fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
isog_Mho(H : {group rT}) : G \isog H -> 'Mho^n(G) \isog 'Mho^n(H).
Proof. exact: gFisog. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | isog_Mho | |
Ohm0G : 'Ohm_0(G) = 1.
Proof.
by apply/trivgP; rewrite /= gen_subG; apply/subsetP=> x /setIdP[_] /[1!inE].
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm0 | |
Ohm_leqm 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm_leq | |
OhmJn 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | OhmJ | |
Mho0G : '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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Mho0 | |
Mho_leqm 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Mho_leq | |
MhoJn 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | MhoJ | |
extend_cyclic_MhoG 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/im... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | extend_cyclic_Mho | |
Ohm1EprimeG : '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 p... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1Eprime | |
abelem_Ohm1p 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelem_Ohm1 | |
Ohm1_abelemp 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_abelem | |
Ohm1_idp 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_id | |
abelem_Ohm1Pp 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelem_Ohm1P | |
TI_Ohm1G 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | TI_Ohm1 | |
Ohm1_eq1G : ('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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_eq1 | |
meet_Ohm1G H : G :&: H != 1 -> G :&: 'Ohm_1(H) != 1.
Proof. by apply: contraNneq => /TI_Ohm1->. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | meet_Ohm1 | |
Ohm1_cent_maxG 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 ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_cent_max | |
Ohm1_cyclic_pgroup_primep 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... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_cyclic_pgroup_prime | |
cyclic_pgroup_dprod_trivgp 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 =>... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | cyclic_pgroup_dprod_trivg | |
piOhm1G : \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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | piOhm1 | |
Ohm1Eexponentp 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/eq... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1Eexponent | |
p_rank_Ohm1p 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | p_rank_Ohm1 | |
rank_Ohm1G : '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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | rank_Ohm1 | |
p_rank_abelianp 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;... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | p_rank_abelian | |
rank_abelian_pgroupp 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | rank_abelian_pgroup | |
abelian_splitsx 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]]... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_splits | |
abelem_splitsp 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] ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelem_splits | |
abelian_type_recn 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_rec | |
abelian_type(A : {set gT}) := abelian_type_rec #|A| <<A>>. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type | |
abelian_type_dvdn_sortedA : 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 => /... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_dvdn_sorted | |
abelian_type_gt1A : 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_gt1 | |
abelian_type_sortedA : 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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_sorted | |
abelian_structureG :
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.... | Theorem | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_structure | |
count_logn_dprod_cyclep 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 dv... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | count_logn_dprod_cycle | |
abelian_type_pgroupp 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/dp... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_pgroup | |
size_abelian_typeG : 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... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | size_abelian_type | |
mul_card_Ohm_Mho_abeliann 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... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | mul_card_Ohm_Mho_abelian | |
grank_abelianG : 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}(bigdpr... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | grank_abelian | |
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 | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | rank_cycle | |
abelian_rank1_cyclicG : 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_cyc... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_rank1_cyclic | |
homocyclicA := abelian A && constant (abelian_type A). | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | homocyclic | |
homocyclic_Ohm_Mhon 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 ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | homocyclic_Ohm_Mho | |
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_co... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm_Mho_homocyclic | |
abelem_homocyclicp G : p.-abelem G -> homocyclic G.
Proof.
move=> abelG; have [_ cGG _] := and3P abelG.
rewrite /homocyclic cGG (@all_pred1_constant _ p) //.
case/abelian_structure: cGG (abelian_type_gt1 G) => b defG <- => b_gt1.
apply/allP=> _ /mapP[x b_x ->] /=; rewrite (abelem_order_p abelG) //.
rewrite -cycle_sub... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelem_homocyclic | |
homocyclic1: homocyclic [1 gT].
Proof. exact: abelem_homocyclic (abelem1 _ 2). Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | homocyclic1 | |
Ohm1_homocyclicPp G : p.-group G -> abelian G ->
reflect ('Ohm_1(G) = 'Mho^(logn p (exponent G)).-1(G)) (homocyclic G).
Proof.
move=> pG cGG; set e := logn p (exponent G); rewrite -subn1.
apply: (iffP idP) => [homoG | ]; first exact: homocyclic_Ohm_Mho.
case: (ltnP 1 e) => [lt1e | ]; first exact: Ohm_Mho_homocyclic.
... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_homocyclicP | |
abelian_type_homocyclicG :
homocyclic G -> abelian_type G = nseq 'r(G) (exponent G).
Proof.
case/andP=> cGG; rewrite -size_abelian_type // /abelian_type.
rewrite -(prednK (cardG_gt0 G)) /=; case: andP => //= _; move: (tag _) => H.
by move/all_pred1P->; rewrite genGid size_nseq.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_homocyclic | |
abelian_type_abelemp G : p.-abelem G -> abelian_type G = nseq 'r(G) p.
Proof.
move=> abelG; rewrite (abelian_type_homocyclic (abelem_homocyclic abelG)).
have [-> | ntG] := eqVneq G 1%G; first by rewrite rank1.
congr nseq; apply/eqP; rewrite eqn_dvd; have [pG _ ->] := and3P abelG.
have [p_pr] := pgroup_pdiv pG ntG; case... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_abelem | |
max_card_abelianG :
abelian G -> #|G| <= exponent G ^ 'r(G) ?= iff homocyclic G.
Proof.
move=> cGG; have [b defG def_tG] := abelian_structure cGG.
have Gb: all [in G] b.
apply/allP=> x b_x; rewrite -(bigdprodWY defG); have [b1 b2] := splitPr b_x.
by rewrite big_cat big_cons /= mem_gen // setUCA inE cycle_id.
have... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | max_card_abelian | |
card_homocyclicG : homocyclic G -> #|G| = (exponent G ^ 'r(G))%N.
Proof.
by move=> homG; have [cGG _] := andP homG; apply/eqP; rewrite max_card_abelian.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | card_homocyclic | |
abelian_type_dprod_homocyclicp K H G :
K \x H = G -> p.-group G -> homocyclic G ->
abelian_type K = nseq 'r(K) (exponent G)
/\ abelian_type H = nseq 'r(H) (exponent G).
Proof.
move=> defG pG homG; have [cGG _] := andP homG.
have /mulG_sub[sKG sHG]: K * H = G by case/dprodP: defG.
have [cKK cHH] := (abelianS ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_dprod_homocyclic | |
dprod_homocyclicp K H G :
K \x H = G -> p.-group G -> homocyclic G -> homocyclic K /\ homocyclic H.
Proof.
move=> defG pG homG; have [cGG _] := andP homG.
have /mulG_sub[sKG sHG]: K * H = G by case/dprodP: defG.
have [abtK abtH] := abelian_type_dprod_homocyclic defG pG homG.
by rewrite /homocyclic !(abelianS _ cGG) /... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | dprod_homocyclic | |
exponent_dprod_homocyclicp K H G :
K \x H = G -> p.-group G -> homocyclic G -> K :!=: 1 ->
exponent K = exponent G.
Proof.
move=> defG pG homG ntK; have [homK _] := dprod_homocyclic defG pG homG.
have [] := abelian_type_dprod_homocyclic defG pG homG.
by rewrite abelian_type_homocyclic // -['r(K)]prednK ?rank_gt0 ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | exponent_dprod_homocyclic | |
isog_abelian_typeG H : isog G H -> abelian_type G = abelian_type H.
Proof.
pose lnO p n gT (A : {set gT}) := logn p #|'Ohm_n.+1(A) : 'Ohm_n(A)|.
pose lni i p gT (A : {set gT}) := \max_(e < logn p #|A| | i < lnO p e _ A) e.+1.
suffices{G} nth_abty gT (G : {group gT}) i:
abelian G -> i < size (abelian_type G) ->
nt... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | isog_abelian_type | |
eq_abelian_type_isogG H :
abelian G -> abelian H -> isog G H = (abelian_type G == abelian_type H).
Proof.
move=> cGG cHH; apply/idP/eqP; first exact: isog_abelian_type.
have{cGG} [bG defG <-] := abelian_structure cGG.
have{cHH} [bH defH <-] := abelian_structure cHH.
elim: bG bH G H defG defH => [|x bG IHb] [|y bH] //... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | eq_abelian_type_isog | |
isog_abelem_cardp G H :
p.-abelem G -> isog G H = p.-abelem H && (#|H| == #|G|).
