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 |
|---|---|---|---|---|---|---|
comm1GA : [~: 1, A] = 1.
Proof. by rewrite commGC commG1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
comm1G
| |
commg_subA B : [~: A, B] \subset A <*> B.
Proof. by rewrite comm_subG // (joing_subl, joing_subr). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_sub
| |
commg_normlG A : G \subset 'N([~: G, A]).
Proof.
apply/subsetP=> x Gx; rewrite inE -genJ gen_subG.
apply/subsetP=> _ /imsetP[_ /imset2P[y z Gy Az ->] ->].
by rewrite -(mulgK [~ x, z] (_ ^ x)) -commMgJ !(mem_commg, groupMl, groupV).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_norml
| |
commg_normrG A : G \subset 'N([~: A, G]).
Proof. by rewrite commGC commg_norml. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_normr
| |
commg_normG H : G <*> H \subset 'N([~: G, H]).
Proof. by rewrite join_subG ?commg_norml ?commg_normr. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_norm
| |
commg_normalG H : [~: G, H] <| G <*> H.
Proof. by rewrite /(_ <| _) commg_sub commg_norm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_normal
| |
normsRlA G B : A \subset G -> A \subset 'N([~: G, B]).
Proof. by move=> sAG; apply: subset_trans (commg_norml G B). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
normsRl
| |
normsRrA G B : A \subset G -> A \subset 'N([~: B, G]).
Proof. by move=> sAG; apply: subset_trans (commg_normr G B). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
normsRr
| |
commg_subrG H : ([~: G, H] \subset H) = (G \subset 'N(H)).
Proof.
rewrite gen_subG; apply/subsetP/subsetP=> [sRH x Gx | nGH xy].
rewrite inE; apply/subsetP=> _ /imsetP[y Ky ->].
by rewrite conjg_Rmul groupMr // sRH // imset2_f ?groupV.
case/imset2P=> x y Gx Hy ->{xy}.
by rewrite commgEr groupMr // memJ_norm (groupV, nGH).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_subr
| |
commg_sublG H : ([~: G, H] \subset G) = (H \subset 'N(G)).
Proof. by rewrite commGC commg_subr. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_subl
| |
commg_subIA B G H :
A \subset 'N_G(H) -> B \subset 'N_H(G) -> [~: A, B] \subset G :&: H.
Proof.
rewrite !subsetI -(gen_subG _ 'N(G)) -(gen_subG _ 'N(H)).
rewrite -commg_subr -commg_subl; case/andP=> sAG sRH; case/andP=> sBH sRG.
by rewrite (subset_trans _ sRG) ?(subset_trans _ sRH) ?commgSS ?subset_gen.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_subI
| |
quotient_cents2A B K :
A \subset 'N(K) -> B \subset 'N(K) ->
(A / K \subset 'C(B / K)) = ([~: A, B] \subset K).
Proof.
move=> nKA nKB.
by rewrite (sameP commG1P trivgP) /= -quotientR // quotient_sub1 // comm_subG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
quotient_cents2
| |
quotient_cents2rA B K :
[~: A, B] \subset K -> (A / K) \subset 'C(B / K).
Proof.
move=> sABK; rewrite -2![_ / _]morphimIdom -!quotientE.
by rewrite quotient_cents2 ?subsetIl ?(subset_trans _ sABK) ?commgSS ?subsetIr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
quotient_cents2r
| |
sub_der1_normG H : G^`(1) \subset H -> H \subset G -> G \subset 'N(H).
Proof.
by move=> sG'H sHG; rewrite -commg_subr (subset_trans _ sG'H) ?commgS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
sub_der1_norm
| |
sub_der1_normalG H : G^`(1) \subset H -> H \subset G -> H <| G.
Proof. by move=> sG'H sHG; rewrite /(H <| G) sHG sub_der1_norm. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
sub_der1_normal
| |
sub_der1_abelianG H : G^`(1) \subset H -> abelian (G / H).
