_cycle", "Mho_sub", "Ohm_dprod", "Ohm_p_cycle", "Ohm_sub", "abelian_structure", "abelian_type", "all", "apply", "big_cons", "big_nil", "cGG", "cardG_gt0", "defG", "dprodP", "eqxx", "exponent", "exponent_dvdn", "genGid", "group", "homocyclic", "...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_Mho_homocyclic (n p : nat) G : abelian G -> p.-group G -> 0 < n < logn p (exponent G) -> 'Ohm_n(G) = 'Mho^(logn p (exponent G) - n)(G) -> homocyclic G.	Proof. set e := exponent G => cGG pG /andP[n_gt0 n_lte] eq_Ohm_Mho. suffices: all (pred1 e) (abelian_type G). by rewrite /homocyclic cGG; apply: all_pred1_constant. case/abelian_structure: cGG (abelian_type_gt1 G) => b defG <-. set H := G in defG eq_Ohm_Mho *; have sHG: H \subset G by []. elim: b H defG sHG eq_Ohm_Mh...	Lemma	Ohm_Mho_homocyclic	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...	[ "Mho", "Mho_dprod", "Mho_p_cycle", "Mho_sub", "Ohm_dprod", "Ohm_p_cycle", "Ohm_sub", "abelian", "abelian_structure", "abelian_type", "abelian_type_gt1", "all", "all_pred1_constant", "apply", "big_cons", "cGG", "cycle_subG", "defG", "divn_mulAC", "dprod1g", "dprodP", "dprod_...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> sED; rewrite inE injm_pElem // card_injm ?inE. Qed.

Lemma

injm_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...

[ "aT", "card_injm", "group", "inE", "injm_pElem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_nElem n (E : {group aT}) : E \subset D -> ((f @* E)%G \in 'E^n(f @* G)) = (E \in 'E^n(G)).

Proof. move=> sED; apply/nElemP/nElemP=> [] [p EpE]; by exists p; rewrite injm_pnElem in EpE *. Qed.

Lemma

injm_nElem

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", "group", "injm_pnElem", "nElemP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. move=> sED; have defE := morphim_invm injf sED. apply/pmaxElemP/pmaxElemP=> [] [EpE maxE]. split=> [|H EpH sEH]; first by rewrite injm_pElem in EpE. have sHD: H \subset D by apply: subset_trans (sGD); case/pElemP: EpH. by rewrite -(morphim_invm injf sHD) [f @* H]maxE ?morphimS ?injm_pElem. rewrite injm_pEl...

Lemma

injm_pmaxElem

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", "fH", "group", "group_inj", "injf", "injm_pElem", "morphimS", "morphim_invm", "morphpreK", "pElemP", "pmaxElemP", "sGD", "sHD", "split", "sub_morphim_pre", "subsetIl", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_grank : 'm(f @* G) = 'm(G).

Proof. by apply/eqP; rewrite eqn_leq -{3}defG !morphim_grank ?morphimS. Qed.

Lemma

injm_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...

[ "apply", "defG", "eqn_leq", "morphimS", "morphim_grank" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_p_rank p : 'r_p(f @* G) = 'r_p(G).

Proof. apply/eqP; rewrite eqn_leq; apply/andP; split. have [fE] := p_rank_witness p (f @* G); move: 'r_p(_) => n Ep_fE. apply/p_rank_geP; exists (f @*^-1 fE)%G. rewrite -injm_pnElem ?subsetIl ?(group_inj (morphpreK _)) //. by case/pnElemP: Ep_fE => sfEG _ _; rewrite (subset_trans sfEG) ?morphimS. have [E] := p_...

Lemma

injm_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...

[ "apply", "eqn_leq", "fE", "group_inj", "injm_pnElem", "morphimS", "morphpreK", "p_rank_geP", "p_rank_witness", "pnElemP", "split", "subsetIl", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_rank : 'r(f @* G) = 'r(G).

Proof. apply/eqP; rewrite eqn_leq; apply/andP; split. by have [p _ ->] := rank_witness (f @* G); rewrite injm_p_rank p_rank_le_rank. by have [p _ ->] := rank_witness G; rewrite -injm_p_rank p_rank_le_rank. Qed.

Lemma

injm_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...

[ "apply", "eqn_leq", "injm_p_rank", "p_rank_le_rank", "rank_witness", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isoGH : G \isog H.

Hypothesis

isoGH

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...

[ "isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_isog : exponent G = exponent H.

Proof. by case/isogP: isoGH => f injf <-; rewrite exponent_injm. Qed.

Lemma

exponent_isog

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", "exponent_injm", "injf", "isoGH", "isogP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_abelem p : p.-abelem G = p.-abelem H.

Proof. by case/isogP: isoGH => f injf <-; rewrite injm_abelem. Qed.

Lemma

isog_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", "injf", "injm_abelem", "isoGH", "isogP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_grank : 'm(G) = 'm(H).

Proof. by case/isogP: isoGH => f injf <-; rewrite [RHS]injm_grank. Qed.

Lemma

isog_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...

[ "injf", "injm_grank", "isoGH", "isogP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_p_rank p : 'r_p(G) = 'r_p(H).

Proof. by case/isogP: isoGH => f injf <-; rewrite injm_p_rank. Qed.

Lemma

isog_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...

[ "injf", "injm_p_rank", "isoGH", "isogP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_rank : 'r(G) = 'r(H).

Proof. by case/isogP: isoGH => f injf <-; rewrite injm_rank. Qed.

Lemma

isog_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...

[ "injf", "injm_rank", "isoGH", "isogP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_quotient G H : exponent (G / H) %| exponent G.

Proof. exact: exponent_morphim. Qed.

Lemma

exponent_quotient

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", "exponent_morphim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_LdivT n H : 'Ldiv_n() / H \subset 'Ldiv_n().

Proof. exact: morphim_LdivT. Qed.

Lemma

quotient_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...

[ "morphim_LdivT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_Ldiv n A H : 'Ldiv_n(A) / H \subset 'Ldiv_n(A / H).

Proof. exact: morphim_Ldiv. Qed.

Lemma

quotient_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...

[ "morphim_Ldiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_abelem G H : p.-abelem G -> p.-abelem (G / H).

Proof. exact: morphim_abelem. Qed.

Lemma

quotient_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", "morphim_abelem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_pElem G H E : E \in 'E_p(G) -> (E / H)%G \in 'E_p(G / H).

Proof. exact: morphim_pElem. Qed.

Lemma

quotient_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...

[ "morphim_pElem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn_quotient G H : logn p #|G / H| <= logn p #|G|.

Proof. exact: logn_morphim. Qed.

Lemma

logn_quotient

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...

[ "logn", "logn_morphim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_pnElem G H n E : E \in 'E_p^n(G) -> {m | m <= n & (E / H)%G \in 'E_p^m(G / H)}.

