statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
commG1 A : [~: A, 1] = 1. | Proof. by apply/commG1P; rewrite centsC sub1G. Qed. | Lemma | commG1 | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"centsC",
"commG1P",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm1G A : [~: 1, A] = 1. | Proof. by rewrite commGC commG1. Qed. | Lemma | comm1G | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commG1",
"commGC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_sub A B : [~: A, B] \subset A <*> B. | Proof. by rewrite comm_subG // (joing_subl, joing_subr). Qed. | Lemma | commg_sub | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"comm_subG",
"joing_subl",
"joing_subr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_norml G 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 | commg_norml | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commMgJ",
"genJ",
"gen_subG",
"groupMl",
"groupV",
"imset2P",
"imsetP",
"inE",
"mem_commg",
"mulgK",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_normr G A : G \subset 'N([~: A, G]). | Proof. by rewrite commGC commg_norml. Qed. | Lemma | commg_normr | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commGC",
"commg_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_norm G H : G <*> H \subset 'N([~: G, H]). | Proof. by rewrite join_subG ?commg_norml ?commg_normr. Qed. | Lemma | commg_norm | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commg_norml",
"commg_normr",
"join_subG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_normal G H : [~: G, H] <| G <*> H. | Proof. by rewrite /(_ <| _) commg_sub commg_norm. Qed. | Lemma | commg_normal | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commg_norm",
"commg_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normsRl A G B : A \subset G -> A \subset 'N([~: G, B]). | Proof. by move=> sAG; apply: subset_trans (commg_norml G B). Qed. | Lemma | normsRl | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commg_norml",
"sAG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normsRr A G B : A \subset G -> A \subset 'N([~: B, G]). | Proof. by move=> sAG; apply: subset_trans (commg_normr G B). Qed. | Lemma | normsRr | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commg_normr",
"sAG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_subr G 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 | commg_subr | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commgEr",
"conjg_Rmul",
"gen_subG",
"groupMr",
"groupV",
"imset2P",
"imset2_f",
"imsetP",
"inE",
"memJ_norm",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_subl G H : ([~: G, H] \subset G) = (H \subset 'N(G)). | Proof. by rewrite commGC commg_subr. Qed. | Lemma | commg_subl | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commGC",
"commg_subr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_subI A 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 | commg_subI | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commgSS",
"commg_subl",
"commg_subr",
"gen_subG",
"sAG",
"subsetI",
"subset_gen",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_cents2 A 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 | quotient_cents2 | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commG1P",
"comm_subG",
"quotientR",
"quotient_sub1",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_cents2r A 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 | quotient_cents2r | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commgSS",
"morphimIdom",
"quotientE",
"quotient_cents2",
"subsetIl",
"subsetIr",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_der1_norm G 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 | sub_der1_norm | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commgS",
"commg_subr",
"sHG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_der1_normal G 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 | sub_der1_normal | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"sHG",
"sub_der1_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_der1_abelian G H : G^`(1) \subset H -> abelian (G / H). | Proof. by move=> sG'H; apply: quotient_cents2r. Qed. | Lemma | sub_der1_abelian | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"abelian",
"apply",
"quotient_cents2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der1_min G H : G \subset 'N(H) -> abelian (G / H) -> G^`(1) \subset H. | Proof. by move=> nHG abGH; rewrite -quotient_cents2. Qed. | Lemma | der1_min | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"abelian",
"nHG",
"quotient_cents2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_abelian n G : abelian (G^`(n) / G^`(n.+1)). | Proof. by rewrite sub_der1_abelian // der_subS. Qed. | Lemma | der_abelian | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"abelian",
"der_subS",
"sub_der1_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_normSl G H K : G \subset 'N(H) -> [~: G, H] \subset 'N([~: K, H]). | Proof. by move=> nHG; rewrite normsRr // commg_subr. Qed. | Lemma | commg_normSl | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commg_subr",
"nHG",
"normsRr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg_normSr G H K : G \subset 'N(H) -> [~: H, G] \subset 'N([~: H, K]). | Proof. by move=> nHG; rewrite !(commGC H) commg_normSl. Qed. | Lemma | commg_normSr | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commGC",