Proof. by move=> sG'H; apply: quotient_cents2r. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
sub_der1_abelian
| |
der1_minG H : G \subset 'N(H) -> abelian (G / H) -> G^`(1) \subset H.
Proof. by move=> nHG abGH; rewrite -quotient_cents2. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der1_min
| |
der_abeliann G : abelian (G^`(n) / G^`(n.+1)).
Proof. by rewrite sub_der1_abelian // der_subS. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_abelian
| |
commg_normSlG H K : G \subset 'N(H) -> [~: G, H] \subset 'N([~: K, H]).
Proof. by move=> nHG; rewrite normsRr // commg_subr. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_normSl
| |
commg_normSrG H K : G \subset 'N(H) -> [~: H, G] \subset 'N([~: H, K]).
Proof. by move=> nHG; rewrite !(commGC H) commg_normSl. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commg_normSr
| |
commMGrG H K : [~: G, K] * [~: H, K] \subset [~: G * H , K].
Proof. by rewrite mul_subG ?commSg ?(mulG_subl, mulG_subr). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commMGr
| |
commMGG H K :
H \subset 'N([~: G, K]) -> [~: G * H , K] = [~: G, K] * [~: H, K].
Proof.
move=> nRH; apply/eqP; rewrite eqEsubset commMGr andbT.
have nRHK: [~: H, K] \subset 'N([~: G, K]) by rewrite comm_subG ?commg_normr.
have defM := norm_joinEr nRHK; rewrite -defM gen_subG /=.
apply/subsetP=> _ /imset2P[_ z /imset2P[x y Gx Hy ->] Kz ->].
by rewrite commMgJ {}defM mem_mulg ?memJ_norm ?mem_commg // (subsetP nRH).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commMG
| |
comm3G1PA B C :
reflect {in A & B & C, forall h k l, [~ h, k, l] = 1} ([~: A, B, C] :==: 1).
Proof.
have R_C := sameP trivgP commG1P.
rewrite -subG1 R_C gen_subG -{}R_C gen_subG.
apply: (iffP subsetP) => [cABC x y z Ax By Cz | cABC xyz].
by apply/set1P; rewrite cABC // !imset2_f.
by case/imset2P=> _ z /imset2P[x y Ax By ->] Cz ->; rewrite cABC.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
comm3G1P
| |
three_subgroupG H K :
[~: G, H, K] :=: 1 -> [~: H, K, G] :=: 1-> [~: K, G, H] :=: 1.
Proof.
move/eqP/comm3G1P=> cGHK /eqP/comm3G1P cHKG.
apply/eqP/comm3G1P=> x y z Kx Gy Hz; symmetry.
rewrite -(conj1g y) -(Hall_Witt_identity y^-1 z x) invgK.
rewrite [X in X ^ z]cGHK ?groupV // [X in X ^ x]cHKG ?groupV //.
by rewrite !conj1g !mul1g conjgKV.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
three_subgroup
| |
der1_joing_cycles(x y : gT) :
let XY := <[x]> <*> <[y]> in let xy := [~ x, y] in
xy \in 'C(XY) -> XY^`(1) = <[xy]>.
Proof.
rewrite joing_idl joing_idr /= -sub_cent1 => /norms_gen nRxy.
apply/eqP; rewrite eqEsubset cycle_subG mem_commg ?mem_gen ?set21 ?set22 //.
rewrite der1_min // quotient_gen -1?gen_subG // quotientU abelian_gen.
rewrite /abelian subUset centU !subsetI andbC centsC -andbA -!abelianE.
rewrite !quotient_abelian ?(abelianS (subset_gen _) (cycle_abelian _)) //=.
by rewrite andbb quotient_cents2r ?genS // /commg_set imset2_set1l imset_set1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der1_joing_cycles
| |
commgACG x y z : x \in G -> y \in G -> z \in G ->
commute y z -> abelian [~: [set x], G] -> [~ x, y, z] = [~ x, z, y].