Proof. exact: morphim_pnElem. Qed.

Lemma

quotient_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...

[ "morphim_pnElem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_grank G H : G \subset 'N(H) -> 'm(G / H) <= 'm(G).

Proof. exact: morphim_grank. Qed.

Lemma

quotient_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...

[ "morphim_grank" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_quotient G H : G \subset 'N(H) -> 'r_p(G) - 'r_p(H) <= 'r_p(G / H).

Proof. move=> nHG; rewrite leq_subLR. have [E EpE] := p_rank_witness p G; have{EpE} [sEG abelE <-] := pnElemP EpE. rewrite -(LagrangeI E H) lognM ?cardG_gt0 //. rewrite -card_quotient ?(subset_trans sEG) // leq_add ?logn_le_p_rank // !inE. by rewrite subsetIr (abelemS (subsetIl E H)). by rewrite quotientS ?quotient_a...

Lemma

p_rank_quotient

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...

[ "LagrangeI", "abelE", "abelemS", "cardG_gt0", "card_quotient", "inE", "leq_add", "leq_subLR", "lognM", "logn_le_p_rank", "nHG", "p_rank_witness", "pnElemP", "quotientS", "quotient_abelem", "subsetIl", "subsetIr", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_dprod K H G : K \x H = G -> 'r_p(K) + 'r_p(H) = 'r_p(G).

Proof. move=> defG; apply/eqP; rewrite eqn_leq -leq_subLR andbC. have [_ defKH cKH tiKH] := dprodP defG; have nKH := cents_norm cKH. rewrite {1}(isog_p_rank (quotient_isog nKH tiKH)) /= -quotientMidl defKH. rewrite p_rank_quotient; first by rewrite -defKH mul_subG ?normG. have [[E EpE] [F EpF]] := (p_rank_witness p K, ...

Lemma

p_rank_dprod

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", "cKH", "centSS", "cents_norm", "defG", "dprodEY", "dprodP", "dprod_abelem", "dprod_card", "eqn_leq", "eqxx", "genGid", "genM_join", "genS", "inE", "isog_p_rank", "leq_subLR", "lognM", "mul_subG", "nKH", "normG", "p_rank_geP", "p_rank_quotient", "p_ra...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_p'quotient G H : (p : nat)^'.-group H -> G \subset 'N(H) -> 'r_p(G / H) = 'r_p(G).

Proof. move=> p'H nHG; have [P sylP] := Sylow_exists p G. have [sPG pP _] := and3P sylP; have nHP := subset_trans sPG nHG. have tiHP: H :&: P = 1 := coprime_TIg (p'nat_coprime p'H pP). rewrite -(p_rank_Sylow sylP) -(p_rank_Sylow (quotient_pHall nHP sylP)). by rewrite (isog_p_rank (quotient_isog nHP tiHP)). Qed.

Lemma

p_rank_p'quotient

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", "coprime_TIg", "group", "isog_p_rank", "nHG", "nat", "p'nat_coprime", "pP", "p_rank_Sylow", "quotient_isog", "quotient_pHall", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_sub G : 'Ohm_n(G) \subset G.

Proof. by rewrite gen_subG; apply/subsetP=> x /setIdP[]. Qed.

Lemma

Ohm_sub

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", "gen_subG", "setIdP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm1 : 'Ohm_n([1 gT]) = 1.

Proof. exact: (trivgP (Ohm_sub _)). Qed.

Lemma

Ohm1

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...

[ "Ohm_sub", "gT", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_id G : 'Ohm_n('Ohm_n(G)) = 'Ohm_n(G).

Proof. apply/eqP; rewrite eqEsubset Ohm_sub genS //. by apply/subsetP=> x /setIdP[Gx oxn]; rewrite inE mem_gen // inE Gx. Qed.

Lemma

Ohm_id

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...

[ "Ohm_sub", "apply", "eqEsubset", "genS", "inE", "mem_gen", "setIdP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_cont rT G (f : {morphism G >-> rT}) : f @* 'Ohm_n(G) \subset 'Ohm_n(f @* G).

Proof. rewrite morphim_gen ?genS //; first by rewrite -gen_subG Ohm_sub. apply/subsetP=> fx /morphimP[x Gx]; rewrite inE Gx /=. case/OhmPredP=> p p_pr xpn_1 -> {fx}. rewrite inE morphimEdom imset_f //=; apply/OhmPredP; exists p => //. by rewrite -morphX // xpn_1 morph1. Qed.

Lemma

Ohm_cont

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...

[ "OhmPredP", "Ohm_sub", "apply", "genS", "gen_subG", "imset_f", "inE", "morph1", "morphX", "morphimEdom", "morphimP", "morphim_gen", "morphism", "p_pr", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

OhmS H G : H \subset G -> 'Ohm_n(H) \subset 'Ohm_n(G).

Proof. move=> sHG; apply: genS; apply/subsetP=> x /[!inE] /andP[Hx ->]. by rewrite (subsetP sHG). Qed.

Lemma

OhmS

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", "genS", "inE", "sHG", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

OhmE p G : p.-group G -> 'Ohm_n(G) = <<'Ldiv_(p ^ n)(G)>>.

Proof. move=> pG; congr <<_>>; apply/setP=> x /[!inE]; apply: andb_id2l => Gx. have [-> | ntx] := eqVneq x 1; first by rewrite !expg1n. by rewrite (pdiv_p_elt (mem_p_elt pG Gx)). Qed.

Lemma

OhmE

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", "eqVneq", "expg1n", "group", "inE", "mem_p_elt", "pG", "pdiv_p_elt", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

OhmEabelian p G : p.-group G -> abelian 'Ohm_n(G) -> 'Ohm_n(G) = 'Ldiv_(p ^ n)(G).

Proof. move=> pG; rewrite (OhmE pG) abelian_gen => cGGn; rewrite gen_set_id //. rewrite -(setIidPr (subset_gen 'Ldiv_(p ^ n)(G))) setIA. by rewrite [_ :&: G](setIidPl _) ?gen_subG ?subsetIl // group_Ldiv ?abelian_gen. Qed.

Lemma

OhmEabelian

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...

[ "OhmE", "abelian", "abelian_gen", "gen_set_id", "gen_subG", "group", "group_Ldiv", "pG", "setIA", "setIidPl", "setIidPr", "subsetIl", "subset_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_p_cycle p x : p.-elt x -> 'Ohm_n(<[x]>) = <[x ^+ (p ^ (logn p #[x] - n))]>.

Proof. move=> p_x; apply/eqP; rewrite (OhmE p_x) eqEsubset cycle_subG mem_gen. rewrite !inE mem_cycle -expgM -expnD addnC -maxnE -order_dvdn. by rewrite -{1}(part_pnat_id p_x) p_part dvdn_exp2l ?leq_maxr. rewrite gen_subG andbT; apply/subsetP=> y /LdivP[x_y ypn]. case: (leqP (logn p #[x]) n) => [|lt_n_x]. by rewr...