"commg_normSl",
"nHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commMGr G H K : [~: G, K] * [~: H, K] \subset [~: G * H , K]. | Proof. by rewrite mul_subG ?commSg ?(mulG_subl, mulG_subr). Qed. | Lemma | commMGr | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commSg",
"mulG_subl",
"mulG_subr",
"mul_subG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commMG G 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 //... | Lemma | commMG | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commMGr",
"commMgJ",
"comm_subG",
"commg_normr",
"eqEsubset",
"gen_subG",
"imset2P",
"memJ_norm",
"mem_commg",
"mem_mulg",
"norm_joinEr",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm3G1P A 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 | comm3G1P | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commG1P",
"gen_subG",
"imset2P",
"imset2_f",
"set1P",
"subG1",
"subsetP",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
three_subgroup G 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 | three_subgroup | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"Hall_Witt_identity",
"apply",
"comm3G1P",
"conj1g",
"conjgKV",
"groupV",
"invgK",
"mul1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 ?(abe... | Lemma | der1_joing_cycles | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"abelian",
"abelianE",
"abelianS",
"abelian_gen",
"apply",
"centU",
"centsC",
"commg_set",
"cycle_abelian",
"cycle_subG",
"der1_min",
"eqEsubset",
"gT",
"genS",
"gen_subG",
"imset2_set1l",
"imset_set1",
"joing_idl",
"joing_idr",
"mem_commg",
"mem_gen",
"norms_gen",
"quoti... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commgAC G 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],... | Lemma | commgAC | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"abelian",
"centsP",
"comm1g",
"commMgJ",
"commgEl",
"commg_normr",
"commute",
"conjMg",
"conjVg",
"conjgM",
"conjg_mulR",
"cyz",
"group1",
"groupMl",
"groupV",
"invMg",
"memJ_norm",
"mem_commg",
"mulgA",
"mulgV",
"set11",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_norm_cent_cent H 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 | comm_norm_cent_cent | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"cKH",
"centsC",
"centsP",
"commGC",
"conjgC",
"conjgCV",
"gen_subG",
"groupV",
"imset2P",
"mem_conjg",
"mulgA",
"nKG",
"normsP"
] | Aschbacher, exercise 3.6 (used in proofs of Aschbacher 24.7 and B & G 1.10 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
charR H 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 | charR | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"Gf",
"apply",
"char",
"charP",
"comm_subG",
"morphimR",
"sHG",
"sKG",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_char n G : G^`(n) \char G. | Proof. by elim: n => [|n IHn]; rewrite ?char_refl // dergSn charR. Qed. | Lemma | der_char | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"char",
"charR",
"char_refl",
"dergSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_sub n G : G^`(n) \subset G. | Proof. by rewrite char_sub ?der_char. Qed. | Lemma | der_sub | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"char_sub",
"der_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_norm n G : G \subset 'N(G^`(n)). | Proof. by rewrite char_norm ?der_char. Qed. | Lemma | der_norm | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"char_norm",
"der_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_normal n G : G^`(n) <| G. | Proof. by rewrite char_normal ?der_char. Qed. | Lemma | der_normal | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"char_normal",
"der_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_subS n G : G^`(n.+1) \subset G^`(n). | Proof. by rewrite comm_subG. Qed. | Lemma | der_subS | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"comm_subG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_normalS n G : G^`(n.+1) <| G^`(n). | Proof. by rewrite sub_der1_normal // der_subS. Qed. | Lemma | der_normalS | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"der_subS",
"sub_der1_normal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_der rT 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 | morphim_der | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"der_sub",
"dergSn",
"morphimR",
"morphism",
"sGD",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dergS n G H : G \subset H -> G^`(n) \subset H^`(n). | Proof. by move=> sGH; elim: n => // n IHn; apply: commgSS. Qed. | Lemma | dergS | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commgSS",
"sGH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_der n G H : G \subset 'N(H) -> G^`(n) / H = (G / H)^`(n). | Proof. exact: morphim_der. Qed. | Lemma | quotient_der | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"morphim_der"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derJ G n x : (G :^ x)^`(n) = G^`(n) :^ x. | Proof. by elim: n => //= n IHn; rewrite !dergSn IHn -conjsRg. Qed. | Lemma | derJ | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"conjsRg",
"dergSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derG1P G : reflect (G^`(1) = 1) (abelian G). | Proof. exact: commG1P. Qed. | Lemma | derG1P | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"abelian",
"commG1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_cont n : GFunctor.continuous (@derived_at n). | Proof. by move=> aT rT G f; rewrite morphim_der. Qed. | Lemma | der_cont | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"aT",
"continuous",
"derived_at",
"morphim_der"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_igFun n | := [igFun by der_sub^~ n & der_cont n]. | Canonical | der_igFun | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"der_cont",
"der_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_gFun n | := [gFun by der_cont n]. | Canonical | der_gFun | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"der_cont"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_mgFun n | := [mgFun by dergS^~ n]. | Canonical | der_mgFun | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"dergS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | isog_der | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"aT",
"gFisog",
"group",
"isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic A | := [exists x, A == <[x]>]. | Definition | cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclicP A : reflect (exists x, A = <[x]>) (cyclic A). | Proof. exact: exists_eqP. Qed. | Lemma | cyclicP | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cyclic",
"exists_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_cyclic x : cyclic <[x]>. | Proof. by apply/cyclicP; exists x. Qed. | Lemma | cycle_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cyclic",
"cyclicP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic1 : cyclic [1 gT]. | Proof. by rewrite -cycle1 cycle_cyclic. Qed. | Lemma | cyclic1 | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cycle1",
"cycle_cyclic",
"cyclic",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zpm (i : 'Z_#[a]) | := a ^+ i. | Definition | Zpm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | ZpmM | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp",
"Zp_cast",
"Zpm",
"eqVneq",
"expg1n",
"expgD",
"expg_mod_order",
"mulg1",
"order_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zpm_morphism | := Morphism ZpmM. | Canonical | Zpm_morphism | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"ZpmM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | im_Zpm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp",
"Zpm",
"a1",
"apply",
"card_Zp",
"cycle_subG",
"eqEcard",
"eq_sym",
"expg1",
"imsetP",
"imset_set1",
"inE",
"leq_imset_card",
"leq_trans",
"morph1",
"morphimEdom",
"order_gt1",
"set11"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | injm_Zpm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zpm",
"apply",
"cardE",
"card_Zp",
"card_uniqP",
"dinjectiveP",
"eq_card",
"im_Zpm",
"imageP",
"imsetP",
"injmP",
"morphimEdom",
"order",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_expg_mod_order m 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) (... | Lemma | eq_expg_mod_order | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp",
"Zp_cast",
"Zpm",
"apply",
"eqVneq",
"eqxx",
"expg1n",
"expg_mod_order",
"inE",
"inZp",
"inj_eq",
"injmP",
"injm_Zpm",
"modn1",
"order1",
"order_gt1",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_expg_ord d (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 | eq_expg_ord | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"eq_expg_mod_order",
"leq_trans",
"modn_small"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgD_Zp d (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 | expgD_Zp | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"d_gt0",
"dvdn1",
"eq_expg_mod_order",
"expgD",
"modn_dvdm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zp_isom : isom (Zp #[a]) <[a]> Zpm. | Proof. by apply/isomP; rewrite injm_Zpm im_Zpm. Qed. | Lemma | Zp_isom | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp",
"Zpm",
"apply",
"im_Zpm",
"injm_Zpm",
"isom",
"isomP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zp_isog : isog (Zp #[a]) <[a]>. | Proof. exact: isom_isog Zp_isom. Qed. | Lemma | Zp_isog | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp",
"Zp_isom",
"isog",
"isom_isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_abelian A : cyclic A -> abelian A. | Proof. by case/cyclicP=> a ->; apply: cycle_abelian. Qed. | Lemma | cyclic_abelian | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"abelian",
"apply",
"cycle_abelian",
"cyclic",
"cyclicP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycleMsub a 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 | cycleMsub | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"addn0",
"apply",
"chinese",
"chinese_modl",
"commute",
"coprime",
"cycleP",
"expg1n",
"expgM",
"expgMn",
"expg_mod_order",
"expg_order",
"mulg1",
"mulnA",
"mulnCA",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycleM a 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 | cycleM | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cent_joinEl",
"cents_cycle",
"commute",
"coprime",
"coprime_sym",
"cycleMsub",
"cycle_id",
"cycle_subG",
"eqEsubset",
"genM_join",
"imset2_f",
"join_subG",