Proof.
move=> Gx Gy Gz cyz /centsP cRxG; pose cx' u := [~ x^-1, u].
have xR3 u v: [~ x, u, v] = x^-1 * (cx' u * cx' v) * x ^ (u * v).
rewrite [X in X * _]mulgA -conjg_mulR conjVg [cx' v]commgEl.
by rewrite [X in X * _]mulgA -invMg -mulgA conjgM -conjMg -!commgEl.
suffices RxGcx' u: u \in G -> cx' u \in [~: [set x], G].
by rewrite !xR3 {}cyz; congr (_ * _ * _); rewrite cRxG ?RxGcx'.
move=> Gu; suffices/groupMl <-: [~ x, u] ^ x^-1 \in [~: [set x], G].
by rewrite -commMgJ mulgV comm1g group1.
by rewrite memJ_norm ?mem_commg ?set11 // groupV (subsetP (commg_normr _ _)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
commgAC
| |
comm_norm_cent_centH G K :
H \subset 'N(G) -> H \subset 'C(K) -> G \subset 'N(K) ->
[~: G, H] \subset 'C(K).
Proof.
move=> nGH /centsP cKH nKG; rewrite commGC gen_subG centsC.
apply/centsP=> x Kx _ /imset2P[y z Hy Gz ->]; red.
rewrite mulgA -[x * _]cKH ?groupV // -!mulgA; congr (_ * _).
rewrite (mulgA x) (conjgC x) (conjgCV z) 2!mulgA [in RHS]mulgA; congr (_ * _).
by rewrite -2!mulgA (cKH y) // -mem_conjg (normsP nKG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
comm_norm_cent_cent
| |
charRH K G : H \char G -> K \char G -> [~: H, K] \char G.
Proof.
case/charP=> sHG chH /charP[sKG chK]; apply/charP.
by split=> [|f infj Gf]; [rewrite comm_subG | rewrite morphimR // chH // chK].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
charR
| |
der_charn G : G^`(n) \char G.
Proof. by elim: n => [|n IHn]; rewrite ?char_refl // dergSn charR. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_char
| |
der_subn G : G^`(n) \subset G.
Proof. by rewrite char_sub ?der_char. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_sub
| |
der_normn G : G \subset 'N(G^`(n)).
Proof. by rewrite char_norm ?der_char. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_norm
| |
der_normaln G : G^`(n) <| G.
Proof. by rewrite char_normal ?der_char. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_normal
| |
der_subSn G : G^`(n.+1) \subset G^`(n).
Proof. by rewrite comm_subG. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_subS
| |
der_normalSn G : G^`(n.+1) <| G^`(n).
Proof. by rewrite sub_der1_normal // der_subS. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_normalS
| |
morphim_derrT D (f : {morphism D >-> rT}) n G :
G \subset D -> f @* G^`(n) = (f @* G)^`(n).
Proof.
move=> sGD; elim: n => // n IHn.
by rewrite !dergSn -IHn morphimR ?(subset_trans (der_sub n G)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
morphim_der
| |
dergSn G H : G \subset H -> G^`(n) \subset H^`(n).
Proof. by move=> sGH; elim: n => // n IHn; apply: commgSS. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
dergS
| |
quotient_dern G H : G \subset 'N(H) -> G^`(n) / H = (G / H)^`(n).
Proof. exact: morphim_der. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
quotient_der
| |
derJG n x : (G :^ x)^`(n) = G^`(n) :^ x.
Proof. by elim: n => //= n IHn; rewrite !dergSn IHn -conjsRg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
derJ
| |
derG1PG : reflect (G^`(1) = 1) (abelian G).
Proof. exact: commG1P. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
derG1P
| |
der_contn : GFunctor.continuous (@derived_at n).