Lemma

Ohm_p_cycle

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...

[ "LdivP", "OhmE", "addnC", "apply", "cardSg", "congr_group", "cycle_subG", "cycle_sub_group", "dvdn_exp2l", "eqEsubset", "expgM", "expnD", "gen_subG", "inE", "leqP", "leq_maxr", "logn", "lognE", "ltnW", "maxnE", "mem_cycle", "mem_gen", "muln_divA", "order_dvdn", "p_par...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_dprod A B G : A \x B = G -> 'Ohm_n(A) \x 'Ohm_n(B) = 'Ohm_n(G).

Proof. case/dprodP => [[H K -> ->{A B}]] <- cHK tiHK. rewrite dprodEY //. - by rewrite (subset_trans (subset_trans _ cHK)) ?centS ?Ohm_sub. - by apply/trivgP; rewrite -tiHK setISS ?Ohm_sub. apply/eqP; rewrite -(cent_joinEr cHK) eqEsubset join_subG /=. rewrite !OhmS ?joing_subl ?joing_subr //= cent_joinEr //= -genM_join...

Lemma

Ohm_dprod

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...

[ "OhmPredP", "OhmS", "Ohm_sub", "apply", "centS", "cent_joinEr", "centsP", "commute", "dprodEY", "dprodP", "eqEsubset", "eq_invg_mul", "expgMn", "genM_join", "genS", "groupV", "groupX", "imset2P", "inE", "invg1", "invgK", "join_subG", "joing_subl", "joing_subr", "mem_g...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_sub G : 'Mho^n(G) \subset G.

Proof. rewrite gen_subG; apply/subsetP=> _ /imsetP[x /setIdP[Gx _] ->]. exact: groupX. Qed.

Lemma

Mho_sub

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...

[ "Mho", "apply", "gen_subG", "groupX", "imsetP", "setIdP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho1 : 'Mho^n([1 gT]) = 1.

Proof. exact: (trivgP (Mho_sub _)). Qed.

Lemma

Mho1

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...

[ "Mho", "Mho_sub", "gT", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_Mho rT D G (f : {morphism D >-> rT}) : G \subset D -> f @* 'Mho^n(G) = 'Mho^n(f @* G).

Proof. move=> sGD; have sGnD := subset_trans (Mho_sub G) sGD. apply/eqP; rewrite eqEsubset {1}morphim_gen -1?gen_subG // !gen_subG. apply/andP; split; apply/subsetP=> y. case/morphimP=> xpn _ /imsetP[x /setIdP[Gx]]. set p := pdiv _ => p_x -> -> {xpn y}; have Dx := subsetP sGD x Gx. by rewrite morphX // Mho_p_elt ...

Lemma

morphim_Mho

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", "Mho", "Mho_p_elt", "Mho_sub", "apply", "constt_p_elt", "eqEsubset", "gen_subG", "groupX", "imsetP", "mem_morphim", "morphX", "morph_constt", "morph_p_elt", "morphimP", "morphim_gen", "morphism", "p_elt_constt", "pdiv", "sGD", "setIdP", "split", "subsetP", "subset...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_cont rT G (f : {morphism G >-> rT}) : f @* 'Mho^n(G) \subset 'Mho^n(f @* G).

Proof. by rewrite morphim_Mho. Qed.

Lemma

Mho_cont

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...

[ "Mho", "morphim_Mho", "morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

MhoS H G : H \subset G -> 'Mho^n(H) \subset 'Mho^n(G).

Proof. move=> sHG; apply: genS; apply: imsetS; apply/subsetP=> x. by rewrite !inE => /andP[Hx]; rewrite (subsetP sHG). Qed.

Lemma

MhoS

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...

[ "Mho", "apply", "genS", "imsetS", "inE", "sHG", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

MhoE p G : p.-group G -> 'Mho^n(G) = <<[set x ^+ (p ^ n) | x in G]>>.

Proof. move=> pG; apply/eqP; rewrite eqEsubset !gen_subG; apply/andP. do [split; apply/subsetP => xpn; case/imsetP => x] => [|Gx ->]; last first. by rewrite Mho_p_elt ?(mem_p_elt pG). case/setIdP=> Gx _ ->; have [-> | ntx] := eqVneq x 1; first by rewrite expg1n. by rewrite (pdiv_p_elt (mem_p_elt pG Gx) ntx) mem_gen /...

Lemma

MhoE

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...

[ "Mho", "Mho_p_elt", "apply", "eqEsubset", "eqVneq", "expg1n", "gen_subG", "group", "imsetP", "imset_f", "last", "mem_gen", "mem_p_elt", "pG", "pdiv_p_elt", "setIdP", "split", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

MhoEabelian p G : p.-group G -> abelian G -> 'Mho^n(G) = [set x ^+ (p ^ n) | x in G].

Proof. move=> pG cGG; rewrite (MhoE pG); rewrite gen_set_id //; apply/group_setP. split=> [|xn yn]; first by apply/imsetP; exists 1; rewrite ?expg1n. case/imsetP=> x Gx ->; case/imsetP=> y Gy ->. by rewrite -expgMn; [apply: (centsP cGG) | apply: imset_f; rewrite groupM]. Qed.

Lemma

MhoEabelian

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...

[ "Mho", "MhoE", "abelian", "apply", "cGG", "centsP", "expg1n", "expgMn", "gen_set_id", "group", "groupM", "group_setP", "imsetP", "imset_f", "pG", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_Mho G : 'Mho^n(G) == 1 -> 'Ohm_n(G) == G.

Proof. rewrite -subG1 gen_subG eqEsubset Ohm_sub /= => Gp1. rewrite -{1}(Sylow_gen G) genS //; apply/bigcupsP=> P. case/SylowP=> p p_pr /and3P[sPG pP _]; apply/subsetP=> x Px. have Gx := subsetP sPG x Px; rewrite inE Gx //=. rewrite (sameP eqP set1P) (subsetP Gp1) ?mem_gen //; apply: imset_f. by rewrite inE Gx; apply: ...

Lemma

trivg_Mho

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...

[ "Mho", "Ohm_sub", "Px", "SylowP", "Sylow_gen", "apply", "bigcupsP", "eqEsubset", "genS", "gen_subG", "imset_f", "inE", "mem_gen", "mem_p_elt", "pP", "p_pr", "pgroup_p", "set1P", "subG1", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_p_cycle p x : p.-elt x -> 'Mho^n(<[x]>) = <[x ^+ (p ^ n)]>.

Proof. move=> p_x. apply/eqP; rewrite (MhoE p_x) eqEsubset cycle_subG mem_gen. by apply: imset_f; apply: cycle_id. rewrite gen_subG andbT; apply/subsetP=> _ /imsetP[_ /cycleP[k ->] ->]. by rewrite -expgM mulnC expgM mem_cycle. Qed.