"mem_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclicM A 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 | cyclicM | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cent1P",
"cent_cycle",
"coprime",
"cycleM",
"cycle_cyclic",
"cycle_subG",
"cyclic",
"cyclicP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclicY K 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 | cyclicY | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cKH",
"cent_joinEr",
"coprime",
"cyclic",
"cyclicM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_dvdn a n : (#[a] %| n) = (a ^+ n == 1). | Proof. by rewrite (eq_expg_mod_order a n 0) mod0n. Qed. | Lemma | order_dvdn | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"eq_expg_mod_order",
"mod0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_inf a n : a ^+ n.+1 == 1 -> #[a] <= n.+1. | Proof. by rewrite -order_dvdn; apply: dvdn_leq. Qed. | Lemma | order_inf | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"dvdn_leq",
"order_dvdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_dvdG G a : a \in G -> #[a] %| #|G|. | Proof. by move=> Ga; apply: cardSg; rewrite cycle_subG. Qed. | Lemma | order_dvdG | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cardSg",
"cycle_subG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expg_cardG G a : a \in G -> a ^+ #|G| = 1. | Proof. by move=> Ga; apply/eqP; rewrite -order_dvdn order_dvdG. Qed. | Lemma | expg_cardG | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"order_dvdG",
"order_dvdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expg_znat G 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 | expg_znat | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cardG_gt1",
"eq_expg_mod_order",
"eqsVneq",
"expg1n",
"modn_dvdm",
"order_dvdG",
"set1P",
"val_Zp_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expg_zneg G 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 | expg_zneg | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"eq_invg_mul",
"eq_sym",
"expgD",
"expg_znat",
"natrD",
"natr_Zp",
"natr_negZp",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nt_gen_prime G 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 | nt_gen_prime | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cycle_eq1",
"cycle_subG",
"eqEsubset",
"prime",
"prime_TIg",
"sXG",
"setD1P",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nt_prime_order p 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 | nt_prime_order | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"order_dvdn",
"order_eq1",
"p_pr",
"prime",
"prime_nt_dvdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderXdvd a n : #[a ^+ n] %| #[a]. | Proof. by apply: order_dvdG; apply: mem_cycle. Qed. | Lemma | orderXdvd | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"mem_cycle",
"order_dvdG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderXgcd a 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_o... | Lemma | orderXgcd | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"divn_mulAC",
"divnn",
"dvd1n",
"dvdn_gcdl",
"dvdn_lcm",
"dvdn_mulr",
"dvdn_pmul2r",
"eqn_dvd",
"eqxx",
"expg1n",
"expgM",
"expg_order",
"gcdn",
"gcdn0",
"mulnC",
"muln_divCA_gcd",
"n_gt0",
"order_dvdn",
"order_gt0",
"posnP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderXdiv a n : n %| #[a] -> #[a ^+ n] = #[a] %/ n. | Proof. by case/dvdnP=> q defq; rewrite orderXgcd {2}defq gcdnC gcdnMl. Qed. | Lemma | orderXdiv | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"dvdnP",
"gcdnC",
"gcdnMl",
"orderXgcd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderXexp p 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 | orderXexp | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"dvdn_exp2l",
"expg1n",
"expgM",
"expg_order",
"expnB",
"expnD",
"expn_gt0",
"leqP",
"ltnW",
"ltn_predK",
"order1",
"orderXdiv",
"order_gt0",
"subnDA",
"subnKC",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderXpfactor p 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}[#[... | Lemma | orderXpfactor | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"divnK",
"dvdn_part",
"expgM",
"expnD",
"leqNgt",
"ltnW",
"mulKn",
"mulnC",
"orderXdiv",
"orderXdvd",
"p'natE",
"p_part",
"p_pr",
"part_pnat",
"partnC",
"pfactor_dvdn",
"pnat_dvd",
"prime",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderXprime p n x :
#[x ^+ p] = n -> prime p -> p %| n -> #[x] = (p * n)%N. | Proof. exact: (@orderXpfactor p 1). Qed. | Lemma | orderXprime | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"orderXpfactor",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderXpnat m 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^'); rew... | Lemma | orderXpnat | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"divnK",
"dvdn_mulr",
"dvdn_partP",
"dvdn_trans",
"expgM",
"logn",
"mem_primes",
"mulnC",
"nat",
"orderXdiv",
"orderXdvd",
"orderXpfactor",
"p_part",
"p_pr",
"partnC",
"pi",
"pnatP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderM a b :
commute a b -> coprime #[a] #[b] -> #[a * b] = (#[a] * #[b])%N. | Proof. by move=> cab co_ab; rewrite -coprime_cardMg -?cycleM. Qed. | Lemma | orderM | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"commute",
"coprime",
"coprime_cardMg",