Proof. by move=> aT rT G f; rewrite morphim_der. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_cont
| |
der_igFunn := [igFun by der_sub^~ n & der_cont n].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_igFun
| |
der_gFunn := [gFun by der_cont n].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_gFun
| |
der_mgFunn := [mgFun by dergS^~ n].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
der_mgFun
| |
isog_der(aT rT : finGroupType) n (G : {group aT}) (H : {group rT}) :
G \isog H -> G^`(n) \isog H^`(n).
Proof. exact: gFisog. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat fintype",
"From mathcomp Require Import bigop finset binomial fingroup morphism",
"From mathcomp Require Import automorphism quotient gfunctor"
] |
solvable/commutator.v
|
isog_der
| |
cyclicA := [exists x, A == <[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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclic
| |
cyclicPA : reflect (exists x, A = <[x]>) (cyclic A).
Proof. exact: exists_eqP. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclicP
| |
cycle_cyclicx : cyclic <[x]>.
Proof. by apply/cyclicP; exists x. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cycle_cyclic
| |
cyclic1: cyclic [1 gT].
Proof. by rewrite -cycle1 cycle_cyclic. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclic1
| |
Zpm(i : 'Z_#[a]) := a ^+ i.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zpm
| |
ZpmM: {in Zp #[a] &, {morph Zpm : x y / x * y}}.
Proof.
rewrite /Zpm; case: (eqVneq a 1) => [-> | nta] i j _ _.
by rewrite !expg1n ?mulg1.
by rewrite /= {3}Zp_cast ?order_gt1 // expg_mod_order expgD.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
ZpmM
| |
Zpm_morphism:= Morphism ZpmM.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zpm_morphism
| |
im_Zpm: Zpm @* Zp #[a] = <[a]>.
Proof.
apply/eqP; rewrite eq_sym eqEcard cycle_subG /= andbC morphimEdom.
rewrite (leq_trans (leq_imset_card _ _)) ?card_Zp //= /Zp order_gt1.
case: eqP => /= [a1 | _]; first by rewrite imset_set1 morph1 a1 set11.
by apply/imsetP; exists 1%R; rewrite ?expg1 ?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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
im_Zpm
| |
injm_Zpm: 'injm Zpm.
Proof.
apply/injmP/dinjectiveP/card_uniqP.
rewrite size_map -cardE card_Zp //= {7}/order -im_Zpm morphimEdom /=.
by apply: eq_card => x; apply/imageP/imsetP=> [] [i Zp_i ->]; exists i.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
injm_Zpm
| |
eq_expg_mod_orderm n : (a ^+ m == a ^+ n) = (m == n %[mod #[a]]).
Proof.
have [->|] := eqVneq a 1; first by rewrite order1 !modn1 !expg1n eqxx.
rewrite -order_gt1 => lt1a; have ZpT: Zp #[a] = setT by rewrite /Zp lt1a.
have: injective Zpm by move=> i j; apply (injmP injm_Zpm); rewrite /= ZpT inE.
move/inj_eq=> eqZ; symmetry; rewrite -(Zp_cast lt1a).
by rewrite -[_ == _](eqZ (inZp m) (inZp n)) /Zpm /= Zp_cast ?expg_mod_order.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
eq_expg_mod_order
| |
eq_expg_ordd (m n : 'I_d) :
d <= #[a]%g -> (a ^+ m == a ^+ n) = (m == n).
Proof.
by move=> d_leq; rewrite eq_expg_mod_order !modn_small// (leq_trans _ d_leq).
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
eq_expg_ord
| |
expgD_Zpd (n m : 'Z_d) : (d > 0)%N ->
#[a]%g %| d -> a ^+ (n + m)%R = a ^+ n * a ^+ m.
Proof.
move=> d_gt0 xdvd; apply/eqP; rewrite -expgD eq_expg_mod_order/= modn_dvdm//.
by case: d d_gt0 {m n} xdvd => [|[|[]]]//= _; rewrite dvdn1 => /eqP->.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
expgD_Zp
| |
Zp_isom: isom (Zp #[a]) <[a]> Zpm.