Lemma

Mho_p_cycle

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...

[ "Mho", "MhoE", "apply", "cycleP", "cycle_id", "cycle_subG", "eqEsubset", "expgM", "gen_subG", "imsetP", "imset_f", "mem_cycle", "mem_gen", "mulnC", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_cprod A B G : A \* B = G -> 'Mho^n(A) \* 'Mho^n(B) = 'Mho^n(G).

Proof. case/cprodP => [[H K -> ->{A B}]] <- cHK; rewrite cprodEY //. by rewrite (subset_trans (subset_trans _ cHK)) ?centS ?Mho_sub. apply/eqP; rewrite -(cent_joinEr cHK) eqEsubset join_subG /=. rewrite !MhoS ?joing_subl ?joing_subr //= cent_joinEr // -genM_join. apply: genS; apply/subsetP=> xypn /imsetP[_ /setIdP[/i...

Lemma

Mho_cprod

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...

[ "Mho", "MhoS", "Mho_p_elt", "Mho_sub", "apply", "centS", "cent_joinEr", "centsP", "commute", "commuteX2", "consttM", "constt_p_elt", "cprodEY", "cprodP", "eqEsubset", "expgMn", "genM_join", "genS", "groupX", "imset2P", "imsetP", "join_subG", "joing_subl", "joing_subr", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_dprod A B G : A \x B = G -> 'Mho^n(A) \x 'Mho^n(B) = 'Mho^n(G).

Proof. case/dprodP => [[H K -> ->{A B}]] defG cHK tiHK. rewrite dprodEcp; last by apply: Mho_cprod; rewrite cprodE. by apply/trivgP; rewrite -tiHK setISS ?Mho_sub. Qed.

Lemma

Mho_dprod

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...

[ "Mho", "Mho_cprod", "Mho_sub", "apply", "cprodE", "defG", "dprodEcp", "dprodP", "last", "setISS", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_igFun i

:= [igFun by Ohm_sub i & Ohm_cont i].

Canonical

Ohm_igFun

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...

[ "Ohm_cont", "Ohm_sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_gFun i

:= [gFun by Ohm_cont i].

Canonical

Ohm_gFun

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...

[ "Ohm_cont" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_mgFun i

:= [mgFun by OhmS i].

Canonical

Ohm_mgFun

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...

[ "OhmS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_igFun i

:= [igFun by Mho_sub i & Mho_cont i].

Canonical

Mho_igFun

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...

[ "Mho_cont", "Mho_sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_gFun i

:= [gFun by Mho_cont i].

Canonical

Mho_gFun

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...

[ "Mho_cont" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_mgFun i

:= [mgFun by MhoS i].

Canonical

Mho_mgFun

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...

[ "MhoS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_char : 'Ohm_n(G) \char G.

Proof. exact: gFchar. Qed.

Lemma

Ohm_char

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...

[ "char", "gFchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_normal : 'Ohm_n(G) <| G.

Proof. exact: gFnormal. Qed.

Lemma

Ohm_normal

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...

[ "gFnormal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_char : 'Mho^n(G) \char G.

Proof. exact: gFchar. Qed.

Lemma

Mho_char

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...

[ "Mho", "char", "gFchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_normal : 'Mho^n(G) <| G.

Proof. exact: gFnormal. Qed.

Lemma

Mho_normal

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...

[ "Mho", "gFnormal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_Ohm (f : {morphism D >-> rT}) : G \subset D -> f @* 'Ohm_n(G) \subset 'Ohm_n(f @* G).

Proof. exact: morphimF. Qed.

Lemma

morphim_Ohm

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...

[ "morphimF", "morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Ohm (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> f @* 'Ohm_n(G) = 'Ohm_n(f @* G).

Proof. by move=> injf; apply: injmF. Qed.

Lemma

injm_Ohm

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", "injf", "injmF", "morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_Ohm (H : {group rT}) : G \isog H -> 'Ohm_n(G) \isog 'Ohm_n(H).

Proof. exact: gFisog. Qed.

Lemma

isog_Ohm

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...

[ "gFisog", "group", "isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_Mho (H : {group rT}) : G \isog H -> 'Mho^n(G) \isog 'Mho^n(H).

Proof. exact: gFisog. Qed.

Lemma

isog_Mho

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...

[ "Mho", "gFisog", "group", "isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm0 G : 'Ohm_0(G) = 1.

Proof. by apply/trivgP; rewrite /= gen_subG; apply/subsetP=> x /setIdP[_] /[1!inE]. Qed.

Lemma

Ohm0

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", "gen_subG", "inE", "setIdP", "subsetP", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_leq m n G : m <= n -> 'Ohm_m(G) \subset 'Ohm_n(G).

Proof. move/subnKC <-; rewrite genS //; apply/subsetP=> y. by rewrite !inE expnD expgM => /andP[-> /eqP->]; rewrite expg1n /=. Qed.

Lemma

Ohm_leq

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", "expg1n", "expgM", "expnD", "genS", "inE", "subnKC", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

OhmJ n G x : 'Ohm_n(G :^ x) = 'Ohm_n(G) :^ x.

Proof. rewrite -{1}(setIid G) -(setIidPr (Ohm_sub n G)). by rewrite -!morphim_conj injm_Ohm ?injm_conj. Qed.

Lemma

OhmJ

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...

[ "Ohm_sub", "injm_Ohm", "injm_conj", "morphim_conj", "setIid", "setIidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho0 G : 'Mho^0(G) = G.

Proof. apply/eqP; rewrite eqEsubset Mho_sub /=. apply/subsetP=> x Gx; rewrite -[x]prod_constt group_prod // => p _. exact: Mho_p_elt (groupX _ Gx) (p_elt_constt _ _). Qed.

Lemma

Mho0

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...

[ "Mho", "Mho_p_elt", "Mho_sub", "apply", "eqEsubset", "groupX", "group_prod", "p_elt_constt", "prod_constt", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_leq m n G : m <= n -> 'Mho^n(G) \subset 'Mho^m(G).

Proof. move/subnKC <-; rewrite gen_subG //. apply/subsetP=> _ /imsetP[x /setIdP[Gx p_x] ->]. by rewrite expnD expgM groupX ?(Mho_p_elt _ _ p_x). Qed.

Lemma

Mho_leq

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...

[ "Mho", "Mho_p_elt", "apply", "expgM", "expnD", "gen_subG", "groupX", "imsetP", "setIdP", "subnKC", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

MhoJ n G x : 'Mho^n(G :^ x) = 'Mho^n(G) :^ x.

Proof. by rewrite -{1}(setIid G) -(setIidPr (Mho_sub n G)) -!morphim_conj morphim_Mho. Qed.

Lemma

MhoJ

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...