Proof.
move=> abelG; apply/idP/andP=> [isoGH | [abelH eqGH]].
by rewrite -(isog_abelem isoGH) (card_isog isoGH).
rewrite eq_abelian_type_isog ?(@abelem_abelian _ p) //.
by rewrite !(@abelian_type_abelem _ p) ?(@rank_abelem _ p) // (eqP... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | isog_abelem_card | |
morphim_rank_abelianG : abelian G -> 'r(f @* G) <= 'r(G).
Proof.
move=> cGG; have sHG := subsetIr D G; apply: leq_trans (rankS sHG).
rewrite -!grank_abelian ?morphim_abelian ?(abelianS sHG) //=.
by rewrite -morphimIdom morphim_grank ?subsetIl.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | morphim_rank_abelian | |
morphim_p_rank_abelianp G : abelian G -> 'r_p(f @* G) <= 'r_p(G).
Proof.
move=> cGG; have sHG := subsetIr D G; apply: leq_trans (p_rankS p sHG).
have cHH := abelianS sHG cGG; rewrite -morphimIdom /=; set H := D :&: G.
have sylP := nilpotent_pcore_Hall p (abelian_nil cHH).
have sPH := pHall_sub sylP.
have sPD: 'O_p(H) \... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | morphim_p_rank_abelian | |
isog_homocyclicG H : G \isog H -> homocyclic G = homocyclic H.
Proof.
move=> isoGH.
by rewrite /homocyclic (isog_abelian isoGH) (isog_abelian_type isoGH).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | isog_homocyclic | |
quotient_rank_abelian: 'r(G / H) <= 'r(G).
Proof. exact: morphim_rank_abelian. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | quotient_rank_abelian | |
quotient_p_rank_abelian: 'r_p(G / H) <= 'r_p(G).
Proof. exact: morphim_p_rank_abelian. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | quotient_p_rank_abelian | |
fin_lmod_pchar_abelemp (R : nzRingType) (V : finLmodType R):
p \in [pchar R]%R -> p.-abelem [set: V].
Proof.
case/andP=> p_pr /eqP-pR0; apply/abelemP=> //.
by split=> [|v _]; rewrite ?zmod_abelian // zmodXgE -scaler_nat pR0 scale0r.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_lmod_pchar_abelem | |
fin_Fp_lmod_abelemp (V : finLmodType 'F_p) :
prime p -> p.-abelem [set: V].
Proof. by move/pchar_Fp/fin_lmod_pchar_abelem->. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_Fp_lmod_abelem | |
fin_ring_pchar_abelemp (R : finNzRingType) :
p \in [pchar R]%R -> p.-abelem [set: R].
Proof. exact: fin_lmod_pchar_abelem R^o. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_ring_pchar_abelem | |
fin_lmod_char_abelem:= (fin_lmod_pchar_abelem) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use fin_ring_pchar_abelem instead.")] | Notation | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_lmod_char_abelem | |
fin_ring_char_abelem:= (fin_ring_pchar_abelem) (only parsing). | Notation | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_ring_char_abelem | |
Definition_ := Finite_isGroup.Build bool addbA addFb addbb. | HB.instance | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Definition | |
Sym: {set {perm T}} := setT. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Sym | |
Sym_group:= Eval hnf in [group of Sym].
Local Notation "'Sym_T" := Sym. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Sym_group | |
sign_morph:= @Morphism _ _ 'Sym_T _ (in2W (@odd_permM _)). | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | sign_morph | |
Alt:= 'ker (@odd_perm T). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt | |
Alt_group:= Eval hnf in [group of Alt].
Local Notation "'Alt_T" := Alt. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_group | |
Alt_evenp : (p \in 'Alt_T) = ~~ p.
Proof. by rewrite !inE /=; case: odd_perm. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_even | |
Alt_subset: 'Alt_T \subset 'Sym_T.
Proof. exact: subsetT. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_subset | |
Alt_normal: 'Alt_T <| 'Sym_T.
Proof. exact: ker_normal. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_normal | |
Alt_norm: 'Sym_T \subset 'N('Alt_T).
Proof. by case/andP: Alt_normal. Qed.