Proof. by apply/isomP; rewrite injm_Zpm im_Zpm. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zp_isom
| |
Zp_isog: isog (Zp #[a]) <[a]>.
Proof. exact: isom_isog Zp_isom. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zp_isog
| |
cyclic_abelianA : cyclic A -> abelian A.
Proof. by case/cyclicP=> a ->; apply: cycle_abelian. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclic_abelian
| |
cycleMsuba b :
commute a b -> coprime #[a] #[b] -> <[a]> \subset <[a * b]>.
Proof.
move=> cab co_ab; apply/subsetP=> _ /cycleP[k ->].
apply/cycleP; exists (chinese #[a] #[b] k 0); symmetry.
rewrite expgMn // -[in LHS]expg_mod_order chinese_modl // expg_mod_order.
by rewrite /chinese addn0 -mulnA mulnCA expgM expg_order expg1n mulg1.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cycleMsub
| |
cycleMa b :
commute a b -> coprime #[a] #[b] -> <[a * b]> = <[a]> * <[b]>.
Proof.
move=> cab co_ab; apply/eqP; rewrite eqEsubset -(cent_joinEl (cents_cycle cab)).
rewrite join_subG {3}cab !cycleMsub // 1?coprime_sym //.
by rewrite -genM_join cycle_subG mem_gen // imset2_f ?cycle_id.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cycleM
| |
cyclicMA B :
cyclic A -> cyclic B -> B \subset 'C(A) -> coprime #|A| #|B| ->
cyclic (A * B).
Proof.
move=> /cyclicP[a ->] /cyclicP[b ->]; rewrite cent_cycle cycle_subG => cab coab.
by rewrite -cycleM ?cycle_cyclic //; apply/esym/cent1P.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclicM
| |
cyclicYK H :
cyclic K -> cyclic H -> H \subset 'C(K) -> coprime #|K| #|H| ->
cyclic (K <*> H).
Proof. by move=> cycK cycH cKH coKH; rewrite cent_joinEr // cyclicM. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclicY
| |
order_dvdna n : #[a] %| n = (a ^+ n == 1).
Proof. by rewrite (eq_expg_mod_order a n 0) mod0n. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
order_dvdn
| |
order_infa n : a ^+ n.+1 == 1 -> #[a] <= n.+1.
Proof. by rewrite -order_dvdn; apply: dvdn_leq. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
order_inf
| |
order_dvdGG a : a \in G -> #[a] %| #|G|.
Proof. by move=> Ga; apply: cardSg; rewrite cycle_subG. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
order_dvdG
| |
expg_cardGG a : a \in G -> a ^+ #|G| = 1.
Proof. by move=> Ga; apply/eqP; rewrite -order_dvdn order_dvdG. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
expg_cardG
| |
expg_znatG x k : x \in G -> x ^+ (k%:R : 'Z_(#|G|))%R = x ^+ k.
Proof.
case: (eqsVneq G 1) => [-> /set1P-> | ntG Gx]; first by rewrite !expg1n.
apply/eqP; rewrite val_Zp_nat ?cardG_gt1 // eq_expg_mod_order.
by rewrite modn_dvdm ?order_dvdG.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
expg_znat
| |
expg_znegG x (k : 'Z_(#|G|)) : x \in G -> x ^+ (- k)%R = x ^- k.
Proof.
move=> Gx; apply/eqP; rewrite eq_sym eq_invg_mul -expgD.
by rewrite -(expg_znat _ Gx) natrD natr_Zp natr_negZp subrr.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
expg_zneg
| |
nt_gen_primeG x : prime #|G| -> x \in G^# -> G :=: <[x]>.
Proof.
move=> Gpr /setD1P[]; rewrite -cycle_subG -cycle_eq1 => ntX sXG.
apply/eqP; rewrite eqEsubset sXG andbT.
by apply: contraR ntX => /(prime_TIg Gpr); rewrite (setIidPr sXG) => ->.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
nt_gen_prime
| |
nt_prime_orderp x : prime p -> x ^+ p = 1 -> x != 1 -> #[x] = p.