[ "Mho", "Mho_sub", "morphim_Mho", "morphim_conj", "setIid", "setIidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extend_cyclic_Mho G p x : p.-group G -> x \in G -> 'Mho^1(G) = <[x ^+ p]> -> forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (p ^ k)]>.

Proof. move=> pG Gx defG1 [//|k _]; have pX := mem_p_elt pG Gx. apply/eqP; rewrite eqEsubset cycle_subG (Mho_p_elt _ Gx pX) andbT. rewrite (MhoE _ pG) gen_subG; apply/subsetP=> ypk; case/imsetP=> y Gy ->{ypk}. have: y ^+ p \in <[x ^+ p]> by rewrite -defG1 (Mho_p_elt 1 _ (mem_p_elt pG Gy)). rewrite !expnS /= !expgM => /...

Lemma

extend_cyclic_Mho

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...

[ "Mho", "MhoE", "Mho_p_elt", "apply", "cycleP", "cycle_subG", "eqEsubset", "expgM", "expnS", "gen_subG", "group", "imsetP", "mem_cycle", "mem_p_elt", "mulnC", "mulnCA", "pG", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm1Eprime G : 'Ohm_1(G) = <<[set x in G | prime #[x]]>>.

Proof. rewrite -['Ohm_1(G)](genD1 (group1 _)); congr <<_>>. apply/setP=> x; rewrite !inE andbCA -order_dvdn -order_gt1; congr (_ && _). apply/andP/idP=> [[p_gt1] | p_pr]; last by rewrite prime_gt1 ?pdiv_id. set p := pdiv _ => ox_p; have p_pr: prime p by rewrite pdiv_prime. by have [_ dv_p] := primeP p_pr; case/pred2P: ...

Lemma

Ohm1Eprime

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", "genD1", "group1", "inE", "last", "order_dvdn", "order_gt1", "p_gt1", "p_pr", "pdiv", "pdiv_id", "pdiv_prime", "pred2P", "prime", "primeP", "prime_gt1", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_Ohm1 p G : p.-group G -> p.-abelem 'Ohm_1(G) = abelian 'Ohm_1(G).

Proof. move=> pG; rewrite /abelem (pgroupS (Ohm_sub 1 G)) //. case abG1: (abelian _) => //=; apply/exponentP=> x. by rewrite (OhmEabelian pG abG1); case/LdivP. Qed.

Lemma

abelem_Ohm1

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...

[ "LdivP", "OhmEabelian", "Ohm_sub", "abelem", "abelian", "apply", "exponentP", "group", "pG", "pgroupS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm1_abelem p G : p.-group G -> abelian G -> p.-abelem ('Ohm_1(G)).

Proof. by move=> pG cGG; rewrite abelem_Ohm1 ?(abelianS (Ohm_sub 1 G)). Qed.

Lemma

Ohm1_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...

[ "Ohm_sub", "abelem", "abelem_Ohm1", "abelian", "abelianS", "cGG", "group", "pG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm1_id p G : p.-abelem G -> 'Ohm_1(G) = G.

Proof. case/and3P=> pG cGG /exponentP Gp. apply/eqP; rewrite eqEsubset Ohm_sub (OhmE 1 pG) sub_gen //. by apply/subsetP=> x Gx; rewrite !inE Gx Gp /=. Qed.

Lemma

Ohm1_id

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...

[ "OhmE", "Ohm_sub", "abelem", "apply", "cGG", "eqEsubset", "exponentP", "inE", "pG", "sub_gen", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_Ohm1P p G : abelian G -> p.-group G -> reflect ('Ohm_1(G) = G) (p.-abelem G).

Proof. move=> cGG pG. by apply: (iffP idP) => [| <-]; [apply: Ohm1_id | apply: Ohm1_abelem]. Qed.

Lemma

abelem_Ohm1P

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...

[ "Ohm1_abelem", "Ohm1_id", "abelem", "abelian", "apply", "cGG", "group", "pG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

TI_Ohm1 G H : H :&: 'Ohm_1(G) = 1 -> H :&: G = 1.

Proof. move=> tiHG1; case: (trivgVpdiv (H :&: G)) => // [[p pr_p]]. case/Cauchy=> // x /setIP[Hx Gx] ox. suffices x1: x \in [1] by rewrite -ox (set1P x1) order1 in pr_p. by rewrite -{}tiHG1 inE Hx Ohm1Eprime mem_gen // inE Gx ox. Qed.

Lemma

TI_Ohm1

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", "Ohm1Eprime", "inE", "mem_gen", "order1", "pr_p", "set1P", "setIP", "trivgVpdiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm1_eq1 G : ('Ohm_1(G) == 1) = (G :==: 1).

Proof. apply/idP/idP => [/eqP G1_1 | /eqP->]; last by rewrite -subG1 Ohm_sub. by rewrite -(setIid G) TI_Ohm1 // G1_1 setIg1. Qed.

Lemma

Ohm1_eq1

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...

[ "Ohm_sub", "TI_Ohm1", "apply", "last", "setIg1", "setIid", "subG1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

meet_Ohm1 G H : G :&: H != 1 -> G :&: 'Ohm_1(H) != 1.

Proof. by apply: contraNneq => /TI_Ohm1->. Qed.

Lemma

meet_Ohm1

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_Ohm1", "apply", "contraNneq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm1_cent_max G E p : E \in 'E*_p(G) -> p.-group G -> 'Ohm_1('C_G(E)) = E.

Proof. move=> EpmE pG; have [G1 | ntG]:= eqsVneq G 1. case/pmaxElemP: EpmE; case/pElemP; rewrite G1 => /trivgP-> _ _. by apply/trivgP; rewrite cent1T setIT Ohm_sub. have [p_pr _ _] := pgroup_pdiv pG ntG. by rewrite (OhmE 1 (pgroupS (subsetIl G _) pG)) (pmaxElem_LdivP _ _) ?genGid. Qed.

Lemma

Ohm1_cent_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...

[ "G1", "OhmE", "Ohm_sub", "apply", "cent1T", "eqsVneq", "genGid", "group", "pElemP", "pG", "p_pr", "pgroupS", "pgroup_pdiv", "pmaxElemP", "pmaxElem_LdivP", "setIT", "subsetIl", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm1_cyclic_pgroup_prime p G : cyclic G -> p.-group G -> G :!=: 1 -> #|'Ohm_1(G)| = p.

Proof. move=> cycG pG ntG; set K := 'Ohm_1(G). have abelK: p.-abelem K by rewrite Ohm1_abelem ?cyclic_abelian. have sKG: K \subset G := Ohm_sub 1 G. case/cyclicP: (cyclicS sKG cycG) => x /=; rewrite -/K => defK. rewrite defK -orderE (abelem_order_p abelK) //= -/K ?defK ?cycle_id //. rewrite -cycle_eq1 -defK -(setIidPr ...