"cycleM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expg_invn A k | := (egcdn k #|A|).1. | Definition | expg_invn | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"egcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgK G 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 | expgK | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"addn0",
"apply",
"chinese",
"chinese_modl",
"coprime",
"eq_expg_mod_order",
"expgM",
"expg_invn",
"modn_dvdm",
"mul1n",
"natexp",
"order_dvdG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_dprod K 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... | Lemma | cyclic_dprod | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"TI_cardMg",
"apply",
"cKH",
"cardG_gt0",
"centsP",
"commute_sym",
"coprime",
"cycle_id",
"cyclic",
"cyclicM",
"cyclicP",
"defG",
"dprodP",
"dvdn1",
"dvdn_lcml",
"dvdn_lcmr",
"dvdn_pmul2l",
"dvdn_trans",
"expgMn",
"last",
"lcmn",
"lcmn_gt0",
"mulg1",
"muln1",
"muln_lc... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
generator (A : {set gT}) a | := A == <[a]>. | Definition | generator | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
generator_cycle a : generator <[a]> a. | Proof. exact: eqxx. Qed. | Lemma | generator_cycle | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"eqxx",
"generator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_generator a x : generator <[a]> x -> x \in <[a]>. | Proof. by move/(<[a]> =P _)->; apply: cycle_id. Qed. | Lemma | cycle_generator | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cycle_id",
"generator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
generator_order a b : generator <[a]> b -> #[a] = #[b]. | Proof. by rewrite /order => /(<[a]> =P _)->. Qed. | Lemma | generator_order | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"generator",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Euler_exp_totient a 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 => /... | Theorem | Euler_exp_totient | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"card_units_Zp",
"coprime",
"coprime_modl",
"coprime_sym",
"inE",
"inZp",
"modn1",
"modnXm",
"n'",
"order_dvdG",
"order_dvdn",
"totient",
"unit",
"unit_Zp_expg",
"val_eqE"
] | Euler's theorem | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eltm & #[y] %| #[x] | := fun x_i => y ^+ invm (injm_Zpm x) x_i. | Definition | eltm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"injm_Zpm",
"invm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvd_y_x : #[y] %| #[x]. | Hypothesis | dvd_y_x | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
eltmE i : 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 /=; first by rewrite /Zp x_gt1 inE.
by rewrite val... | Lemma | eltmE | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp",
"apply",
"cycle_eq1",
"cycle_id",
"dvd_y_x",
"dvdn1",
"dvdn_trans",
"eltm",
"eq_expg_mod_order",
"expg_znat",
"inE",
"invmE",
"leqP",
"modn1",
"modn_dvdm",
"order_eq1",
"trivg_card_le1",
"val_Zp_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eltm_id : eltm dvd_y_x x = y. | Proof. exact: (eltmE 1). Qed. | Lemma | eltm_id | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"dvd_y_x",
"eltm",
"eltmE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | eltmM | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cycleP",
"dvd_y_x",
"eltm",
"eltmE",
"expgD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eltm_morphism | := Morphism eltmM. | Canonical | eltm_morphism | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"eltmM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_eltm : eltm dvd_y_x @* <[x]> = <[y]>. | Proof. by rewrite morphim_cycle ?cycle_id //= eltm_id. Qed. | Lemma | im_eltm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cycle_id",
"dvd_y_x",
"eltm",
"eltm_id",
"morphim_cycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | ker_eltm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Lagrange",
"apply",
"card_morphim",
"cycle_subG",
"dvd_y_x",
"dvdn_indexg",
"eltm",
"eltmE",
"eqEcard",
"eq_sym",
"eqxx",
"expg_order",
"im_eltm",
"inE",
"indexg_gt0",
"ker",
"leq_divRL",
"mem_cycle",
"orderE",
"orderXdiv",
"setIid",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_eltm : 'injm (eltm dvd_y_x) = (#[x] %| #[y]). | Proof. by rewrite ker_eltm subG1 cycle_eq1 -order_dvdn. Qed. | Lemma | injm_eltm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cycle_eq1",
"dvd_y_x",
"eltm",
"ker_eltm",
"order_dvdn",
"subG1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 -... | Lemma | cycle_sub_group | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Lagrange",
"apply",
"cardSg",
"cycleP",
"cycleX",
"cycle_id",
"divnK",
"divnMr",
"dvdnP",
"dvdn_gt0",
"dvdn_pmul2r",
"eqEcard",
"eqEsubset",
"eqxx",
"expgM",
"gT",
"gcdnC",
"gcdnMr",
"gen_subG",
"group",
"groupX",
"inE",
"in_set1",
"leqnn",
"mulKn",
"mulnC",
"mul... | Gorenstein, 1.3.1 (i) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cycle_subgroup_char a (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 | cycle_subgroup_char | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cardSg",
"card_isog",
"char",
"cycle_sub_group",
"eqxx",
"gT",
"group",
"inE",
"lone_subgroup_char",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Dx : x \in D. | Hypothesis | Dx | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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