Proof.
move=> p_pr xp ntx; apply/prime_nt_dvdP; rewrite ?order_eq1 //.
by rewrite order_dvdn xp.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
nt_prime_order
| |
orderXdvda n : #[a ^+ n] %| #[a].
Proof. by apply: order_dvdG; apply: mem_cycle. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
orderXdvd
| |
orderXgcda n : #[a ^+ n] = #[a] %/ gcdn #[a] n.
Proof.
apply/eqP; rewrite eqn_dvd; apply/andP; split.
rewrite order_dvdn -expgM -muln_divCA_gcd //.
by rewrite expgM expg_order expg1n.
have [-> | n_gt0] := posnP n; first by rewrite gcdn0 divnn order_gt0 dvd1n.
rewrite -(dvdn_pmul2r n_gt0) divn_mulAC ?dvdn_gcdl // dvdn_lcm.
by rewrite order_dvdn mulnC expgM expg_order eqxx dvdn_mulr.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
orderXgcd
| |
orderXdiva n : n %| #[a] -> #[a ^+ n] = #[a] %/ n.
Proof. by case/dvdnP=> q defq; rewrite orderXgcd {2}defq gcdnC gcdnMl. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
orderXdiv
| |
orderXexpp m n x : #[x] = (p ^ n)%N -> #[x ^+ (p ^ m)] = (p ^ (n - m))%N.
Proof.
move=> ox; have [n_le_m | m_lt_n] := leqP n m.
rewrite -(subnKC n_le_m) subnDA subnn expnD expgM -ox.
by rewrite expg_order expg1n order1.
rewrite orderXdiv ox ?dvdn_exp2l ?expnB ?(ltnW m_lt_n) //.
by have:= order_gt0 x; rewrite ox expn_gt0 orbC -(ltn_predK m_lt_n).
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
orderXexp
| |
orderXpfactorp k n x :
#[x ^+ (p ^ k)] = n -> prime p -> p %| n -> #[x] = (p ^ k * n)%N.
Proof.
move=> oxp p_pr dv_p_n.
suffices pk_x: p ^ k %| #[x] by rewrite -oxp orderXdiv // mulnC divnK.
rewrite pfactor_dvdn // leqNgt; apply: contraL dv_p_n => lt_x_k.
rewrite -oxp -p'natE // -(subnKC (ltnW lt_x_k)) expnD expgM.
rewrite (pnat_dvd (orderXdvd _ _)) // -p_part // orderXdiv ?dvdn_part //.
by rewrite -{1}[#[x]](partnC p) // mulKn // part_pnat.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
orderXpfactor
| |
orderXprimep n x :
#[x ^+ p] = n -> prime p -> p %| n -> #[x] = (p * n)%N.
Proof. exact: (@orderXpfactor p 1). 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
orderXprime
| |
orderXpnatm n x : #[x ^+ m] = n -> \pi(n).-nat m -> #[x] = (m * n)%N.
Proof.
move=> oxm n_m; have [m_gt0 _] := andP n_m.
suffices m_x: m %| #[x] by rewrite -oxm orderXdiv // mulnC divnK.
apply/dvdn_partP=> // p; rewrite mem_primes => /and3P[p_pr _ p_m].
have n_p: p \in \pi(n) by apply: (pnatP _ _ n_m).
have p_oxm: p %| #[x ^+ (p ^ logn p m)].
apply: dvdn_trans (orderXdvd _ m`_p^'); rewrite -expgM -p_part ?partnC //.
by rewrite oxm; rewrite mem_primes in n_p; case/and3P: n_p.
by rewrite (orderXpfactor (erefl _) p_pr p_oxm) p_part // dvdn_mulr.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
orderXpnat
| |
orderMa b :
commute a b -> coprime #[a] #[b] -> #[a * b] = (#[a] * #[b])%N.