Lemma

Ohm1_cyclic_pgroup_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...

[ "Ohm1_abelem", "Ohm_sub", "TI_Ohm1", "abelem", "abelem_order_p", "apply", "contraNneq", "cycle_eq1", "cycle_id", "cyclic", "cyclicP", "cyclicS", "cyclic_abelian", "group", "orderE", "pG", "sKG", "setIid", "setIidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cyclic_pgroup_dprod_trivg p A B C : p.-group C -> cyclic C -> A \x B = C -> A = 1 /\ B = C \/ B = 1 /\ A = C.

Proof. move=> pC cycC; case/cyclicP: cycC pC => x ->{C} pC defC. case/dprodP: defC => [] [G H -> ->{A B}] defC _ tiGH; rewrite -defC. have [/trivgP | ntC] := eqVneq <[x]> 1. by rewrite -defC mulG_subG => /andP[/trivgP-> _]; rewrite mul1g; left. have [pr_p _ _] := pgroup_pdiv pC ntC; pose K := 'Ohm_1(<[x]>). have prK ...

Lemma

cyclic_pgroup_dprod_trivg

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...

[ "Ohm1_cyclic_pgroup_prime", "TI_Ohm1", "apply", "cards1", "cycle_cyclic", "cyclic", "cyclicP", "dprodP", "eqVneq", "group", "last", "mul1g", "mulG_subG", "mulG_subl", "mulG_subr", "mulg1", "pgroup_pdiv", "pr_p", "prime", "prime_subgroupVti", "sKG", "setIidPl", "subsetI", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

piOhm1 G : \pi('Ohm_1(G)) = \pi(G).

Proof. apply/eq_piP => p; apply/idP/idP; first exact: (piSg (Ohm_sub 1 G)). rewrite !mem_primes !cardG_gt0 => /andP[p_pr /Cauchy[] // x Gx oxp]. by rewrite p_pr -oxp order_dvdG //= Ohm1Eprime mem_gen // inE Gx oxp. Qed.

Lemma

piOhm1

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", "Ohm1Eprime", "Ohm_sub", "apply", "cardG_gt0", "eq_piP", "inE", "mem_gen", "mem_primes", "order_dvdG", "p_pr", "pi", "piSg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm1Eexponent p G : prime p -> exponent 'Ohm_1(G) %| p -> 'Ohm_1(G) = 'Ldiv_p(G).

Proof. move=> p_pr expG1p; have pG: p.-group G. apply: sub_in_pnat (pnat_pi (cardG_gt0 G)) => q _. rewrite -piOhm1 mem_primes; case/and3P=> q_pr _; apply: pgroupP q_pr. by rewrite -pnat_exponent (pnat_dvd expG1p) ?pnat_id. apply/eqP; rewrite eqEsubset {2}(OhmE 1 pG) subset_gen subsetI Ohm_sub. by rewrite sub_Ldiv...

Lemma

Ohm1Eexponent

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...

[ "OhmE", "Ohm_sub", "apply", "cardG_gt0", "eqEsubset", "exponent", "group", "mem_primes", "pG", "p_pr", "pgroupP", "piOhm1", "pnat_dvd", "pnat_exponent", "pnat_id", "pnat_pi", "prime", "sub_LdivT", "sub_in_pnat", "subsetI", "subset_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_Ohm1 p G : 'r_p('Ohm_1(G)) = 'r_p(G).

Proof. apply/eqP; rewrite eqn_leq p_rankS ?Ohm_sub //. apply/bigmax_leqP=> E /setIdP[sEG abelE]. by rewrite (bigmax_sup E) // inE -{1}(Ohm1_id abelE) OhmS. Qed.

Lemma

p_rank_Ohm1

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...

[ "Ohm1_id", "OhmS", "Ohm_sub", "abelE", "apply", "bigmax_leqP", "bigmax_sup", "eqn_leq", "inE", "p_rankS", "setIdP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_Ohm1 G : 'r('Ohm_1(G)) = 'r(G).

Proof. apply/eqP; rewrite eqn_leq rankS ?Ohm_sub //. by have [p _ ->] := rank_witness G; rewrite -p_rank_Ohm1 p_rank_le_rank. Qed.

Lemma

rank_Ohm1

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...

[ "Ohm_sub", "apply", "eqn_leq", "p_rank_Ohm1", "p_rank_le_rank", "rankS", "rank_witness" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank_abelian p G : abelian G -> 'r_p(G) = logn p #|'Ohm_1(G)|.

Proof. move=> cGG; have nilG := abelian_nil cGG; case p_pr: (prime p); last first. by apply/eqP; rewrite lognE p_pr eqn0Ngt p_rank_gt0 mem_primes p_pr. case/dprodP: (Ohm_dprod 1 (nilpotent_pcoreC p nilG)) => _ <- _ /TI_cardMg->. rewrite mulnC logn_Gauss. rewrite prime_coprime // -p'natE // -/(pgroup _ _). exact: ...

Lemma

p_rank_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...

[ "Ohm1_abelem", "Ohm_dprod", "Ohm_sub", "TI_cardMg", "abelian", "abelianS", "abelian_nil", "apply", "cGG", "dprodP", "eqn0Ngt", "last", "logn", "lognE", "logn_Gauss", "mem_primes", "mulnC", "nilpotent_pcoreC", "nilpotent_pcore_Hall", "p'natE", "p_pr", "p_rank_Ohm1", "p_ran...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_abelian_pgroup p G : p.-group G -> abelian G -> 'r(G) = logn p #|'Ohm_1(G)|.

Proof. by move=> pG cGG; rewrite (rank_pgroup pG) p_rank_abelian. Qed.

Lemma

rank_abelian_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...

[ "abelian", "cGG", "group", "logn", "pG", "p_rank_abelian", "rank_pgroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_splits x G : x \in G -> #[x] = exponent G -> abelian G -> [splits G, over <[x]>].

Proof. move=> Gx ox cGG; apply/splitsP; have [n] := ubnP #|G|. elim: n gT => // n IHn aT in x G Gx ox cGG * => /ltnSE-leGn. have: <[x]> \subset G by [rewrite cycle_subG]; rewrite subEproper. case/predU1P=> [<- | /properP[sxG [y]]]. by exists 1%G; rewrite inE -subG1 subsetIr mulg1 /=. have [m] := ubnP #[y]; elim: m y ...

Lemma

abelian_splits

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", "abelian", "apply", "cGG", "card_quotient", "centsP", "complP", "coset", "cycleP", "cycle_eq1", "cycle_id", "cycle_subG", "cycle_sub_group", "divnA", "dvdn_exponent", "dvdn_leq", "eqEsubset", "eqn_dvd", "eqxx", "expgM", "expgMn", "expgVn", "exponent", "exponentP",...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem_splits p G H : p.-abelem G -> H \subset G -> [splits G, over H].