Let n := #|T|. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_norm | |
Alt_index: 1 < n -> #|'Sym_T : 'Alt_T| = 2.
Proof.
move=> lt1n; rewrite -card_quotient ?Alt_norm //=.
have : ('Sym_T / 'Alt_T) \isog (@odd_perm T @* 'Sym_T) by apply: first_isog.
case/isogP=> g /injmP/card_in_imset <-.
rewrite /morphim setIid=> ->; rewrite -card_bool; apply: eq_card => b.
apply/imsetP; case: b => /=; l... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_index | |
card_Sym: #|'Sym_T| = n`!.
Proof.
rewrite -[n]cardsE -card_perm; apply: eq_card => p.
by apply/idP/subsetP=> [? ?|]; rewrite !inE.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | card_Sym | |
card_Alt: 1 < n -> (2 * #|'Alt_T|)%N = n`!.
Proof.
by move/Alt_index <-; rewrite mulnC (Lagrange Alt_subset) card_Sym.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | card_Alt | |
Sym_trans: [transitive^n 'Sym_T, on setT | 'P].
Proof.
apply/imsetP; pose t1 := [tuple of enum T].
have dt1: t1 \in n.-dtuple(setT) by rewrite inE enum_uniq; apply/subsetP.
exists t1 => //; apply/setP=> t; apply/idP/imsetP=> [|[a _ ->{t}]]; last first.
by apply: n_act_dtuple => //; apply/astabsP=> x; rewrite !inE.
ca... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Sym_trans | |
Alt_trans: [transitive^n.-2 'Alt_T, on setT | 'P].
Proof.
case n_m2: n Sym_trans => [|[|m]] /= tr_m2; try exact: ntransitive0.
have tr_m := ntransitive_weak (leqW (leqnSn m)) tr_m2.
case/imsetP: tr_m2; case/tupleP=> x; case/tupleP=> y t.
rewrite !dtuple_on_add 2![x \in _]inE inE negb_or /= -!andbA.
case/and4P=> nxy ntx... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_trans | |
aperm_faithful(A : {group {perm T}}) : [faithful A, on setT | 'P].
Proof.
by apply/faithfulP=> /= p _ np1; apply/eqP/perm_act1P=> y; rewrite np1 ?inE.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | aperm_faithful | |
trivial_Alt_2(T : finType) : #|T| <= 2 -> 'Alt_T = 1.
Proof.
rewrite leq_eqVlt => /predU1P[] oT.
by apply: card_le1_trivg; rewrite -leq_double -mul2n card_Alt oT.
suffices Sym1: 'Sym_T = 1 by apply/trivgP; rewrite -Sym1 subsetT.
by apply: card1_trivg; rewrite card_Sym; case: #|T| oT; do 2?case.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | trivial_Alt_2 | |
simple_Alt_3(T : finType) : #|T| = 3 -> simple 'Alt_T.
Proof.
move=> T3; have{T3} oA: #|'Alt_T| = 3.
by apply: double_inj; rewrite -mul2n card_Alt T3.
apply/simpleP; split=> [|K]; [by rewrite trivg_card1 oA | case/andP=> sKH _].
have:= cardSg sKH; rewrite oA dvdn_divisors // !inE orbC /= -oA.
case/pred2P=> eqK; [righ... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | simple_Alt_3 | |
not_simple_Alt_4(T : finType) : #|T| = 4 -> ~~ simple 'Alt_T.
Proof.
move=> oT; set A := 'Alt_T.
have oA: #|A| = 12 by apply: double_inj; rewrite -mul2n card_Alt oT.
suffices [p]: exists p, [/\ prime p, 1 < #|A|`_p < #|A| & #|'Syl_p(A)| == 1%N].
case=> p_pr pA_int; rewrite /A; case/normal_sylowP=> P; case/pHallP.
r... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | not_simple_Alt_4 | |
simple_Alt5_base(T : finType) : #|T| = 5 -> simple 'Alt_T.
Proof.
move=> oT.
have F1: #|'Alt_T| = 60 by apply: double_inj; rewrite -mul2n card_Alt oT.
have FF (H : {group {perm T}}): H <| 'Alt_T -> H :<>: 1 -> 20 %| #|H|.