Proof. by move=> cab co_ab; rewrite -coprime_cardMg -?cycleM. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
orderM
| |
expg_invnA k := (egcdn k #|A|).1.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
expg_invn
| |
expgKG k :
coprime #|G| k -> {in G, cancel (natexp^~ k) (natexp^~ (expg_invn G k))}.
Proof.
move=> coGk x /order_dvdG Gx; apply/eqP.
rewrite -expgM (eq_expg_mod_order _ _ 1) -(modn_dvdm 1 Gx).
by rewrite -(chinese_modl coGk 1 0) /chinese mul1n addn0 modn_dvdm.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
expgK
| |
cyclic_dprodK H G :
K \x H = G -> cyclic K -> cyclic H -> cyclic G = coprime #|K| #|H| .
Proof.
case/dprodP=> _ defKH cKH tiKH cycK cycH; pose m := lcmn #|K| #|H|.
apply/idP/idP=> [/cyclicP[x defG] | coKH]; last by rewrite -defKH cyclicM.
rewrite /coprime -dvdn1 -(@dvdn_pmul2l m) ?lcmn_gt0 ?cardG_gt0 //.
rewrite muln_lcm_gcd muln1 -TI_cardMg // defKH defG order_dvdn.
have /mulsgP[y z Ky Hz ->]: x \in K * H by rewrite defKH defG cycle_id.
rewrite -[1]mulg1 expgMn; last exact/commute_sym/(centsP cKH).
apply/eqP; congr (_ * _); apply/eqP; rewrite -order_dvdn.
exact: dvdn_trans (order_dvdG Ky) (dvdn_lcml _ _).
exact: dvdn_trans (order_dvdG Hz) (dvdn_lcmr _ _).
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclic_dprod
| |
generator(A : {set gT}) a := A == <[a]>.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
generator
| |
generator_cyclea : generator <[a]> a.
Proof. exact: 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
generator_cycle
| |
cycle_generatora x : generator <[a]> x -> x \in <[a]>.
Proof. by move/(<[a]> =P _)->; apply: cycle_id. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cycle_generator
| |
generator_ordera b : generator <[a]> b -> #[a] = #[b].
Proof. by rewrite /order => /(<[a]> =P _)->. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
generator_order
| |
Euler_exp_totienta n : coprime a n -> a ^ totient n = 1 %[mod n].
Proof.
(case: n => [|[|n']] //; [by rewrite !modn1 | set n := n'.+2]) => co_a_n.
have{co_a_n} Ua: coprime n (inZp a : 'I_n) by rewrite coprime_sym coprime_modl.
have: FinRing.unit 'Z_n Ua ^+ totient n == 1.
by rewrite -card_units_Zp // -order_dvdn order_dvdG ?inE.
by rewrite -2!val_eqE unit_Zp_expg /= -/n modnXm => /eqP.
Qed.
|
Theorem
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Euler_exp_totient
| |
eltmof #[y] %| #[x] := fun x_i => y ^+ invm (injm_Zpm x) x_i.
Hypothesis dvd_y_x : #[y] %| #[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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
eltm
| |
eltmEi : eltm dvd_y_x (x ^+ i) = y ^+ i.
Proof.
apply/eqP; rewrite eq_expg_mod_order.
have [x_le1 | x_gt1] := leqP #[x] 1.
suffices: #[y] %| 1 by rewrite dvdn1 => /eqP->; rewrite !modn1.
by rewrite (dvdn_trans dvd_y_x) // dvdn1 order_eq1 -cycle_eq1 trivg_card_le1.
rewrite -(expg_znat i (cycle_id x)) invmE /=; last by rewrite /Zp x_gt1 inE.
by rewrite val_Zp_nat // modn_dvdm.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
eltmE
| |
eltm_id: eltm dvd_y_x x = y. Proof. exact: (eltmE 1). 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
eltm_id
| |
eltmM: {in <[x]> &, {morph eltm dvd_y_x : x_i x_j / x_i * x_j}}.