Proof. have [m] := ubnP #|G|; elim: m G H => // m IHm G H /ltnSE-leGm abelG sHG. have [-> | ] := eqsVneq H 1. by apply/splitsP; exists G; rewrite inE mul1g -subG1 subsetIl /=. case/trivgPn=> x Hx ntx; have Gx := subsetP sHG x Hx. have [_ cGG eGp] := and3P abelG. have ox: #[x] = exponent G. by apply/eqP; rewrite eq...

Lemma

abelem_splits

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_order_p", "abelian_splits", "apply", "cGG", "complP", "cycle_eq1", "cycle_subG", "defG", "dvdn_exponent", "eqn_dvd", "eqsVneq", "exponent", "group_modl", "inE", "leq_trans", "ltnSE", "mul1g", "mulG_subr", "mulgA", "properP", "proper_card", "...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_type_subproof G : {H : {group gT} & abelian G -> {x | #[x] = exponent G & <[x]> \x H = G}}.

Proof. case cGG: (abelian G); last by exists G. have [x Gx ox] := exponent_witness (abelian_nil cGG). case/splitsP/ex_mingroup: (abelian_splits Gx (esym ox) cGG) => H. case/mingroupp/complP=> tixH defG; exists H => _. exists x; rewrite ?dprodE // (sub_abelian_cent2 cGG) ?cycle_subG //. by rewrite -defG mulG_subr. Qed.

Fact

abelian_type_subproof

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_nil", "abelian_splits", "cGG", "complP", "cycle_subG", "defG", "dprodE", "ex_mingroup", "exponent", "exponent_witness", "gT", "group", "last", "mingroupp", "mulG_subr", "splitsP", "sub_abelian_cent2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_type_rec n G

:= if n is n'.+1 then if abelian G && (G :!=: 1) then exponent G :: abelian_type_rec n' (tag (abelian_type_subproof G)) else [::] else [::].

Fixpoint

abelian_type_rec

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_type_subproof", "exponent", "n'" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_type (A : {set gT})

:= abelian_type_rec #|A| <<A>>.

Definition

abelian_type

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_type_rec", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_type_dvdn_sorted A : sorted [rel m n | n %| m] (abelian_type A).

Proof. set R := SimplRel _; pose G := <<A>>%G; pose M := G. suffices: path R (exponent M) (abelian_type A) by case: (_ A) => // m t /andP[]. rewrite /abelian_type -/G; have: G \subset M by []. elim: {A}#|A| G M => //= n IHn G M sGM. case: andP => //= -[cGG ntG]; rewrite exponentS ?IHn //=. case: (abelian_type_subproof ...

Lemma

abelian_type_dvdn_sorted

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_type", "abelian_type_subproof", "cGG", "dprodP", "exponent", "exponentS", "mulG_subr", "path", "rel", "sorted" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_type_gt1 A : all [pred m | m > 1] (abelian_type A).

Proof. rewrite /abelian_type; elim: {A}#|A| <<A>>%G => //= n IHn G. case: ifP => //= /andP[_ ntG]; rewrite {n}IHn. by rewrite ltn_neqAle exponent_gt0 eq_sym -dvdn1 -trivg_exponent ntG. Qed.

Lemma

abelian_type_gt1

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_type", "all", "dvdn1", "eq_sym", "exponent_gt0", "ltn_neqAle", "trivg_exponent" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_type_sorted A : sorted geq (abelian_type A).

Proof. have:= abelian_type_dvdn_sorted A; have:= abelian_type_gt1 A. case: (abelian_type A) => //= m t; elim: t m => //= n t IHt m /andP[]. by move/ltnW=> m_gt0 t_gt1 /andP[n_dv_m /IHt->]; rewrite // dvdn_leq. Qed.

Lemma

abelian_type_sorted

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_type", "abelian_type_dvdn_sorted", "abelian_type_gt1", "dvdn_leq", "geq", "ltnW", "sorted" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_structure G : abelian G -> {b | \big[dprod/1]_(x <- b) <[x]> = G & map order b = abelian_type G}.

Proof. rewrite /abelian_type genGidG; have [n] := ubnPleq #|G|. elim: n G => /= [|n IHn] G leGn cGG; first by rewrite leqNgt cardG_gt0 in leGn. rewrite [in _ && _]cGG /=; case: ifP => [ntG|/eqP->]; last first. by exists [::]; rewrite ?big_nil. case: (abelian_type_subproof G) => H /= [//|x ox xdefG]; rewrite -ox. have...

Theorem

abelian_structure

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", "abelian", "abelianS", "abelian_type", "abelian_type_subproof", "big_cons", "big_nil", "cGG", "cardG_gt0", "defG", "dprod", "dprodP", "dvdn1", "eq_sym", "genGidG", "last", "leqNgt", "leq_trans", "ltnS", "ltn_Pmull", "ltn_neqAle", "map", "mulG_subr", "order"...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

count_logn_dprod_cycle p n b G : \big[dprod/1]_(x <- b) <[x]> = G -> count [pred x | logn p #[x] > n] b = logn p #|'Ohm_n.+1(G) : 'Ohm_n(G)|.

Proof. have sOn1 H: 'Ohm_n(H) \subset 'Ohm_n.+1(H) by apply: Ohm_leq. pose lnO i (A : {set gT}) := logn p #|'Ohm_i(A)|. have lnO_le H: lnO n H <= lnO n.+1 H. by rewrite dvdn_leq_log ?cardG_gt0 // cardSg ?sOn1. have lnOx i A B H: A \x B = H -> lnO i A + lnO i B = lnO i H. move=> defH; case/dprodP: defH (defH) => {A ...

Lemma

count_logn_dprod_cycle

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...

[ "Ohm_dprod", "Ohm_leq", "Ohm_p_cycle", "Ohm_sub", "TI_cardMg", "addn1", "addnBA", "addnC", "apply", "big_cons", "big_nil", "cardG_gt0", "cardSg", "centsP", "consttC", "coprime", "coprime_TIg", "coprime_partC", "count", "cycleM", "cycle_abelian", "cycle_subG", "defG", "d...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_type_pgroup p b G : p.-group G -> \big[dprod/1]_(x <- b) <[x]> = G -> 1 \notin b -> perm_eq (abelian_type G) (map order b).

Proof. rewrite perm_sym; move: b => b1 pG defG1 ntb1. have cGG: abelian G. elim: (b1) {pG}G defG1 => [_ <-|x b IHb G]; first by rewrite big_nil abelian1. rewrite big_cons; case/dprodP=> [[_ H _ defH]] <-; rewrite defH => cxH _. by rewrite abelianM cycle_abelian IHb. have p_bG b: \big[dprod/1]_(x <- b) <[x]> = G -...

Lemma

abelian_type_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...