- move=> Hh1 Hh3.
have [x _]: exists x, x \in T by apply/existsP/eqP; rewrite oT.
have F2 := A... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | simple_Alt5_base | |
T':= {y | y != x}. | Notation | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | T' | |
rfd_funP(p : {perm T}) (u : T') :
let p1 := if p x == x then p else 1 in p1 (val u) != x.
Proof.
case: (p x =P x) => /= [pxx | _]; last by rewrite perm1 (valP u).
by rewrite -[x in _ != x]pxx (inj_eq perm_inj); apply: (valP u).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_funP | |
rfd_funp := [fun u => Sub ((_ : {perm T}) _) (rfd_funP p u) : T']. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_fun | |
rfdPp : injective (rfd_fun p).
Proof.
apply: can_inj (rfd_fun p^-1) _ => u; apply: val_inj => /=.
rewrite -(can_eq (permK p)) permKV eq_sym.
by case: eqP => _; rewrite !(perm1, permK).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfdP | |
rfdp := perm (@rfdP p).
Hypothesis card_T : 2 < #|T|. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd | |
rfd_morph: {in 'C_('Sym_T)[x | 'P] &, {morph rfd : y z / y * z}}.
Proof.
move=> p q; rewrite !setIA !setIid; move/astab1P=> p_x; move/astab1P=> q_x.
apply/permP=> u; apply: val_inj.
by rewrite permE /= !permM !permE /= [p x]p_x [q x]q_x eqxx permM /=.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_morph | |
rfd_morphism:= Morphism rfd_morph. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_morphism | |
rgd_fun(p : {perm T'}) :=
[fun x1 => if insub x1 is Some u then sval (p u) else x]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rgd_fun | |
rgdPp : injective (rgd_fun p).
Proof.
apply: can_inj (rgd_fun p^-1) _ => y /=.
case: (insubP _ y) => [u _ val_u|]; first by rewrite valK permK.
by rewrite negbK; move/eqP->; rewrite insubF //= eqxx.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rgdP | |
rgdp := perm (@rgdP p). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rgd | |
rfd_odd(p : {perm T}) : p x = x -> rfd p = p :> bool.
Proof.
have rfd1: rfd 1 = 1.
by apply/permP => u; apply: val_inj; rewrite permE /= if_same !perm1.
have [n] := ubnP #|[set x | p x != x]|; elim: n p => // n IHn p le_p_n px_x.
have [p_id | [x1 Hx1]] := set_0Vmem [set x | p x != x].
suffices ->: p = 1 by rewrite ... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_odd | |
rfd_iso: 'C_('Alt_T)[x | 'P] \isog 'Alt_T'.
Proof.
have rgd_x p: rgd p x = x by rewrite permE /= insubF //= eqxx.
have rfd_rgd p: rfd (rgd p) = p.
apply/permP => [[z Hz]]; apply/val_eqP; rewrite !permE.
by rewrite /= [rgd _ _]permE /= insubF eqxx // permE /= insubT.
have sSd: 'C_('Alt_T)[x | 'P] \subset 'dom rfd.
... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_iso | |
simple_Alt5(T : finType) : #|T| >= 5 -> simple 'Alt_T.
Proof.
suff F1 n: #|T| = n + 5 -> simple 'Alt_T by move/subnK/esym/F1.
elim: n T => [| n Hrec T Hde]; first exact: simple_Alt5_base.
have oT: 5 < #|T| by rewrite Hde addnC.
apply/simpleP; split=> [|H Hnorm]; last have [Hh1 nH] := andP Hnorm.
rewrite trivg_card1 -... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | simple_Alt5 | |
gen_tperm_circular_shift(X : finType) x y c : prime #|X| ->
x != y -> #[c]%g = #|X| ->
<<[set tperm x y; c]>>%g = ('Sym_X)%g.
Proof.
move=> Xprime neq_xy ord_c; apply/eqP; rewrite eqEsubset subsetT/=.
have c_gt1 : (1 < #[c]%g)%N by rewrite ord_c prime_gt1.
have cppSS : #[c]%g.-2.+2 = #|X| by rewrite ?prednK ?ltn_pr... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | gen_tperm_circular_shift |
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