Proof.
move=> _ _ /cycleP[i ->] /cycleP[j ->].
by apply/eqP; rewrite -expgD !eltmE expgD.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
eltmM
| |
eltm_morphism:= Morphism eltmM.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
eltm_morphism
| |
im_eltm: eltm dvd_y_x @* <[x]> = <[y]>.
Proof. by rewrite morphim_cycle ?cycle_id //= eltm_id. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
im_eltm
| |
ker_eltm: 'ker (eltm dvd_y_x) = <[x ^+ #[y]]>.
Proof.
apply/eqP; rewrite eq_sym eqEcard cycle_subG 3!inE mem_cycle /= eltmE.
rewrite expg_order eqxx (orderE y) -im_eltm card_morphim setIid -orderE.
by rewrite orderXdiv ?dvdn_indexg //= leq_divRL ?indexg_gt0 ?Lagrange ?subsetIl.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
ker_eltm
| |
injm_eltm: 'injm (eltm dvd_y_x) = (#[x] %| #[y]).
Proof. by rewrite ker_eltm subG1 cycle_eq1 -order_dvdn. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
injm_eltm
| |
cycle_sub_group(a : gT) m :
m %| #[a] ->
[set H : {group gT} | H \subset <[a]> & #|H| == m]
= [set <[a ^+ (#[a] %/ m)]>%G].
Proof.
move=> m_dv_a; have m_gt0: 0 < m by apply: dvdn_gt0 m_dv_a.
have oam: #|<[a ^+ (#[a] %/ m)]>| = m.
apply/eqP; rewrite [#|_|]orderXgcd -(divnMr m_gt0) muln_gcdl divnK //.
by rewrite gcdnC gcdnMr mulKn.
apply/eqP; rewrite eqEsubset sub1set inE /= cycleX oam eqxx !andbT.
apply/subsetP=> X; rewrite in_set1 inE -val_eqE /= eqEcard oam.
case/andP=> sXa /eqP oX; rewrite oX leqnn andbT.
apply/subsetP=> x Xx; case/cycleP: (subsetP sXa _ Xx) => k def_x.
have: (x ^+ m == 1)%g by rewrite -oX -order_dvdn cardSg // gen_subG sub1set.
rewrite {x Xx}def_x -expgM -order_dvdn -[#[a]](Lagrange sXa) -oX mulnC.
rewrite dvdn_pmul2r // mulnK // => /dvdnP[i ->].
by rewrite mulnC expgM groupX // cycle_id.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cycle_sub_group
| |
cycle_subgroup_chara (H : {group gT}) : H \subset <[a]> -> H \char <[a]>.
Proof.
move=> sHa; apply: lone_subgroup_char => // J sJa isoJH.
have dvHa: #|H| %| #[a] by apply: cardSg.
have{dvHa} /setP Huniq := esym (cycle_sub_group dvHa).
move: (Huniq H) (Huniq J); rewrite !inE /=.
by rewrite sHa sJa (card_isog isoJH) eqxx => /eqP<- /eqP<-.
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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cycle_subgroup_char
| |
morph_order: #[f x] %| #[x].
Proof. by rewrite order_dvdn -morphX // expg_order morph1. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
morph_order
| |
morph_generatorA : generator A x -> generator (f @* A) (f x).
Proof. by move/(A =P _)->; rewrite /generator morphim_cycle. 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
morph_generator
| |
cyclicSG H : H \subset G -> cyclic G -> cyclic H.
Proof.
move=> sHG /cyclicP[x defG]; apply/cyclicP.
exists (x ^+ (#[x] %/ #|H|)); apply/congr_group/set1P.
by rewrite -cycle_sub_group /order -defG ?cardSg // inE sHG 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 bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclicS
|
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