[ "abelian", "abelian1", "abelianM", "abelian_structure", "abelian_type", "abelian_type_gt1", "add0n", "all", "allP", "all_cat", "apply", "big_cons", "big_nil", "cGG", "count", "count_logn_dprod_cycle", "count_map", "cycle_abelian", "cycle_eq1", "defG", "dprod", "dprodP", "...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_abelian_type G : abelian G -> size (abelian_type G) = 'r(G).

Proof. move=> cGG; have [b defG def_t] := abelian_structure cGG. apply/eqP; rewrite -def_t size_map eqn_leq andbC; apply/andP; split. have [p p_pr ->] := rank_witness G; rewrite p_rank_abelian //. by rewrite -indexg1 -(Ohm0 G) -(count_logn_dprod_cycle _ _ defG) count_size. case/lastP def_b: b => // [b' x]; pose p :...

Lemma

size_abelian_type

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...

[ "Ohm0", "abelian", "abelian_structure", "abelian_type", "abelian_type_dvdn_sorted", "abelian_type_gt1", "all", "allP", "all_cat", "all_count", "apply", "cGG", "cat_path", "cat_rcons", "cats1", "count_logn_dprod_cycle", "count_size", "defG", "dvdn_trans", "eqn_leq", "headI", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul_card_Ohm_Mho_abelian n G : abelian G -> (#|'Ohm_n(G)| * #|'Mho^n(G)|)%N = #|G|.

Proof. case/abelian_structure => b defG _. elim: b G defG => [_ <-|x b IHb G]. by rewrite !big_nil (trivgP (Ohm_sub _ _)) (trivgP (Mho_sub _ _)) !cards1. rewrite big_cons => defG; rewrite -(dprod_card defG). rewrite -(dprod_card (Ohm_dprod n defG)) -(dprod_card (Mho_dprod n defG)) /=. rewrite mulnCA -!mulnA mulnCA mu...

Lemma

mul_card_Ohm_Mho_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...

[ "Mho", "Mho_dprod", "Mho_p_cycle", "Mho_sub", "Ohm_dprod", "Ohm_p_cycle", "Ohm_sub", "abelian", "abelian_structure", "apply", "big_cons", "big_nil", "card_pgroup", "cards1", "cent1P", "cent_cycle", "commute", "commuteX2", "consttC", "coprime", "coprime_TIg", "coprime_partC"...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

grank_abelian G : abelian G -> 'm(G) = 'r(G).

Proof. move=> cGG; apply/eqP; rewrite eqn_leq; apply/andP; split. rewrite -size_abelian_type //; case/abelian_structure: cGG => b defG <-. suffices <-: <<[set x in b]>> = G. by rewrite (leq_trans (grank_min _)) // size_map cardsE card_size. rewrite -{G defG}(bigdprodWY defG). elim: b => [|x b IHb]; first by...

Lemma

grank_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...

[ "Mho", "Mho_p_elt", "Mho_sub", "abelian", "abelian_structure", "apply", "big_cons", "big_nil", "bigdprodWY", "cGG", "cardG_gt0", "card_quotient", "card_size", "cards1", "cardsD1", "cardsE", "cent_joinEl", "coset", "coset_id", "cycle_subG", "defG", "divgS", "dvdn_Pexp2l", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank_cycle (x : gT) : 'r(<[x]>) = (x != 1).

Proof. have [->|ntx] := eqVneq x 1; first by rewrite cycle1 rank1. apply/eqP; rewrite eqn_leq rank_gt0 cycle_eq1 ntx andbT. by rewrite -grank_abelian ?cycle_abelian //= -(cards1 x) grank_min. Qed.

Lemma

rank_cycle

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", "cycle1", "cycle_abelian", "cycle_eq1", "eqVneq", "eqn_leq", "gT", "grank_abelian", "grank_min", "rank1", "rank_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelian_rank1_cyclic G : abelian G -> cyclic G = ('r(G) <= 1).

Proof. move=> cGG; have [b defG atypG] := abelian_structure cGG. apply/idP/idP; first by case/cyclicP=> x ->; rewrite rank_cycle leq_b1. rewrite -size_abelian_type // -{}atypG -{}defG unlock. by case: b => [|x []] //= _; rewrite ?cyclic1 // dprodg1 cycle_cyclic. Qed.

Lemma

abelian_rank1_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...

[ "abelian", "abelian_structure", "apply", "cGG", "cycle_cyclic", "cyclic", "cyclic1", "cyclicP", "defG", "dprodg1", "leq_b1", "rank_cycle", "size_abelian_type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

homocyclic A

:= abelian A && constant (abelian_type A).

Definition

homocyclic

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_type", "constant" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

homocyclic_Ohm_Mho n p G : p.-group G -> homocyclic G -> 'Ohm_n(G) = 'Mho^(logn p (exponent G) - n)(G).

Proof. move=> pG /andP[cGG homoG]; set e := exponent G. have{pG} p_e: p.-nat e by apply: pnat_dvd pG; apply: exponent_dvdn. have{homoG}: all (pred1 e) (abelian_type G). move: homoG; rewrite /abelian_type -(prednK (cardG_gt0 G)) /=. by case: (_ && _) (tag _); rewrite //= genGid eqxx. have{cGG} [b defG <-] := abelian...

Lemma

homocyclic_Ohm_Mho

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...

[ "Mho", "Mho_dprod", "Mho_p_cycle", "Mho_sub", "Ohm_dprod", "Ohm_p_cycle", "Ohm_sub", "abelian_structure", "abelian_type", "all", "apply", "big_cons", "big_nil", "cGG", "cardG_gt0", "defG", "dprodP", "eqxx", "exponent", "exponent_dvdn", "genGid", "group", "homocyclic", "...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_Mho_homocyclic (n p : nat) G : abelian G -> p.-group G -> 0 < n < logn p (exponent G) -> 'Ohm_n(G) = 'Mho^(logn p (exponent G) - n)(G) -> homocyclic G.

Proof. set e := exponent G => cGG pG /andP[n_gt0 n_lte] eq_Ohm_Mho. suffices: all (pred1 e) (abelian_type G). by rewrite /homocyclic cGG; apply: all_pred1_constant. case/abelian_structure: cGG (abelian_type_gt1 G) => b defG <-. set H := G in defG eq_Ohm_Mho *; have sHG: H \subset G by []. elim: b H defG sHG eq_Ohm_Mh...

Lemma

Ohm_Mho_homocyclic

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...

[ "Mho", "Mho_dprod", "Mho_p_cycle", "Mho_sub", "Ohm_dprod", "Ohm_p_cycle", "Ohm_sub", "abelian", "abelian_structure", "abelian_type", "abelian_type_gt1", "all", "all_pred1_constant", "apply", "big_cons", "cGG", "cycle_subG", "defG", "divn_mulAC", "dprod1g", "dprodP", "dprod_...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d